Analysis of Changes in the Great Lakes
Net Basin Supply (NBS) Components and
Explanatory Variables
T.B.M.J. Ouarda1, E. Ehsanzadeh1, H. M. Saley1, N. Khaliq, O. Seidou2, C. Charron1,
A. Pietroniro3 and D. Lee4
1
Hydro-Quebec/NSERC
Chair in Statistical Hydrology,
Canada Research Chair on the Estimation of Hydrological Variables,
University of Quebec, INRS-ETE,
490, de la Couronne, QC, Canada G1K 9A9
Tel : (418) 654-3842, Fax : (418) 654-2600 .
2
Dept. of Civil Engineering,
University of Ottawa,
161 Louis Pasteur Office A113
Ottawa, ON K1N 6N5, Canada
Email: [email protected]
3
Environment Canada,
Saskatoon, SK, Canada
4
US Army Corps of Engineers,
Cincinnati, OH, USA.
1
Executive summary
An analysis of changes in the Net Basin Supply (NBS) and hydro-climatic variables of the Great Lakes
was performed in order to investigate the impacts of possible climate change on the natural processes in
the Great Lakes basin. The main focus was on the NBS and water levels; however, other variables such as
runoff, precipitation, evaporation, change in storage, connecting channel flows, diversions, mean,
maximum, and minimum over basin temperature, and water temperatures were analyzed as well. Making
use of a nonparametric Mann-Kendall trend test, the time series were tested for trends under
independence, short term persistence (STP), and long term persistence (LTP) hypotheses. As the principal
variables under study, a shift detection analysis was performed on the NBS and water level time series as
well. The statistical significance of trends was investigated at 5 and 10 percent but results presented in this
summary are based on 5% significance level.
Under the independence assumption, 20% of monthly and annual NBS time series showed significant
upward trends. While Net Basin Supply in Lake Superior did not show any trend in any period of the year,
other lakes showed some trends in some periods of the year. The maximum number of significant trends
in NBS was observed in Lakes St. Clair and Ontario. Modification on the MK test to account for
autocorrelations did not have significant impacts on the test results. There was some but not
overwhelming evidence of LTP in NBS time series where about 11% of the time series could be assumed
to be consistent with the LTP hypothesis. This, however, reduced the percentage of significant trends from
20% (independence assumption) to 13%. Based on results obtained from the Bayesian change point
detection method, at least one change point was observed in approximately half (47%) of the studied NBS
time series (in more than 91% of cases only one change point was found). Changes were observed in the
Lake Superior (1960), Michigan (1966), Erie (1981), and Ontario (1992).
Based on original MK test, water levels in Lakes STC and ERI experienced significant upward trends for
monthly and annual time series, inclusively. While no significant trends were detected in monthly or
annual water levels for lakes SUP and MHG, annual and some monthly water levels experienced
significant upward trends in Lake ONT. Due to extremely correlated records, the number of significant
trends experienced a dramatic drop after modifications on the MK test where, except in few cases, no
evidence of significant trends was observed in annual or monthly water levels. Statistical analysis suggests
presence of very strong LTP in all time series of monthly and annual water levels where no significant
trends in water levels were identified under the LTP hypothesis. Trend analysis in water levels was
repeated considering a common change point in 1972 for all lakes. For the period 1918-1972, based on
original MK test, water levels experienced significant upward trends in Lakes St-Clair and Erie in monthly
and annual scales, inclusively. However, no significant trend was detected in Lakes Superior, Ontario, and
MHG. For the mentioned period, no significant trends were detected in either monthly or an annual scale
after MK test was modified to account for autocorrelations. For the 1973-2007 period, except Lake
Ontario (in some periods of the year), water levels experienced significant downward trends in all lakes in
both monthly and annual scales. However, under a short term persistence assumption, trends were only
significant in monthly and annual scale in Lake SUP, and in annual scale in Lake Ontario. Interestingly,
the direction of trends for the second period of observations is opposite to that of the whole period of
observations.
Runoff experienced significant trends in 34% of time series under independence assumption (trends were
upward except in the Lake Superior). Under STP and LTP assumptions 28 and 16 percent of runoff time
series experienced significant trends, respectively. Nine percent of precipitation time series showed
2
significant trends under the independence/STP assumptions where annual precipitation in lakes HGB,
MHG, and ONT experienced upward trends. Unlike NBS and precipitation, evaporation experienced both
upward and downward trends in 15% of time series with a slight domination of upward trends. The
number of significant trends decreased by 41% when the MK test was corrected to account for
autocorrelations. Under the independence and STP assumptions, respectively, 21 and 19 percent of change
in storage (CIS) time series showed significant trends where the majority of observed significant trends
had downward direction.
Under independence and STP assumptions, connecting channel discharges in 38 and 18 percent of time
series, respectively, experienced significant trends where significant trends occurred mostly in the cold
portion of the year. Due to strong evidence of long term memory, no significant trend was detected in
connecting channel discharges under the LTP assumption. While trends were significant and upward in
LongLac-Ogoki diversion in annual and some monthly observations, they were significant and downward
in Chicago diversion in annual and monthly observations, inclusively, under independence assumption.
No significant evidence of trends was observed under STP assumption. Annual mean, maximum, and
minimum over basin temperatures did not experience any significant change under independence or STP
assumptions except upward trend in minimum temperature in GB under independence assumption.
Annual as well as monthly water temperature (except in few cases) experienced upward trends in Lakes
Superior, Michigan, and St-Clair under the independence assumption. Under this assumption, no
remarkable evidence of trend was found in annual or monthly water temperature in lakes HGB, ERI, and
ONT. Under an STP assumption, the number of significant trends at a 5% level decreased remarkably;
however, this was not the case for trends significant at a 10% level.
3
Chapter 1 Introduction
1.1
Great Lakes Basin
The Great Lakes System is one of the major lake systems in the world. It is a combination of a series of
five interconnected lakes (Superior, Michigan-Huron, St. Clair, Erie, and Ontario) that are connected
through four interconnecting channels (St. Marys River, St. Clair River, Detroit River, and Niagara River).
The Laurentian Great Lakes contain 23,000 km3 of water (about 20% of the world's fresh surface water)
and, with their surrounding basins, cover 770,000 km2 in the United States and Canada. Their surface
areas comprise about one-third of the total basin area. The basin extends over 3,200 km from the western
edge of the Lake Superior to the Moses-Saunders Power Dam on the St. Lawrence River. The water
surface cascades over this distance more than 182 meters to sea level. The most upstream, largest, and
deepest lake is Lake Superior. This lake has two interbasin diversions of water into the system from the
Hudson Bay basin: the Ogoki and Long Lake diversions. Lake Superior water flows through the lock and
compensating works at Sault Ste. Marie, Michigan and down the St. Marys River into Lake Huron where
it is joined by water flowing from Lake Michigan through the Straits of Mackinac. The water from Lake
Huron flows through the St. Clair River, Lake St. Clair, and Detroit River system into Lake Erie. From
Lake Erie the flow continues through the Niagara River and Welland Canal diversion into Lake Ontario.
The Welland Canal diversion is an intrabasin diversion bypassing Niagara Falls and is used for navigation
and hydropower production. From Lake Ontario, the water flows through the St. Lawrence River to the
Gulf of St. Lawrence and the Atlantic Ocean (Croley et al., 2001). The hydrologic cycle of the Great
Lakes basin and meteorology determine water supplies to the lakes. Runoff comprises a significant part of
the Great Lakes water supply, particularly during the snowmelt season, late March through early June.
Because the lakes are so large, lake precipitation and evaporation are of the same order of magnitude as
runoff. On a monthly scale, precipitation is fairly uniformly distributed throughout the year. Lake
evaporation typically has the greatest effect on water supplies during the winter months as dry air and
warm water result in massive evaporation. Condensation on the cool lake surface from the wet overlying
air occurs in the summer. Net groundwater flows to each of the Great Lakes are generally ignored. Net
basin supplies (NBS), which is comprised of runoff and precipitation less evaporation, typically reach a
maximum in late spring and a minimum in late fall (Croley et al., 2001).
In this study, based on the variable and the way lakes Huron, Michigan, and Georgian Bay are dealt with,
the analysis of changes in the NBS and other variables was performed on seven lakes/combinations of
lakes (Superior, Sainte-Clair, Michigan, Michigan-Huron-Georgian Bay, Huron-Georgian Bay, Erie, and
Ontario). Although Lake St. Clair is small compared to other four lakes, due to its location in the middle
of the system, it is of considerable importance. The name of the lakes and studied variables and
abbreviations used to introduce the lakes/variables are given in Table 1.1.
Table 1.1. List of abbreviations used in the study
Abbreviation
Lake/Variable
ERI
Lake Erie
Evp
Evaporation
GB
Georgian Bay
CIS
Change In Storage
GRT
Great Lakes
HGB
Lake Huron (with Georgian Bay)
HUR
Lake Huron
4
MHG
MIC
NBS
ONT
PrcLd
PrcLk
Run
STC
SUP
Lake Michigan-Huron and Georgian Bay
Lake Michigan
Net Basin Supply
Lake Ontario
Overland Precipitation
Overlake Precipitation
Runoff from land surface expressed
Lake St. Clair
Lake Superior
While Lakes Superior and Ontario have been regulated for the past several decades, the intermediate lakes
are not regulated. However, modifications in the connecting channels have impacted the lakes’ outflows
(Quinn, 1979). Figure 1.1 illustrates the structure of the system, the drainage area for the Great Lakes, and
the regulation points.
Figure 1.1. The Great Lakes and the drainage area of the lakes (Clites and Quinn, 2003).
5
1.2 Data
The data used in this study were provided by the Great Lakes Environmental Research Laboratory
(GLERL). These data were made available by GLERL and were obtained as of Dec, 2008. Since these
data are sometimes modified, it is important that the date they were obtained be noted. The data base has
been coordinated between the offices of the Great Lakes Hydraulics and Hydrology Office, Detroit
District, U.S. Army Corps of Engineers (USACE) and the Great Lakes-St. Lawrence Regulation Office,
Environment Canada (EC), Cornwall, Ontario. For the purpose of trend detection in the Great Lakes’ NBS
and other variables impacting the NBS, a number of data sets were analysed and tested to identify any
existing trends. A list of analysed data sets along with the period of available records is presented in Table
1.2.
Table 1.2. Studied variables and the record periods.
GLERL data obtained as of Dec, 2008
Variable
Record period
Great Lakes component Net Basin Supply (NBS)
1948-2005
Great Lakes Runoff (Run)
1900-2005
Great Lakes Precipitation Over Land (PrcLd)
1900-2005
Great Lakes Evaporation (Evp)
1948-2005
Great Lakes Average Water Levels
1918-2007
Great Lakes Change in Storage (CIS)
1900-2006
Great Lakes Connecting Channel Flows
1900-2006
Great Lake Diversions
1900-2006
Great Lakes Maximum Air Temperature – Over Basin
1948-2005
Great Lakes Minimum Air Temperature – Over Basin
1948-2005
Great Lakes Mean Air Temperature – Over Basin
1948-2005
Great Lakes Water Temperature
1948-2005
Each of the variables has monthly and annual observations for the record period. Considering seven lakes
(combination of lakes), 12 variables, and 13 time series for each variable (monthly and annual
observations), overall more than 1200 time series were tested to detect any monotonic trends in the Great
Lakes NBS/NBS components and other explanatory variables. Although NBS components have different
record lengths, a common period have been used to calculate component NBS for each of the lakes.
Therefore, trend detection was performed on both the total record length and the common periods of
Runoff and Precipitation. Evaporation records are the shortest and have been used as the base for common
period study. A definition and also description of each variable analyzed in this study is presented in the
following subsections.
1.2.1 Great Lakes Net Basin Supply (NBS)
The term Net Basin Supply (NBS) is used to describe the amount of water that is contributed to or lost
from a lake within the confines of its natural drainage basin. Net basin supply includes water which a lake
6
receives from precipitation on its surface, and runoff from its own land drainage basin, less the
evaporation of water from the lakes surface:
NBS = P + R – E
(1.1)
where P is the over-lake precipitation, R is the runoff into the lake, and E is lake evaporation (Brinkmann,
2000). However, the NBS can be computed indirectly as a residual of the water balance for a lake:
NBS = ∆S + O – I ± D
(1.2)
where ∆S is the change in water storage, or CIS, computed from the difference in lake levels over a time
interval, such as the beginning and the end of a month, O is the outflow, I is the inflow, and D is the total
diversion over the time interval. There are some differences between the NBS derived from the estimated
values of its components (Equation 1.1), referred to as component NBS, and the NBS computed as a
residual (Equation 1.2), referred to as residual NBS, because of uncertainties in the independent variables
in both equations (Croley & Lee, 1993). The focus of this study is on the component NBS and no
statistical analysis are performed on the residual NBS.
1.2.2 Great Lakes Runoff (Run)
Watershed runoff estimates are the summation of stream flow records from major rivers, available from
the U.S. Geological Survey (Showen, 1980) for U.S. streams and the Inland Waters Directorate of
Environment Canada for Canadian streams (Inland Waters Directorate, 1980). Complete years of
historical daily runoff data begin in 1908 (Superior), 1910 (Michigan), 1915 (Huron), 1935 (St. Clair),
1914 (Erie), and 1916 (Ontario).
1.2.3 Great Lakes Precipitation over Land (PrcLd)
The precipitation used in NBS calculations represents the amount of water that is estimated to have fallen
on the surface of the lakes. The terms "overland" and "overlake" are used to designate which method was
used to estimate that quantity. Precipitation estimates are all based on station sites located on land
somewhere. These stations may be very close to the lake shore or they may be located at a fair distance
away from the lakeshore. Thiessen polygons are used to compute weighting factors for each station's
contribution to the average precipitation over a designated area. Overlake precipitation is estimated using
mostly stations that are close to the lakeshore. Computation of the overland precipitation will use many
more of the stations that are not near the lakeshore. In each case an average depth (per day) of
precipitation over the designated area is computed. Stations on the lee (downwind) side of the lake may
measure a significantly larger precipitation amount than actually occurred over the lake surface just a short
distance away (Tim Hunter, GLERL, personal communications). By using the estimates from all of the
gauges in/near the watershed of the lake, any orographic effects on the average precipitation estimates are
diminished and this is the rational behind using overland precipitation for change detection in NBS
components in this study. Monthly over-land precipitation data begin in 1882 (Superior), 1883
(Michigan), 1883 (Huron), 1900 (St. Clair), 1882 (Erie), and 1883 (Ontario).
1.2.4 Great Lakes Evaporation (Evp)
Monthly evaporation estimates are from daily evaporation estimates generated by the Great Lakes
Evaporation Model. This is a lumped-parameter surface flux and heat-storage model. It uses arealaverage daily air temperature, wind speed, humidity, precipitation, and cloud cover. These data are
available since 1948 (1953 for Georgian Bay).
7
1.3 Great Lakes Average Water Levels
Under the sponsorship of the Coordinating Committee on Great Lakes Basic Hydraulic and Hydrologic
Data a set of lake-wide average levels have been developed using water levels recorded at a network of
gauges on each lake. All water levels are in meters and referenced to International Great Lakes Datum
1985. Lake Superior average level is represented by a 5-gauge average. The five gauges in this network
are: Pt.Iroquois, Thunder Bay, Marquette, Michipicoten, and Duluth. The lake average level of Lakes
Michigan-Huron is represented by a 6-gauge average. The six gauges in this network are: Harbor Beach,
Mackinaw City, Tobermory, Ludington, Thessalon, and Milwaukee. The lake-wide average level of Lake
St. Clair is presently represented by a 2-gauge average. The two gauges in this network are: St. Clair
Shores and Belle River. The lake-wide average level of Lake Erie is presently represented by a 4-gauge
average. The four gauges in this network are: Fairport, Pt. Colborn, Toledo & St. Stanley. The lake-wide
average level of Lake Ontario is presently represented by a 6-gauge average. The six gauges in this
network are: Oswego, Rochester, Toronto, Kingston, Port Weller, and Cobourg.
1.4 Great Lakes change in storage (CIS)
Beginning of month levels (BOM) are used to determine the change in storage on a lake over the
course of a month. Monthly changes in storage for each lake are computed by multiplying the difference
between two consecutive beginning-of-month levels (BOM2 - BOM1) by the area of the lake. The
beginning-of-month level for a gauge is defined as the level at 12:00:00 a.m. (midnight) on the first day of
that month. Since this value is generally unknown for practical reasons, it is standard practice to compute
the value by averaging the daily mean level for the last day of the previous month with the daily mean
level for the first day of the current month, when both are available. For example, the April Change in
Storage would be determined by subtracting the April BOM level from the May BOM level.
1.5 Great Lakes connecting channel discharges
The connecting channels of the Great Lakes system consist of the St. Marys, St. Clair, Detroit, Niagara,
and St. Lawrence Rivers. The St. Marys River is the outlet from Lake Superior. The monthly
average flows in the St. Marys River are those reported by the International Lake Superior Board
of Control’s (ILSBC). The St. Clair River is the natural outlet from Lake Michigan-Huron (a
diversion is also made out of this lake at Chicago, IL). The monthly average flows in the St.
Clair River are computed using stage discharge relationships, water balance equations and
unsteady flow models. Final values are coordinated between the EC, USACE and the Great
Lakes Environmental Research Lab (GLERL) under the auspices of the Coordinating Committee.
The data prior to 1990 was originally coordinated in thousands of cubic feet per second (tcfs) and
has since been converted to the nearest 10 m3/s.
The Detroit River is the outlet from Lake St. Clair. Monthly average flows in the Detroit River are
computed using stage discharge relationships, water balance equations and unsteady flow models. Final
values are coordinated between the EC, USACE and the Great Lakes Environmental Research Lab
(GLERL) under the auspices of the Coordinating Committee. The data prior to 1990 was originally
coordinated in tcfs and has since been converted to the nearest 10 m3/s. The Niagara River is the natural
outlet from Lake Erie (a diversion is also made out of Lake Erie into the Welland Canal). The monthly
average flows in the Niagara River are provided by the International Niagara River Board of Control
(INRBC), and compiled by the CCGLBHHD. Until 1993, Niagara flows were reported in tcfs. Historic
flow has been converted to the nearest 10 m3/s.
8
1.6 Great Lakes Diversions
Major man-made diversions of water occur at five points on the Great Lakes basin. Water is diverted into
the Great Lakes basin from the Albany River via the Long Lac and the Ogoki Diversion projects. Both
these diversions enter Lake Superior. Water is diverted out of Lake Michigan into the Illinois River at
Chicago, Illinois. Water is diverted between Great Lakes basins - Lake Erie into Lake Ontario - through
the Welland Canal and through the New York State Barge Canal. The New York State Barge Canal
diversion extracts a small amount of water from the Niagara River at Tonawanda, New York for use in the
New York State Barge Canal. Most of the water is returned to Lake Ontario at Oswego, New York.
Because this diversion is taken from the river and not the lake, it is not considered in the computation of
Lake Erie's net basin supplies. A short description of each of Great Lake diversions is presented in the
following subsections.
I. Long Lac and Ogoki diversions
The amount of diversion into Lake Superior through the Long Lac and Ogoki projects are reported by
Ontario Hydro. Previously compiled and coordinated data for 1900-1989 are converted from cubic feet
per second (cfs) to m3/s and re-coordinated. The monthly averages of these diversions are provided
individually and combined. The Long Lac diversion began in 1939. Water from Long Lac, which
naturally drained into James Bay via the Kenogami and Albany Rivers, is diverted through a series of
small lakes into the Aguasabon River, a tributary to Lake Superior. The Ogoki diversion was begun in
1943. Water from the Ogoki River, another triburary of the Albany River, is diverted in to Lake Nipigon
and held there until required by hydro-electric plants on the Nipigon River which drains into Lake
Superior.
II. Chicago diversion
The Chicago diversion out of Lake Michigan is monitored and reported by the Metropolitan Sanitary
District of Greater Chicago. The Chicago Sanitary and Ship Canal, between the Chicago River and the
Des Plaines River, forms part of the Illinois Waterway connecting Lake Michigan and the Mississippi
River. The flow in this canal is controlled by a dam and gates at Lockport, Illinois. The diversion amounts
take several years to be finalized and reported; therefore preliminary values had to be used for the period
October 2003 – December 2006.
III. Welland Canal
The Welland Canal diversion from Lake Erie is presently reported by the St. Lawrence Seaway
Management Corp (SLSMC). Historic Welland Canal diversion was compiled for previous studies from a
number of sources. The previously compiled and coordinated data for 1900-1989 is converted from cubic
feet per second (cfs) to m3/s.
1.7 Station Water Level Data Sets
The data used in the study have been mainly obtained from the TWG's SharePoint website. The first set of
data was supplied by Dr. Alain Pietroniro which consists of water levels and falls for a few numbers of
stations and is referred to as the “Multi-station” dataset hereafter. Table 1.3 presents the specification of
water level time series of Multi-station dataset. As it can be seen from this table, all time series end in
2006. The record length of the provided time series varies from 43 to 107 years. These time series are the
average of June to September observations. As it can be seen from the first column of Table 1.3, the
sample data can be categorized in three classes: (1) recorded water levels at certain stations in Lakes
Michigan-Huron and Erie; (2) the difference between water levels in predefined stations; and (3) the
difference between water falls presented in class (2).
9
Table 1.3. Water level time series in Multi-station dataset1.
Record
Record
Seasonal observation
Class
Water level data
Length
period
period
(Year)
Cleveland
107
1900-2006
June - September
Gibraltar
70
1937-2006
June - September
Water level
Harbour Beach
107
1900-2006
June - September
St-Clair Shore
107
1900-2006
June - September
GLB- CLV
70
1937-2006
June - September
GLB- Fermi
43
1964-2006
June - September
HB – SCS
107
1900-2006
June - September
Fall between
HB- CLV
107
1900-2006
June - September
stations/lakes
HB – GIB
69
1938-2006
June - September
SCS - CLV
107
1900-2006
June - September
SCS – GIB
69
1938-2006
June - September
107
1900-2006
June - September
Difference in (HB - SCS) - (SCS-CLV)
falls
(HB - SCS) - (SCS-GLB)
59
1948-2006
June - September
1
The abbreviations used in this are as follows: GLB (Gibraltar), CLV (Cleveland), HB (Harbour Beach), SCS (StClair Shore).
The second set of data, obtained from the TWG's SharePoint website, concern again water level
observations (and their differences) for few stations in Lakes Michigan-Huron and Erie which were
separately analyzed for changes as they were specified as special cases by the Great Lakes responsibilities.
Table 1.4 presents the specifications of the second set of time series. There is one station (Harbour Beach)
which is common between Tables 1.3 and 1.4. This is because the record length of this time series in the
TWG's SharePoint website is considerably longer compared to that reported by Dr. Alain Pietroniro (see
Table 1.4) and as such the change analysis was performed for the longer sample data reported in Table 1.4
as well. The record length of the sample data in Table 1.4 varies from 51 to 147 years where all time series
end in 2006. The data series in this table are the average of observations for June – September period.
Figure 1.2 depicts the Great Lakes and the location of water level stations described in Tables 1.3 and 1.4.
Table 1.4. Water level time series obtained from the TWG's SharePoint website.
Record Length
Seasonal observation
Water level station
Record period
(Year)
period
Buffalo
120
1887-2006
June - September
Harbour Beach
147
1860-2006
June - September
Lakeport
51
1956-2006
June - September
Lakeport - Buffalo
51
1956-2006
June - September
Harbour Beach - Buffalo
120
1887-2006
June - September
10
Figure 1.2. Great Lakes and the location of described water level stations in Tables 1.3 and 1.4.
1.8 Great Lakes Residual Net Basin Supply (RNBS)
The RNBS data were coordinated between the offices of the Great Lakes Hydraulics and Hydrology
Office, Detroit District, U.S. Army Corps of Engineers (USACE) and the Great Lakes-St. Lawrence
Regulation Office, Environment Canada (EC), Cornwall, Ontario. The term net basin supply (NBS) is
used to describe the amount of water that is contributed to or lost from a lake within the confines of its
natural drainage basin. Net basin supply includes water which a lake receives from precipitation on its
surface, runoff from its own land drainage basin, groundwater inflow, and condensation on the lake
surface; less the evaporation of water from the lakes surface and consumptive use. Until recently most of
these factors could not be determined individually with any degree of accuracy. When the need arose to
determine a coordinated set of historic NBS for hydrologic studies in the 1960’s, the Coordinating
Committee on Great Lakes Basic Hydraulic and Hydrologic Data chose a method which used readily
available water level, outflow and diversion records. These NBS, known as residual NBS, are computed
using the relationship in Equation 1.3:
NBS = k ∆S + O – I ± D
(1.3)
Where,
k = conversion from meters to cubic feet per second (m3/s)
11
∆S = change in storage in meters
I = inflow, m3/s
O = outflow, m3/s
D = diversion into or out of a lake,[(-),in;(+),out],m3/s
To convert the change in storage in meters (m) to change in storage in m3/s the following values were
used:
Superior:
m x 31,380 = m3/s-month
Michigan-Huron: m x 44,670 = m3/s-month
St. Clair:
m x 427 = m3/s-month
Erie:
m x 9,770 = m3/s-month
These are based on the area of the lake surface and a standard month. Table 1.5 presents the record length,
record period, and the scale of studied RNBS data.
Table 1.5. The Great Lakes RNBS sample data used in the study
Lake
Record Length (Year)
Record period
Scale
Superior
109
1900-2008
Monthly/Annual
Michigan-Huron
109
1900-2008
Monthly/Annual
St-Clair
1900-2008
Monthly/Annual
Erie
109
109
1900-2008
Monthly/Annual
Ontario
109
1900-2008
Monthly/Annual
1.9
Great Lakes Net Total Supply (NTS)
Net Total Supply (NTS) is defined using Equation 1.4:
NTS = RNBS + Inflow
(1.4)
In this equation RNBS is the Residual Net Basin Supply, and Inflow is the discharge of the upstream
connecting channel emptying in the lake for which the NTS is being estimated. The Great Lakes NTS
were estimated using the residual NBS and the connecting channel discharges using the relation provided
by Equation (1.4). Table 1.6 provides more information on the NTS data used in the study.
Variable
Erie NTS
Table 1.6. Annual NTS estimates for Lakes Michigan-Huron and Erie
Record Length
Record period
Scale
(Year)
109
1900-2008
Annual
Michigan-Huron NTS
109
1900-2008
Annual
Erie NTS – Michigan Huron NTS
109
1900-2008
Annual
12
Chapter 2 Statistical Approaches
2.1 Mann-Kendall Trend Test
2.1.1 General Theory
The nonparametric Mann–Kendall (MK) statistical test (Mann, 1945; Kendall, 1975) has been frequently
used to quantify the significance of trends in hydro-meteorological time series such as water quality,
streamflow, temperature, and precipitation and as such is used to identify monotonic trends in this study.
The comparison made by different researchers supports the superiority of Mann-Kendall test over other
parametric and nonparametric tests when dealing with hydro-climatic variables (Ehsanzadeh and
Adamowski, 2007). The main reason for using non-parametric statistical test is that compared with
parametric statistical tests, the non-parametric tests are thought to be more suitable for non-normally
distributed data and censored data, which are frequently encountered in hydro-meteorological time series.
Moreover, among nonparametric tests, Mann-Kendall (MK) test due to unbiased estimation of population
parameters is preferred to other tests and as such is used in this study to investigate trends in Great Lakes
NBS and hydro-climatic variables. The serial independence of a time series is still required in nonparametric tests. Therefore some modifications are required to account for the impact of autocorrelation
on the MK test.
The null and the alternative hypothesis of the MK test are:
H0
: Prob [x j > x i ] = 0.5 where j > i
HA
: Prob [x j > x i ] ≠ 0.5 (two sided test)
(2.1)
The Mann-Kendall test statistic S is calculated using the formula (Yue et. al., 2002):
n −1
S =∑
n
∑ sgn
i =1 j =i +1
( x j − xi )
(2.2)
Where xj and xi are the data values in years j and i, respectively, with j > i, and sgn(xj -xi) is the sign
function as:
⎧1 if x j - x i > 0
⎪
sgn( x j − xi ) = ⎨0 if x j - x i = 0
⎪
⎩−1 if x j - x i < 0
(2.3)
The distribution of MK S statistic can be approximated well by a normal distribution for large n, with
mean ( µ s ) and standard deviation ( σ s ) given by:
µs = o
(2.4)
13
m
σs =
n(n − 1)(2n + 5) − ∑ ti (i )(i − 1)(2i + 5)
i =1
18
(2.5)
Equation (2.5) estimates the standard deviation of S statistic with the correction for ties in data (ti denotes
the number of ties of extent i). For n larger than 10, the standard normal test statistic ZS for hypothesis
testing is:
⎧ S −1
if S > 0
⎪ σ
s
⎪⎪
Z s = ⎨0
if S = 0
⎪ S +1
⎪
if S < 0
⎩⎪ σ s
⎫
⎪
⎪⎪
⎬
⎪
⎪
⎭⎪
(2.6)
Zs has a standard normal distribution (Kendall, 1975). Local (at-site) significance levels (p-values) for
each trend test can be obtained from (Douglas et al., 2000).
p = 2 [1 − Φ ( Z s )]
(2.7)
Where
1
Φ ( Zs ) =
2π
Z
∫e
t2
− θ
2
dt
(2.8)
0
If the P value is small enough, the trend is quite unlikely to be caused by random sampling. At the
significance level of 0.05, if p ≤ 0.05, then the existing trend is assessed to be statistically significant. The
examples using the Mann-Kendall test for detecting monotonic trends in hydrological time series may be
found in Hirsch & Slack (1984), Burn (1994), Lettenmaier et al. (1994), Gan (1998), Lins & Slack (1999),
Douglas et al. (2000), Zhang et al. (2000, 2001), Yue et al. (2002), Burn & Hag Elnur (2002), Adamowski
and Bougadis (2003), Ehsanzadeh and Adamowski (2007), and others.
2.1.2 Effects of time dependence on the MK test
Hydrological time series may frequently display statistically significant serial correlation. In such cases
the existence of serial correlation will increase the probability that the MK test detects a significant trend.
This leads to a disproportionate rejection of the null hypothesis, whereas the null hypothesis is actually
true. The existence of positive serial correlation in a time series does not alter the normality of the MK
statistic S or the location of the centre of the distribution or the mean of S. However, the presence of
positive serial correlation changes the scattering of the distribution. For a time series with negative serial
correlation, opposite to the positive case, this results in underestimation of significant trends and
consequently accepting the null hypothesis of no trend when it is false (Yue and Wang, 2002; Ehsanzadeh
and Adamowski, 2007).
2.1.3 Modifications on Mann-Kendall (MK) test
Positive serial correlation may increase the variance of the MK statistic, and this in turn leads to an
increase in the rejection rate of the null hypothesis (the test may detect more trends compared to what in
reality exists). A negative autocorrelation has an opposite impact on the MK test (it reduces the rejection
14
rate of the test). Prewhitening (removing autocorrelation prior to applying the trend test) can effectively
remove the AR(1) component .On the other hand, however, the existence of a trend influences the
magnitude of the estimate of serial correlation. Therefore, removing a positive AR (1) will also remove a
portion of trend and the magnitude of trend after pre-whitening is smaller than that before pre-whitening.
Yue et al. (2002) demonstrated that detrending the time series prior to pre-whitening provides a more
accurate estimate of the true AR (1) and introduced an alternative approach, termed trend free prewhitening (TFPW). To implement the new approach of dealing with autocorrelation in the MK test, they
defined the following steps:
1- The slope of trend in the sample data is estimated using the nonparametric Sen slope estimator
(Sen, 1968):
β = Median(
Where
x j − xi
j −i
)
(2.9)
i <j and β is the estimate of the slope of the trend.
2- If the slope is zero then it is assumed that no trend exists in the time series. Otherwise, it is
assumed that the existing trend is monotonic and the sample data is detrended using the following
equation:
X t′ = X t − Tt = X t − β t
(2.10)
3- The lag-1 autocorrelation of the detrended time series ( X t′ ) is estimated using the rank
correlation coefficient estimator by replacing the sample data by their ranks in the following
equation (Salas et al., 1980):
1 n− j
∑ ( X i − X )( X i + j − X )
n − j i =1
rj =
1 n
( X i − X )2
∑
n i =1
Where
rj is
the lag-j autocorrelation coefficient and
(2.11)
X
is the mean sample data. Then, the
estimated lag-1 autocorrelation is removed from the time series using the following equation:
X t′′ = X t′ − r1 X t′−1
(2.12)
It is assumed that the prewhitened series ( X ′′ ) is an independent residual.
4- The removed trend in step (2) is added to the independent residual ( X ′′ ) using the following
equation:
X t′′′= X t′′ + Tt
Where
(2.13)
X ′′′ is the sample data with the true trend which is not being affected by autocorrelation.
15
5- The MK test is applied to the new sample data ( X ′′′ ) to investigate the existence of trend in the
time series.
2.2 Bayesian multiple change point detection procedure
Classical tests of hypothesis do not provide any information on the uncertainty or the nature of the given
date of a change. To cope with this issue, a second generation of statistical methods has been developed in
a Bayesian framework. In this study, the Bayesian method proposed by Seidou and Ouarda (2007) is
applied to data series modeled as a linear combination of relevant hydro-meteorological variables. The
method handles an unknown number of changes and displays the complete probability distribution of the
dates of the change. A brief description of the method is given hereafter.
Let Y = ( y1 , y2 ..., yn ) be the n-sample of observations representing the response variable, m the unknown
0 < τ 1 < τ 2 < ... < τ m < n the set of change-points (with the
convention τ 0 = 0 and τ m +1 = n ). Let Yt:s = ( yt , yt +1 , yt + 2 ,..., ys ) (t ≤ s). Then, for k = 1, …, m+1, the kth
segment is the set of data observed between τ k −1 + 1 and τ k . A parameter Θ k is associated to the kth
segment and π( Θ k ) denotes the prior distribution of Θ k . As established by Fearnhead (2006), the
number
of
change-points,
posterior probability of change-points is given by:
⎧Pr(τ 1 | Y1:n ) = P(1,τ 1 )Q(τ 1 + 1) g 0 (τ 1 ) / Q(1)
⎪
⎨
⎪Pr(τ | τ ,Y ) = P(τ + 1,τ )Q(τ + 1) g (τ − τ ) / Q(τ + 1),
k
k −1
1:n
k −1
k
k
k
k −1
k −1
⎩
(2.14)
for
k = 2,..., m
Where g(.) is the probability distribution of the time interval between two consecutive change points and
g0(.) is the probability distribution of the first change point. For s ≥ t and yi ∈ Yt:s ,
s
P (t , s ) = ∫ ∏ f ( yi | Θ)π ( Θ ) dΘ is the probability that t and s belong to the same segment. Q (t) is the
i =t
likelihood of the segment Yt:n given a change point at t − 1.
Q(t) is derived from a recursive relation using P(t, s) and both g and g0 (see Theorem 1, Fearnhead, 2006).
Now, let X = ( x1 j , x2 j ,..., xnj ), j = 1,..., d * denotes the set of the d * explanatory vectors (including
intercept if any), the multiple linear relation can thus be written as:
d*
yi = ∑ θ j xij + ε i , i = 1,..., n
or Y = Xθ + ε
(2.15)
j =1
Where θ=(θ1 , θ 2 ,..., θ d * ) is the vector of regression parameters and ε=(ε1 , ε 2 ,..., ε d * ) is the Gaussian
vector of residuals with mean zero and variance σ2. Note that relation (2.15) changes after each changepoint and is re-calculated on each segment. On a given segment, the parameter vector Θ is defined as:
16
Θ = ⎡⎣θ1 ,θ 2 ,...,θ d * , σ ⎤⎦
(2.16)
And it follows that:
⎛
d*
⎛
⎜
⎜ yi − ∑ θ j xij
1
⎜
j =1
exp ⎜ −0.5 ⎜
f ( yi | Θ) =
⎜
σ
σ 2π
⎜
⎜⎜
⎜
⎝
⎝
⎞
⎟
⎟
⎟
⎟⎟
⎠
2
⎞
⎟
⎟
⎟
⎟
⎟
⎠
(2.17)
In the current study, the prior distribution that will be used depends only on the scale parameter σ and as
such:
⎛ c ⎞
σ − a exp ⎜ − 2 ⎟
2σ ⎠
(2.18)
a > 1, c > 0
π 1 (Θ) = π 1 (σ ) = p(σ | a, c) = a −3 a −1⎝
−
a
−
1
⎛
⎞
2 2 c 2 Γ⎜
⎟
⎝ 2 ⎠
Where a and c are the hyperparameters. Hence, as shown in Seidou and Ouarda (2007), the posterior
probability of the change-points displayed in equation (2.14) is given in this setting by:
d*
P ( t , s ) = ( 2π ) 2
(
π ( ε tT:s ε t:s + c )
( cπ )
−
a −1
2
)
( s −t + a −1)
2
⎛ s − t + a − d* ⎞
Γ⎜
⎟
2
⎝
⎠
1
2
−
a
1
⎛
⎞
XtT:s Xt:s Γ ⎜
⎟
2
⎝
⎠
−
for s ≥ t
(2.19)
In the current study, the parameter a given in Equation (2.19) is fixed at 2. For a = 2, the variance of the
parameter σ is infinite so that the prior distribution is non-informative (but still proper since
∫
+∞
0
π 1 (σ ) dσ = 1 always holds for a = 2).
The Bayesian method described above performs in three main steps. The posterior distribution of
probability of the number of changes is first computed and the number of changes observed in the
response variable is estimated. Then, in the second step, conditional on the number of detected changes,
the posterior probability of the position of each change is derived following Equation (2.14) and the dates
of changes are therefore located. At the final step, the estimation of the simulated mean value functions
(before and after each change point) of the response variable and the estimation of the magnitude of the
detected changes are presented. The identified changes can either represent shifts in the mean or changes
in the trend of data series or a combination of both. Precision on the exact nature of the identified change
points (shift in the mean, presence of local trend or trend change, etc.) can be achieved after a visual
examination of the time series plot and its discontinuous time-regression lines. In the current study,
graphical outputs of the Bayesian techniques are presented and the histograms summarize the discrete
posterior distributions. The mode of a given histogram represents either the most probable number of
observed changes or the most probable date of a change, depending on the considered case.
17
2.3 Long Term Persistence (LTP)
Different potential driving mechanisms have to be considered when analyzing hydrological time series,
for example, anthropogenic influences, and natural short and long term variability. Indeed, interpretation
of trend results under independent and dependent assumptions may differ significantly. Long term
persistence (LTP) was studied first by Hurst (1951). This phenomenon, also known as scaling behaviour,
is a tendency of hydro-climatic variables to exhibit clustering behaviour in certain periods of time (i.e.
draughts). The presence of LTP is usually investigated by estimating the Hurst exponent H, which ranges
between 0 and 1. The range 0.5 < H < 1 corresponds to a persistent process and the range 0 < H < 0.5
corresponds to an independent process, and the value H = 0.5 corresponds to a purely random process.
The scaling behaviour has been identified in several hydrological time series by a number of investigators
including (to mention a few of the more recent studies) Koutsoyianis (2002), Koutsoyiannis (2003a),
Cohn and Lins (2005), Koutsoyiannis and Montanari (2007), Khaliq et al. (2008), and Hamed (2008). It is
hypothesized that LTP may reflect the long-term variability of several factors such as solar forcing,
volcanic activity and so on. It is well known that the presence of LTP has significant impact on the
interpretation of trends identified under the independence or short-term persistence (STP) assumptions.
A number of trend tests are available that can accommodate LTP. Hosking (1984) proposed a unified
approach for modeling fractional Gaussian noise as a generalization of ARIMA models known as
fractional autoregressive integrated moving average (FARIMA) modeling approach. Other commonly
used techniques are: adjusted likelihood ratio test (ALRT) proposed by Cohn and Lins (2005), rescaled
adjusted range statistic (RARS) (Mielniczuk and Wojdyłło, 2007), and aggregated standard deviation
(ASD) (Koutsoyiannis, 2003a, 2003b, 2006).
In this work, the commonly used technique of fractional autoregressive integrated moving average
(FARIMA ( p, d , q ) ) modeling approach (Hosking, 1984) is utilised where p and q respectively stand for
the number of autoregressive and moving average parameters and d = H − 0.5 is the fractional
differencing parameter. Because of finite sample sizes of hydro-climatic variables, the criterion H = 0.5
is subjected to sampling errors. Therefore, it becomes important to study the sampling distribution of H
for diagnostic purposes (Couillard and Davison, 2005). To establish whether the value of H estimated with
selected method for a given sample is significantly different from 0.5, Monte Carlo simulated distribution
of H is developed by generating 5,000 random samples, each of size equal to the observed one, from a
white noise process (i.e., normally distributed values with zero mean and unit variance) and estimating H
for each of the simulated samples. For the FARIMA ( p, d , q) modeling method, the FARIMA (0, d ,0)
model is used to develop confidence intervals. For this method, the ‘fracdiff’ package of the ‘R’
computing environment is used.
18
Chapter 3 - Net Basin Supply (NBS), Net
Total Supply (NTS), and Component
Variables
3.1 Statistical analysis for Trends
In order to detect any trends in the Great Lakes hydro-meteorological variables it was decided to apply the
trend test considering two different significance levels of 5 and 10 percent. Testing the time series for
trends at a 10 percent significance level will provide a better understanding of the variability in the time
series where trends are not significant at a 5% significance level. Serial correlation is another issue that
needs to be dealt with in interpretation of the results. For the purpose of this study, the first order
autocorrelations were estimated and removed if they were larger than a predefined threshold level (0.05).
In this study, results of the MK trend test on the original sample data are presented as original MK test
results and the results obtained after removing autocorrelation are presented as TFPW_MK test results.
3.1.1 Trend detection in the Great Lakes NBS
Mann-Kendall nonparametric trend test was applied to the monthly and annual NBS time series for
different lakes. Table 3.1(a) presents the trend test results using original MK test for the Great Lakes NBS.
This table shows that out of 91 tested time series (monthly and annual), 26 time series (28%) experienced
significant trends at a 10% significance level; however, at a 5% significance level, 18 time series (20%)
showed significant trends. While Net Basin Supply in the Lake Superior did not show any trend in any
period of the year, other lakes showed some trends in some periods. According to Table 3.1(a), the
maximum number of significant trends in NBS was observed in the Lakes St. Clair and Ontario.
November, followed by September and January, had the highest number of significant trends for all lakes.
It can be seen that no significant trends were observed in the NBS time series of Great Lakes in February,
March, and April.
It is also noteworthy that except one case, all significant trends in the NBS have upward directions.
Although autocorrelations in annual time series are positive for all lakes, the monthly NBS time series are
negatively correlated for some lakes in some periods of the year. Out of 91 time series, 27 time series
(30%) are contaminated by negative autocorrelation and the rest of the time series are dominated by
positive autocorrelations. The results of trend detection in pre-whitened NBS sample data (after removing
autocorrelation) are presented in Table 3.1(b). This table shows that out of 91 tested time series (monthly
and annual), 14 time series (15%) showed significant trends at a 5% significance level where all observed
significant trends (at this significance level) had upward directions. While the NBS in Lake Superior
(SUP) did not show trends in annual or monthly scales (except a downward trend at 10% significance
level in August), other lakes showed some trends in some periods of the year. The maximum numbers of
significant trends were observed in Lakes St. Clair (STC) and Ontario (ONT). November, followed by
September, had the highest number of significant trends for all lakes. It can be seen that no significant
trends were observed in January, February, March, and April. While annual NBS experienced significant
upward trend in Lake Ontario, there was no evidence of significant trends in annual NBS in other lakes.
19
Table 3.1. Trend detection results for Great lakes NBS
Lake
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
DEC
Annual
a) original MK test
SUP
0
HGB
∆
∆
∆
MHG
MIC
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
∆
0
0
0
0
0
∆
∆
∆
0
0
0
0
0
0
0
0
0
0
∆
0
0
0
0
0
∇
0
0
0
0
∆
∆ ∆
∆
0
0
0
∆
0
∆
STC
0
0
0
0
0
ERI
0
0
0
0
0
0
0
∆
ONT
0
0
0
0
0
∆
0
0
∆ ∆
∆ 0
∆ ∆
∆
∆
∆
∆
∆
b) Trend-free pre-whitening (TFPW_MK) test
0
0
∇
∆
0
0
0
0
0
0
0
0
0
0
0
∇
0
0
0
0
0
0
0
0
0
0
0
0
0
0
SUP
0
0
0
0
HGB
0
0
0
0
MHG
0
0
0
MIC
0
0
STC
0
ERI
ONT
∆ ∆
0
0
∆
0
0
0
0
0
0
0
∆
∆
0
0
0
0
0
0
0
0
0
0
0
∆
∆
∆
∆
∆
∆
∆
∆
0
0
∆
∆
0
0
0
∆
∆
0
∆
c) Assumption of LTP using the MK-FARIMA approach
SUP
HGB
∆
∆
∆
MHG
MIC
STC
ERI
ONT
∆
∆ ∆ ∆
∆
∆
∆ ∆
∆
∆ ∆
∆
∆
∆
∆
A comparison between the percentages of detected significant trends using the two trend detection
methods reveals that removing autocorrelation had slight impacts on the test results. That is, at a 5%
significance level, the number of significant trends decreased from 18 to 14 whereas they decreased from
26 to 21 at a 10% significance level. Except in Lake Ontario, no significant trends were observed in
annual NBS time series using TFPW_MK test. Further, unlike original MK test, there were no significant
trends in January time series when the sample data were tested for trends using TFPW_MK approach.
20
3.1.2 Trend detection in the Great Lakes Precipitation (PrcLd)
The precipitation sample data are estimated using over basin gauging stations (precipitation is one of the
input components of component NBS). Original MK test was applied to the Great Lakes precipitation time
series and Table 3.2 presents trend detection results for precipitation in annual and monthly scales. It can
be seen from Table 3.2(a), that out of 91 tested time series, 16 time series (18%) showed significant trends
in precipitation at a 10% significance level. Moreover, 8 of observed trends (50%) were significant at a
5% significance level. Precipitation did not show any upward/downward trend in any period of the year
for Lake Michigan. Similar to the NBS, all but one of observed significant trends had upward directions.
There were no significant trends in precipitation in winter (NOV, DEC, JAN, and FEB) and summer
(JUN, JUL, AUG) for any of the lakes. Annual precipitations in ONT, HGB, and MHG experienced
significant upward trends whereas no significant trends were observed in annual precipitation for lakes
SUP, MIC, STC, and ERI. No evidence of significant trends in the lake MIC and similarity between the
number and the direction of detected trends in HGB and MHG implies that the detected trends in these
two combinations of lakes are mainly due to trends in the Georgian Bay and Huron lake precipitation time
series.
Table 3.2. Trend detection results for the Great Lakes Precipitation
Lake
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
DEC
Annual
0
a) original MK test, 1948-2005
0
0
0
0
∆
0
0
0
0
0
0
∆
0
0
0
0
0
0
∆
0
0
0
0
0
0
0
0
0
0
0
0
∆
0
0
0
0
0
0
0
∇
∆
0
∆
0
0
0
∆
0
0
0
0
0
0
0
0
0
0
∆
∆
0
0
∆
0
SUP
0
0
0
0
HGB
0
0
0
0
MHG
0
0
0
0
MIC
0
0
0
0
STC
0
0
0
ERI
0
0
ONT
0
0
0
∆
∆
∆
∆
b) Trend-free pre-whitening (TFPW_MK) test, 1949-2005
0
0
0
0
∆
0
0
0
0
0
0
∆
0
0
0
0
0
0
∆
0
0
0
0
0
0
0
0
0
0
0
0
∆
0
0
0
∆
0
0
0
0
0
0
0
0
0
0
∆
0
0
0
0
0
0
0
0
0
0
∆
∆
0
0
∆
SUP
0
0
0
0
HGB
0
0
0
0
MHG
0
0
0
0
MIC
0
0
0
0
STC
0
0
0
ERI
0
0
ONT
0
0
0
∆
∆
∆
∆
First order autocorrelations were estimated for precipitation time series in an annual and monthly scale. It
was observed, similar to the NBS time series, that January precipitation had the highest level of
autocorrelations compared to other periods of the year. Out of 91 tested precipitation time series, 29 time
series (32%) were negatively auto-correlated and the rest were dominated by positive autocorrelations.
Annual precipitation time series in all lakes but Lake Superior were positively auto-correlated. Lakes
Ontario and Superior had the highest level of autocorrelations; however, unlike Lake Superior, the sign of
autocorrelation in Lake Ontario was positive. The highest number of negative autocorrelations in different
21
lakes was observed in October followed by September. In order to account for autocorrelations, the
precipitation time series were pre-whitened and Table 3.2(b) presents trend detection results after
modifications and shows that at a 5% significance level, precipitation over the Great Lakes experienced a
significant trend in 8 (9%) of the time series . However, trends were significant in 14 time series (15%) at
a 10% significance level. The detected trends had upward direction in all cases. No significant trend was
observed in the precipitation for Lake Michigan.
Although the NBS components (Precipitation, Evaporation, and Runoff) have different record lengths, the
common period of these observations starting in 1948 and ending in 2005 were used to calculate the Great
Lakes NBS. However, in order to use as much information as available to detect any changes in the Great
Lakes variables, it was essential to analyse the NBS component time series with actual record lengths as
well. Therefore, besides trend analysis on the common period, Great Lakes precipitation time series were
tested for monotonic trends using the whole record length. Table 3.3 presents the results of trend detection
performed on the whole records of precipitation sample data using original and modified MK test.
Table 3.3. Trend detection results for the Great Lakes precipitation (whole record length)
Lake
JAN FEB MAR APR MAY JUN
JUL AUG SEP OCT NOV DEC Annual
a) original MK test
∆
0
0
SUP-1882
∆
∆
0
0
∆
HGB-1883
0
0
0
MHG-1883
0
MIC-1883
∇
STC-1900
0
0
0
ERI-1882
∆
∆
∆
ONT-1883
0
0
0
GRT-19001
∇
∇
0
0
∆ 0
∆ ∆
∆ ∆
∆ ∆
∆ 0
∆
∆
∆
∆
∆
0
∆
0
0
∆
0
0
0
0
∆
0
0
0
∆
∆
0
0
0
0
0
0
∆
0
0
0
0
0
0
0
0
0
0
∆
∆
∆
∆
∆
∆
∆
0
0
0
0
0
∆
0
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
0
∆
∆
∆
∆
∆
∆
∆
∆
∆
0
∆
0
0
∆
0
0
0
0
∆
∆
∆
∆
∆
b) Trend-free pre-whitening (TFPW_MK test)
2
∆ 0
∆ ∆
∆ ∆
∆ ∆
∆ 0
GRT-19002
0
0
0
∆
0
0
0
SUP-1882
∆
0
0
∆
∆
∆
0
HGB-1883
0
0
0
0
0
∆
MHG-1883
0
0
0
0
MIC-1883
0
0
0
0
STC-1900
0
0
0
0
0
0
0
0
0
0
ERI-1882
0
∆
∆
∆
∆
∆
∆
∆
0
ONT-1883
0
0
0
∆
∆
0
0
0
0
∆
0
∇
∇
0
0
∆
∆
∆
0
∆
∆
∆
∆
0
∆
∆
∆
∆
The numbers represent the first year of the observation period. All time series end in 2007.
22
Table 3.3(a) shows that 53 out of 91 precipitation time series (58%) experienced significant trends at a
10% significance level. At a 5% significance level, 41 time series (45%) showed significant trends where
all but two of detected significant trends had upward directions. It was observed that annual precipitation
experienced an upward trend at a 5% significance level in the Great Lakes except in Lake St-Clair for the
whole observational period. Table 3.3(b) shows that 45 out of 91 tested time series (50%) experienced
significant trends at a 10% significance level. At a 5% significance level, 38 time series (42%) showed
significant trends where all but two of detected trends (in February) had upward directions.
Table 3.4 compares the trend detection results for the common period and the whole record length in the
Great Lakes precipitation after modifications to account for autocorrelation. Table 3.4 shows that at a 10%
significance level, the number of significant trends increased by 214% when the whole precipitation
record length was tested instead of the common period of records (1948-2005). This increase in the
detected trends due to the extended record length was 337 percent considering a 5% significance level.
The highest increase in the percentage of significant trends due to record length increase was observed in
the Lake SUP and the Lake ERI whereas the percentage of significant trends in the Lake STC did not
experience any change.
Table 3.4. Comparison of precipitation test results for the common periods and total record lengths3
(TFPW_MK)
Variable
Tested period
PrcLd
Common period
(1948-2005)
Total
record
length (- 2007)
SUP
HGB
MHG
1/1
3/2
3/2
10/8
5/4
6/5
MIC
STC
ERI
ONT
0
2/1
1/0
4/2
4/4
2/1
12/8
5/5
3.1.3 Trend detection in the Great Lakes runoff (Run)
The Great Lakes annual and monthly runoff time series were tested to detect any monotonic trends using
MK test for the common period (1948-2005) as well as whole record period. As the first step, runoff
observations were tested using original MK test regardless of serial correlation. Table 3.5(a) presents trend
test results for the common period of annual and monthly runoff discharges at 5 and 10 percent
significance levels. According to Table 3.5, 45 out of 91 tested time series (49%) showed significant
trends at a 10% significance level using original MK test. At a 5% significance level, however, 31 time
series (34%) showed significant trends. Out of 31 time series with significant trends at a 5% significance
level, only 4 time series (13%) showed downward trends while the rest (27 time series) showed upward
trends. Runoff did not experience significant trends in spring (MAR, APR, and MAY) in any of the Lakes.
At a 10% significance level, annual runoff experienced significant trends in all lakes except Lakes
Superior and St-Clair.
3
The figures before slash represent the number of significant trends at a 10% significance level whereas the figures
after the slash represent the number of significant trends at a 5% significance level.
23
Table 3.5. Trend detection results for the Great Lakes runoff
Lake
SUP
HGB
MHG
JAN
FEB
∇ ∇
∆ ∆
∆ ∆
MAR
APR
MAY
JUN
JUL
NOV
DEC
Annual
0
0
0
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
0
0
∆
0
∆
∆
∆
∆
0
0
∇
∆
∆
∆
0
∆
0
∆
0
0
∆
0
0
∆
0
∆
∆
0
∆
0
0
0
∇
∇
∇
0
0
0
0
∆
0
0
0
0
0
0
0
∆
0
0
0
∆
∆
0
∆
0
∆
∆
∆
∆
∆
∆
0
0
0
STC
0
0
0
0
0
ERI
0
0
0
0
0
ONT
0
0
0
0
0
∆ ∆
∆ ∆
∆
0
0
b) (TFPW_MK, 1948-2005).
∇
∆
∆
∇
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
MIC
0
0
0
0
0
∆
STC
0
0
0
0
0
ERI
0
0
0
0
0
ONT
0
0
0
0
0
MHG
OCT
0
∆
HGB
SEP
a) (Original MK test, 1948-2005)
MIC
SUP
AUG
∇ ∇
∇
0
0
0
0
0
0
0
∆
0
0
0
0
∆
∆
∆
∆
∆
∆
∆
∆
∆ ∆
∆ ∆
∆
∆
∆
∆
∆
0
0
c) Assumption of LTP using the MK-FARIMA approach
SUP
HGB
MHG
∇
∇
∇
∇
∆
∆
∆
∆
∆
∆
∆
MIC
STC
ERI
ONT
∆ ∆ ∆
∆ ∆ ∆
∆
∆
∆
∆
∆
∆
∆
∆
Analysis of autocorrelation of these data shows that annual runoff time series are positively correlated in
all lakes. It was found that the autocorrelations are highest in January for all lakes. Runoff time series are
negatively correlated in the second portion of the calendar year in some lakes. In order to evaluate the
impacts of autocorrelations on the test results, the time series were tested using TFPW_MK test. Table
3.5(b) presents the results of trend detection in the Great Lakes runoff after performing modifications to
account for autocorrelation. According to Table 3.5(b), out of 91 tested time series, 33 time series (36%)
showed significant trends at a 10% significance level. At a 5% significance level, 24 time series (26%)
showed significant trends. Out of 24 observed trends (significant at a 5% significance level), 5 time series
24
had downward trends and the rest showed upward trends. Interestingly, all significant downward trends
were observed in Lake Superior and no downward trends were detected for runoff in other lakes. The
results indicate that runoff did not experience significant trends in the spring (March, April, and May) in
any of the lakes. A comparison between trend test results reveals that the number of significant trends at a
10% significance level decreased by 26% when the time series were tested using TFPW_MK approach.
This decrease due to modifications to account for autocorrelation was 22% for trends significant at a 5%
significance level.
Besides trend detection in a common period of runoff records (used in NBS estimates), trend detection
was performed on the whole record length of runoff sample data using original MK test as well as
TFPW_MK test. The results of this part of trend analysis are presented in Table 3.6.
Table 3.6. Trend detection results for the runoff time series (whole record length)
Lake
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
DEC
ANN
∆
a) Original MK test
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
0
ERI
∆
ONT
0
SUP
HGB
MHG
MIC
STC
∆ ∆
∇ 0
0
0
∇
∇
0
0
0
0
0
0
0
0
∇
0
∇
0
0
∆
0
0
∆
∆
0
0
0
0
∆ ∆
∆ ∆
0
∇
0
0
0
0
∆
∆
∆
∆
0
0
0
∆
∆
∆
0
∆ ∆
∆ ∆
∆ ∆
∆ ∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆
∆ ∆
∆
∆
∆
∆
0
0
0
∆
∆
0
b) TFPW_MK test
SUP-1908
HGB-1916
MHG-1916
MIC-1902
STC-1933
∆ ∆
∆ ∆
∆ ∆
0
∆
∆ ∆
∆
∆
0
0
0
∆
0
0
∇
0
0
0
0
0
0
0
0
0
0
0
0
∆
∆
∆
∆
0
0
∇
0
0
0
0
0
0
∆
0
0
0
∆
∆
∆
∆
∆
∆
∆
∆
∆
ERI-1914
0
0
0
∆
∆
ONT-1901
0
0
0
0
0
∆ ∆
∆ ∆
0
∇
∆
∆
0
∆ ∆
∆ ∆
∆
0
0
0
0
The results of statistical analysis for the whole period of observations presented in Table 3.6 show that the
percentage of significant trends at a 5% significance level increased by 55% (67%) based on original MK
(TFPW_MK) test when trend detection was performed on whole record length. A great majority of
detected significant trends had upward directions. The most significant difference in trend analysis results
for the two different record observations was observed in Lake Superior. Interestingly, opposite to the
common period, Lake Superior runoff showed increasing trends in annual and most of monthly time series
for the whole record period.
A comparison of results for the common period and the whole record length using original and modified
MK test is performed in Table 3.7. According to this table, at a 10% significance level, the number of
significant trends increased by 24 and 45 percent using original and TFPW_MK test, respectively, when
25
the testing period was extended to the whole period of observations. At a 5% significance level, this
increase was 55 and 66 percent for original and TFPW_MK test, respectively. This table also shows that
the highest difference in the number of significant trends with different record lengths was observed in
Lakes Superior and St-Clair.
Table 3.7. Comparison of the number of significant trends in runoff for different record periods
MK test
Tested period
Original
TFPW_MK
Common period
(1948-2005)
Total
record
length
Common period
(1948-2005)
Total
record
length
SUP
HGB
5/4
6/4
11/11
MHG
MIC
STC
ERI
ONT
7/5
8/2
6/5
7/6
6/5
7/6
5/5
6/3
11/11
11/8
5/4
6/5
3/3
4/2
2/0
6/5
7/5
5/4
10/10
5/4
5/5
3/1
11/10
10/8
4/2
3.1.4 Trend detection in the Great Lakes evaporation (Evp)
Great Lakes evaporation records were tested to detect monotonic trends in observations using original MK
and TFPW_MK test. Unlike precipitation and runoff observations, the whole record length of available
runoff measurements (1948-2005) were used to estimate NBS and also tested for trends. Table 3.8
summarises obtained results from applying MK test on the evaporation time series. Table 3.8(a) shows
that unlike NBS and precipitation, evaporation experienced both upward and downward trends in the
Great Lakes for the observational period based on original MK test. Out of 91 tested time series, 22 time
series (24%) showed significant trends at a 10% significant level. However, 14 time series (15%)
experienced trends at a 5% significance level. Out of 22 observed significant trends, 9 time series (41%)
experienced downward trends and the rest experienced upward trends. It was observed that annual
evaporation did not undergo any significant trends for any of the lakes. While upward trends were
dominant for most of the lakes, Lake St-Claire evaporation time series were dominated by downward
trends. The analysis showed that autocorrelations for annual evaporations are positive for all Great Lakes
except Lake Ontario where annual evaporation is negatively autocorrelated. Moreover, the magnitude of
autocorrelation in annual evaporation is much smaller compared to precipitation and runoff. Trend free
prewhitening was performed to account for autocorrelations observed in evaporation time series and the
TFPW_MK test results are presented in table 3.8(b).
Table 3.8. Trend detection results for the Great Lakes evaporation (Evp) (1948-2005).
Lake
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
DEC
Annual
0
0
0
0
0
0
a) original MK test
SUP
0
∇
0
0
0
∆
0
0
0
HGB
0
0
0
0
0
0
0
0
0
MHG
0
0
0
0
0
0
0
∆
0
0
0
MIC
0
0
0
0
0
0
0
∇
∆
∇
STC
∇
∇
0
∇
∆
∇
0
∇
0
0
0
∆
∆ ∆
∆ ∆
0
0
∆
∆
26
ERI
0
0
0
∆
0
0
0
0
0
0
ONT
0
0
0
0
0
0
0
0
0
∆
∇
0
0
0
0
0
0
0
0
0
0
0
b) TFPW_MK test
∆
∆
SUP
0
0
0
0
0
0
0
0
0
HGB
0
0
0
0
0
0
0
0
0
MHG
0
0
0
0
0
∆
0
0
0
∆
0
0
0
MIC
0
0
0
0
∆
∆
0
0
0
0
0
∆
0
0
0
0
∇
∇
∇
∇
∆
0
0
0
STC
∇ ∇
ERI
0
0
0
0
0
0
0
0
0
0
∇
0
0
ONT
0
0
0
0
0
0
0
0
0
∆
0
0
0
The results of TFPW_MK test indicate that evaporation experienced a significant trend in 16 out of 91
time series (17%) at a 10% significant level. At a 5% significance level, 9 time series (10%) showed
significant trends. Evaporation experienced both significant upward and downward trends where 7 out of
16 observed significant trends (43%) had downward directions. The highest number of significant trends
for different lakes was observed in October. A comparison between methods shows that at a 10% (5%)
significance level, the number of significant trends decreased by 27% (35%) when autocorrelations were
removed from the time series.
3.2 Change point detection in the Great Lakes NBS time series
The question of change-point detection in the mean or the trend of times series is considered. Databases
and sites of interests have already been presented in Chapter 1. A Bayesian technique is applied to The
Net Basin Supply (NBS) time series in order to detect changes that could reflect climate variability. An
illustration of the used methodology is given through two case studies. Then, the findings for the overall
data series are summarized in Table 3.9.
Abrupt changes that occur in a hydrologic data series can roughly be split into two classes: the first ones
result from errors inferred by measuring instruments (errors of grading, change of technology …) whereas
the second ensue from change of dynamics over time resulting from climate change impact or human
activities effect. To detect eventual change in the parameters (e.g., mean, variance, or trend) of a time
series, different statistical methods such as classical hypothesis testing (e.g. Man-Kendall test for trend
detection, see Chapter 2) could be used. Statistical tests enable assessing time series non-stationarity and
are based on the acceptance or rejection of the null hypothesis: «No change in the specified parameter ».
They also give the most probable time-position of the change (the date of the change also referred as
change-point) but do not provide any information on the uncertainty or the nature of the given date of
change. To handle this aspect, a second generation of statistical methods has been developed in a Bayesian
framework. In light of the observed data (prior information), Bayesian models provide updated
information about the change point-position as they display the full posterior probability distribution of
the date of change. A large range of Bayesian change-point detection methodologies involving multiple
linear regressions have been designed. Indeed, the multivariate linear regression setting is used to point
out the influence of sudden environmental or climatic variability on hydrologic time series. Two
approaches have been developed recently:
-
The method of Seidou et al. (2007): it has been designed to detect single change-point in the
parameters of a linear combination of explanatory variables. It requires informative priors and
involves relatively long Monte-Carlo Markov Chain simulations. Missing data and several response
27
-
variables can be handled by this approach. This method is more flexible and covers a substantial
larger scope than those presented in precursor works (Rasmussen (2001), Perreault et al. (2000a,
2000b)).
The method of Seidou and Ouarda (2007): it is an adaptation of the method developed by Fearnhead
(2006) to a multivariate linear regression relationship. The method deals with an unknown number of
changes and allows detecting multiple change points. It also handles with non-informative priors.
Note that both methodologies cope well with almost all change-point detection problems encountered in
hydrology. To study climatic variability effect on NBS and water level fluctuations, the Bayesian method
of Seidou and Ouarda (2007) is applied to data series modeled as a linear combination of relevant hydrometeorological variables. The method developed by Seidou and Ouarda (2007) is performed in three main
steps:
- The first step consists of evaluation of the number of the most probable shifts that occur in the mean
or the trend of time series.
- Then secondly, the posterior probability distribution of the date of change conditional on the number
of changes is displayed. The most probable position of change is therefore retained as the date of
change.
- The final results consist of both the estimation of the simulated mean values (before and after the
breakpoints) of the data series and the estimation of the magnitude of the detected shifts.
Furthermore, trend analysis can be performed by visual inspection of plotted graphs.
3.2.1 Change point detection in NBS time series
The period of study spans from 1948 to 2005. NBS data are monthly recorded and there is no missing data
during the whole period of observation. Data had been collected on seven sites namely:
1. Lake Superior (SUP)
2. Lake Michigan (MIC)
3. Lake Michigan – Huron (MHG)
4. Lake Huron – Georgian Bay (HGB)
5. Lake St. Clair (STC)
6. Lake Erie (ERI)
7. Lake Ontario (ONT)
The Bayesian change-point detection method of Seidou and Ouarda (2007) will first be applied to yearly
average data series. Then, each monthly NBS time series (the time series composed with data collected
during the same month) will be analysed. The obtained results will be compared and discussed at the end
of the section.
3.2.2 Change point detection in NBS data series: the multivariate linear
regression framework
The Bayesian method of Seidou & Ouarda (2007) is applied to NBS data series modeled as a linear
combination of selected hydro-meteorological variables. Let us recall that NBS is derived from the
relation: NBS = Runoff + Precipitation – Evaporation. This relation provides the three main explanatory
variables that will represent NBS time series. However, as it is well known that variables such as water
28
levels and temperatures influence NBS indirectly, these variables are also selected. Thus, the data set of
predictors that will be used in the current study is represented by:
ƒ Precipitation over land (mms)
ƒ Runoff (mms)
ƒ Evaporation (mms)
ƒ Water levels (m)
ƒ Water temperature (Co)
ƒ Air temperature over basin (Co)
Note that the selection of water levels and temperature variables in the set of explanatory variables would
not affect relation (1.1). Their contribution in the linear regression equation will just be useful for
providing a term of error essential for the method of analysis to perform. In the following part, a complete
description of the results derived from the application of the Bayesian change-point method to the annual
average NBS data series for Lake Superior is presented.
3.2.3 Case study: Lake Superior (annual average data series)
a- Number of changes
The first result derived from the application of the Bayesian method to the data series observed in Lake
Superior NBS data series is the discrete probability distribution of the number of changes. A probability of
occurrence (that is a number between 0 and 1) is associated to each possible number of changes. Then, the
most probable number of changes (i.e. the one displaying the highest probability) is retained. As shown in
Figure 3.1(a), the existence of one change is quite certain (with a probability of 100%). Thus, in this case,
only one position (the date of change) and the mean values of NBS before and after the change should be
estimated.
b- Posterior probability of change-point
Conditional on the number of detected changes, it will therefore be interesting to locate the position of this
change. This is the purpose of the second result provided by the method of analysis. Figure 3.1(b) displays
the posterior probability distribution of the position of the detected change. A small weight is attributed to
1963 (less than 2%). Hence, the most probable date of change is 1966 (with a probability of 98%). Note
that when several change points are detected, a date of change is estimated for each detected change-point.
c- Estimation of simulated mean NBS and trend analysis
Figure 3.1(c) shows a sudden change in the direction of the trend after 1966. Indeed, an upward trend
before this date is followed by a downward trend for the period 1967-2005.
P os terior probability of the num ber of c hangepoints - Lak e S uperior (A nnual A verage)
1
0.9
0.8
(a)
0.7
Pr(m)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
m
2
29
Location of change point and probability of occurence- Lake Superior(Annual Average)
1
(b)
0.9
0.8
0.7
Pr(τc )
0.6
0.5
0.4
0.3
0.2
0.1
0
1950
1955
1960
1965
1970
1975
1980
Years
1985
1990
1995
2000
2005
Trend analysis in NBS - Lake Superior (Annual Average)
110
100
(c)
*-----*--- Observed NBS
Simulated mean NBS
90
NBS(m3/s)
80
70
60
50
40
30
1940
1950
1960
1970
1980
1990
2000
2010
Years
Figure 3.1. Lake Superior –annual average NBS: (a) Distribution of probability of the number of change
points; (b) A posterior distribution of probability of the position of change point; (c) Trend change analysis in
NBS time series (Lake Superior –annual average NBS)
Mean NBS before and after the change have been estimated and plotted. The magnitude of the shift
observed in NBS mean values can be visually inspected.
3.2.4 Analysis of obtained results
Table 3.9 summarizes the results derived from the application of the Bayesian change point detection
method to the NBS monthly and yearly data series for the seven lakes evoked in section 3.2.1. Overall, 91
NBS data series were investigated and at least one change point was observed in approximately half
(47%) of the studied time series. However, in almost all the cases (more than 91%) only one change point
was found. Moreover, in only four cases two change points were detected. The biggest number of changes
(12) was detected in NBS records observed in the Michigan –Huron Lake. More precisely, four lakes
present single change points in NBS annual average data series. Most of the changes were observed in 60s
(Superior (1960), Michigan (1966)) however the most recent ones are located in 1981 (Erie) and 1992
(Ontario).
More than 2/3 of the change-points detected in monthly data series were located in the interval-time
spanning from 1959 to 1979: the highest number of change points (more than 20%) occurred in 19591961 period followed by the periods of 1964-1967, 1976-1979, and 1992-1995 (more than 16%). For most
30
of the lakes, April data displayed at least one change-point (except for STC and HGB) contrary to May
and January data when change points were observed in only one or two lakes.
Table 3.9. Detected change points in NBS data series
Lake Superior
Lake Michigan
Lake Michigan- Huron
Lake Huron -Georgian Bay
Lake St. Clair
Lake Érié
Lake Ontario
Jan.
Feb.
1961
1981
1986
1964
March
April
May
June
July
Aug.
Sept.
Oct.
1971
1967
1977
1960- 1982
1967
1979
1965
1989
1979
1971
1959
1960
1995
1961
1995
1976
1979
1970
1994
1994
1978
1972
1993
1964 - 1985
1961
1994
1987
1974
1960
1971
Detected
Annual
average changepoints
1966
6
1960
9
1964 - 1985 1994
12
1959
2
1966
4
1960- 1978 1981
7
1992
7
Nov.
Dec.
3.2.5 Trend analysis in NBS considering a number of common change points
An exploratory analysis was performed by comparing Bayesian change point detection results and a visual
inspection of time evolution of NBS sample data. It was found that there are three change points that are
more common for all lakes including: 1965, 1975, and 1980. Therefore, all annual/monthly NBS time
series were partitioned and tested for each of the change points individually and the number of detected
trends for each change point was calculated. The obtained results for common change points in 1965,
1975, and 1980 are presented in Tables 3.10, 3.11, and 3.12, respectively.
Table 3.10. Trend detection in NBS considering a change point in 19651
Lake
SUP
MIC
MHG
HGB
STC
ERI
ONT
1
Period
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
48-65
0
0
0
0
0
0
0
66→
0
∇
∇
∇
0
0
∇
0
48-65
0
0
0
0
0
∇
∇
0
66→
0
0
0
0
0
48-65
0
0
∇
0
0
0
66→
0
0
0
48-65
0
0
∇
0
66→
0
0
48-65
0
66→
SEP
OCT
NOV
DEC
ANN
0
0
0
0
0
0
0
0
∆
0
0
0
0
0
0
0
0
0
0
0
∇
0
0
∆
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
∇
0
0
0
0
0
0
0
0
0
0
0
∇
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
48-65
0
∇
∇
0
0
0
0
0
0
0
0
0
0
∇
66→
0
0
0
0
0
0
0
0
0
0
0
0
0
48-65
∇
0
0
0
0
0
0
0
0
0
0
0
∇
66→
0
0
0
0
0
0
0
0
0
0
0
0
0
∆
∇ ∇
SUM
8
2
3
1
3
2
2
All NBS observations end in 2005
Table 3.11. Trend detection in NBS considering a change point in 19751
31
Lake
Period
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
DEC
ANN
SUM
48-75
∆
0
0
0
0
0
0
0
0
0
0
0
76→
0
∇
0
0
0
0
0
∇
0
0
0
∇
∆
5
48-751
0
0
0
0
∆
0
0
0
∆
0
∆
0
∆
76→
∆
0
0
∇
0
0
0
∆
0
0
0
0
0
48-751
0
0
0
0
∆
0
0
∆
0
0
0
0
∆
0
0
0
1
0
∇
0
0
0
0
1
SUP
MIC
MHG
HGB
STC
ERI
ONT
1
0
76→
∆
0
∇
∇
48-751
0
0
0
0
∆
0
0
∆
0
0
0
0
0
76→
0
0
0
0
0
0
0
∇
0
0
0
0
0
48-751
0
0
0
0
0
0
0
0
0
0
0
0
0
76→
∆
0
∇
01
0
0
0
0
0
0
0
0
0
48-751
0
0
0
0
0
0
0
0
∆
0
0
0
0
76→
0
0
∇
0
0
0
∇
0
0
0
0
0
0
48-751
0
0
0
0
0
∆
0
0
0
0
∆
∆
0
76→
0
0
0
0
0
0
0
0
0
0
0
0
0
7
7
2
3
3
3
All NBS observations end in 2005
Table 3.12. Trend detection in NBS considering a change point in 19801
Lake
SUP
MIC
MHG
HGB
STC
ERI
ONT
1
Period
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
DEC
ANN
48-80
∆
0
∆
0
0
0
0
0
0
0
0
0
∆
0
0
0
0
0
0
0
0
0
81→
∇ ∇
1
∇ ∇
0
∆
48-80
0
0
0
0
0
0
0
∆
∆
0
0
81→
0
0
0
0
0
0
0
0
0
0
0
∇
48-80
0
0
0
0
∆
0
0
∆
0
∆
0
0
0
∆
81→
0
0
0
0
0
0
0
0
0
0
0
0
∇
48-80
0
0
0
0
∆
0
0
0
0
0
0
∆
81→
0
0
0
0
0
0
0
∇
0
0
0
0
48-80
0
∇
∇
0
0
0
0
0
∆
∆
0
0
0
0
81→
0
0
0
0
0
0
0
∇
∇
0
0
0
0
0
0
0
0
∆
∇
48-80
∇
0
0
0
0
0
81→
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
∆
0
48-80
∆
0
81→
0
0
0
0
0
0
0
0
0
∆
∆
0
∆
∆
0
∆
0
0
∆
0
SUM
5
3
4
4
5
3
7
All NBS observations end in 2005
It is hypothesized that the change point that results in detection of the largest number of trends explains
the position (timing) of trend change point with the highest certainty. In order to decide which common
change point explains the variability in the NBS time series the best, the number of detected trends by
each change point were counted (see last columns). However, only one significant trend at either side of
the change point was considered. It was found that a change point in 1965 explains the observed
32
variability in Lake Superior the best; however, 1980 is the year that explains the variability in trends in
Lake Ontario the best. It is noteworthy that different combinations of Lakes Huron, Michigan, and
Georgian Bay have been used in this part of the study and this can cause some bias in the analysis results.
When only one combination of these lakes (MHG) is considered as the representative of these
combinations, the number of detected trends considering a change point in 1980 describes the variability
in the NBS the best. There is also another justification for choosing this year as the common change point:
annual NBS time series have the maximum number of significant trends when 1980 is considered as the
common change point.
For the first segment of the data (1948-1980), over all, 19 out of 65 time series (considering MHG as the
representative of lakes Michigan, Huron, and Georgian Bay) had significant trends. Out of 19 significant
trends only tow time series showed downward trends and the rest (17 time sires) showed upward trends.
For the second segment of the time series (1981-2005), 8 time series showed significant trends where all
detected trends had downward directions. It can be concluded that the Great Lakes NBS experienced an
upward trend in some lakes in some portions of the year until 1980. After a period of upward trends, the
NBS time series experienced a plateau and even in some lakes they started decreasing in some portions of
the year. Considering the annual NBS time series, these decreasing trends are more visible in Lakes
Superior and Michigan-Huron. There is less evidence of decreasing trends in Lakes St-Clair, Erie, and
Ontario. Given that the results of considering a change point in 1975 are quite similar to those of
considering a change point in 1980, it can be hypothesized that a general change in trends has occurred in
the second half of 1970s. Consequently, the Great Lakes NBS have been constant or in a decreasing
course after this period. These finding can be verified through considering the same change points for the
components of the NBS (Run, Precipitation, and Evaporation). It is also suggested that the seasonality
analysis performed on different variables for the whole observation periods be repeated considering
defined change points. Observed changes in trend directions might be coincided with some changes in
seasonality of explanatory variables.
3.3 Net Total Supply
3.3.1 Trend Analysis of NTS
The results of trend analysis on the NTS time series are presented in Table 3. According to this table, the
NTS time series did not experience significant trends in any of the lakes in annual scale. In seasonal scale,
however, the NTS experienced significant upward trends in January and February in Lakes St-Clair, Erie,
and Ontario. Upward trends were also significant in November and December for Lake Erie and in March
and April for Lake Ontario.
Table 3.13. Trend detection in the Great Lakes NTS
Lake
MIC-HUR
STC
ERI
ONT
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
DEC
Annual
0
0
0
0
0
0
0
0
∇
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
∆
∆
0
∆
∆
0
0
0
0
0
0
0
0
0
∆
∆
∆
∆
∆
∆
Results of trend analysis on special cases of the Net Total Supply (NTS) time series provided by the Great
Lakes Authorities are presented in Table 3.. This table shows that annual NTS experienced an upward
trend in Lake Erie; however, observed trend was significant only at a 10% significance level. As it can be
seen, this time series was highly dominated by autocorrelation. On the other hand, Lake Michigan-Huron
33
NTS time series did not show any significant trend. The estimated lag-1 autocorrelation for this time series
was much smaller compared to that of Lake Erie. According to the Table, the difference between Lake
Erie and Lake Michigan-Huron NTS (Erie NTS – Michigan_Huron NTS) experienced a very significant
upward trend for the observational period.
Time series
ERI NTS
Table 3.14. Annual NTS estimates for Lakes Michigan-Huron and Erie
Standard Normal Autocorrelation
P-values
Variate (Z)
(Lag-1)
(TFPW_MK)
1.77
0.709
0.077
Michigan-Huron NTS
0.26
0.279
0.789
Erie NTS – Michigan_Huron NTS
3.6
0.22
2E-04
3.3.2 Change point detection in Net Total Supply (NTS)
Table 3.15 presents the results of change detection in a selected number of NTS time series. According to
this table, Lake Erie annual NTS experienced four abrupt changes in 1923, 1942, 1957, and 1969. The
annual NTS time series of Lake Michigan-Huron, however, experienced only one change point in 1916.
The difference in the annual NTS for Lakes Michigan-Huron and Erie was also analysed for abrupt
changes and it was observed that this time series experienced two change points in 1926 and 1972. Figure
3.2 presents more details on the specifications of detected change points in the selected NTS time series.
Table 3.15. Results of change detection in a selected number of NTS time series
Water level time series
ERI NTS
Michigan-Huron NTS
Erie NTS – Michigan_Huron NTS
Beginning
of period
1900
1900
1900
Position of observed abrupt changes
1923
1942
1957
1969
1916
1926
1972
34
NTS−ANN−ERI
800
(a)
750
ANN−NTS−MichHur
(b)
800
750
700
700
650
650
600
600
550
500
550
450
500
400
450
350
400
1900
1920
1940
1960
Year
1980
2000
300
1900
2020
1920
1940
1960
Year
1980
2000
2020
Stand−NTS(ERI−MHU)
3
(c)
2
1
0
−1
−2
−3
1900
1920
1940
1960
Year
1980
2000
2020
Figure 3.2. Detected change points in a selected number of NTS time series
Figure 3.2(a) shows that NTS in Lake Erie experienced four abrupt changes during the observational
period: a decreasing shift in 1923, an increasing shift in 1942, a decreasing shift in 1957, and finally an
increasing shift in 1969. Despite the identified short term variability during the record period, the most
obvious change is observed in 1969 where after a significant increasing shift, the time series show a
consistent decreasing trend for the rest of observational period. Figure 3.2(b) shows that the minor
decreasing shift in Lake Michigan-Huron NTS in 1916 did not result in substantial variability in the time
series mean and no significant change in direction of trend for NTS is detected in this lake. The result of
change detection in the difference between Lake Michigan-Huron and Lake Erie NTS is depicted in
Figure 3.2(c). It can be seen that this time series experienced a decreasing shift in 1926 and an increasing
shift in 1972. While the first change point did not have significant impact on the direction of trend, the
observed shift in 1972 coincides with beginning of a consistent downward trend for the rest of record
period.
35
3.4 The seasonality evaluation of the NBS and Component
variables
In this part of the study, the seasonality in the hydro-meteorological variables in the Great Lakes for
different periods of the year is investigated. Although the main focus of the present study is the Great
Lakes NBS, the analysis of seasonal behaviour of explanatory hydro-climatic variables such as
temperature and evaporation is discussed first. This is because the interpretation of seasonality in the NBS
and the assessment of causes and effects of such behaviour will be considerably easier after the
seasonality of the NBS components and other contributors is addressed beforehand.
3.4.1 Seasonality assessment in the Great Lakes air temperature
Figure 3.3 compares maximum air temperature for different lakes in different periods of the year. It can be
seen that the highest maximum temperatures occurred in the first portion of the calendar year (between
NOV and MAY) whereas the lowest maximum air temperatures were observed in the second portion of
the year (from MAY to JUN). The highest maximum temperatures are also observed over the Lake Erie
basin. This can be attributed to the geography of the Lake Erie which has the lowest latitudes compared to
the other lakes. The Lake Superior, on the other hand, is located at the most north location compared to
the other lakes and this, probably, makes it the coldest basin with the lowest observed maximum
temperatures.
Superior
Michigan
Huron
Erie
Ontario
MCG-HUR
HUR-GB
JUN
St-Clair
Georgian Bay
Great Lakes
30
Superior
Michigan
Huron
Erie
Ontario
MCG-HUR
JUN
HUR-GB
St-Clair
Georgian Bay
20
Great Lakes
APR
MAY
MAR
FEB
20
10
APR
MAY
FEB
10
0
0
JUL
JAN
AUG
DEC
SEP
NOV
OCT
Figure 3.3. Seasonality in maximum air
temperature over the Great Lakes
MAR
-10
JUL
JAN
AUG
DEC
SEP
NOV
OCT
Figure 3.4. Seasonality in the minimum air
temperature over the Great Lakes
36
Superior
Michigan
Huron
Erie
Ontario
MCG-HUR
JUN
HUR-GB
St-Clair
Georgian Bay
25
Great Lakes
APR
MAY
MAR
FEB
15
5
-5
JUL
JAN
AUG
DEC
SEP
NOV
OCT
Figure 3.5. Seasonality in over basin mean air temperature in the Great Lakes
Seasonality in the minimum temperature records over the Great Lakes was investigated and the results are
presented in a polar plot in Figure 3.4. It can be seen that the observed seasonality in maximum
temperature is repeated in minimum temperature and no significant departures from explained patterns
were found. The highest and the lowest minimum temperatures were found in the Lake Erie and the Lake
Superior, respectively. Similar to maximum temperature, the variability of minimum temperature in
different lakes can be attributed to the geographical distribution (latitude) of the lakes.
Mean temperatures over the Great Lakes which are the averages of maximum and minimum temperatures
were analysed for seasonality and the results are presented in Figure 3.5. As it was expected, there is no
difference in seasonality polar plots for mean temperature compared to maximum and minimum
temperature as in all plots the variability in mean temperature correspond to the same periods of the year
observed for maximum and minimum temperature.
3.4.2 Seasonality assessment in Great Lakes water temperature
Water temperature in the Great Lakes was analysed for seasonality behaviour and the results are presented
in a polar plot in Figure 3.6. The observed seasonality in the air temperature over the Great lakes is more
or less observed in the water temperature as well. However, there exists a shift in timing of highest/lowest
water temperatures toward later dates compared to those of air temperature. That is, the highest water
temperatures were observed in the late summer and early fall and the lowest water temperatures were
registered in the early spring which shows water temperatures lag air temperatures by at least one month.
In respect to the magnitude of the recorded water temperatures, there exists some variability for different
lakes. The observed water temperatures in the Lake Superior are significantly inferior to the average of the
Great lakes water temperatures. This can be explained by the greatness of the water mass (more
particularly water depth) of the Lake Superior compared to other lakes. Since the main source of energy
for warming up the mass of water in the lakes is the solar energy, the larger the water mass (more
specifically the depth) is the less energy is absorbed by the water mass unit. On the other hand, water
temperature in the lakes St-Clair and Erie are considerably higher compared to other Great Lakes. The
opposite reasoning as what explained for the Lake Superior applies for superiority of water temperature in
these lakes, as they are the smallest among the Great Lakes. The significant high water temperature in the
Lake Erie and the Lake St-Clair can also be attributed to the geography (latitude) of these lakes as they
have the lowest latitudes among the Great Lakes.
37
Superior
Michigan
Huron
Erie
Ontario
HUR-GB
St-Clair
JUN
Georgian Bay
APR
MAY
MAR
FEB
30
20
10
JUL
JAN
AUG
DEC
SEP
NOV
OCT
Figure 3.6. Seasonality in modeled water temperature in the Great Lakes
3.4.3 Seasonality assessment in Great Lakes evaporation
Seasonality in evaporation in the Great Lakes was analyzed and a polar plot of obtained results is
presented in Figure 3.7. Interestingly, for most of the lakes the maximum evaporation occurs in cold
portion of the year (DEC, JAN, and FEB) whereas the minimum evaporation occurs in early summer.
However, this is not the case for the lakes St-Clair and Erie where the highest evaporation occurs in the
summer and early fall, respectively, in these lakes. While the behaviour that the timing of the max/min
evaporation seems non-intuitive relative to temperatures, it is correct. The key concept to remember is
that evaporation rate depends on a combination of temperature gradient (water to air) and the potential
humidity of the air. The Great Lakes have a very large thermal mass, which results in a temperature
response lag between the air temperature changes and those of water temperature. Evaporation is very
low (or even negative) in May and June because the water is still relatively cold while warm moist air
passes over it. This results in condensation and/or minimal evaporation. The opposite is true in the late
fall and winter months where water is relatively warm and cold air passes over it (Tim Hunter, GLERL,
personal communications). The smaller the water mass the smaller the lag between air and water
temperature changes. That is why, compared to the larger lakes, there is a minimal lag between the
max/min of air temperature and those of evaporation in the Lake St-Clair and to a certain extent in the
Lake Erie. For the same reason (impact of the size) the magnitude of evaporation is larger in theses two
lakes compared to the other lakes as there is a positive correlation between the mass of water and
temperature and eventually the amount of evaporation.
38
Superior
Michigan
Huron
Erie
Ontario
MCG-HUR
HUR-GB
JUN
St-Clair
Georgian Bay
Great Lakes
2× 102
APR
MAY
MAR
FEB
1.5
1
0.5
0
JUL
JAN
AUG
DEC
SEP
NOV
OCT
Figure 3.7. Seasonality in evaporation on the Great Lakes
3.4.4 Seasonality assessment in Great Lakes precipitation
The observations for precipitation over the Great Lakes were analysed to detect any seasonality behaviour
of this variable in the Great Lakes. It is noteworthy that the recorded data represent the precipitation over
lake surfaces; however, the measurements are based on the gauging stations located on the land all over
each of the Great Lakes basins. Figure 3.8 compares precipitations in different lakes using a polar plot of
records for the observation period. It can be seen from this figure that the magnitude of precipitation is
lowest in the winter and it is the highest in the summer and early fall. Although summer precipitations in
different lakes have the same (or fairly close) magnitudes, there is up to 50% difference in winter
precipitations for different lakes. While Lake Superior followed by the Michigan Lake has the lowest
amount of precipitation in the winter, the Georgian Bay followed by the Lake Ontario has the highest level
of precipitations in the cold portion of the year. The highest recorded precipitation over the lakes was
observed in the Georgian Bay in the month of October.
39
Superior
Michigan
Huron
Erie
Ontario
MCG-HUR
HUR-GB
JUN
St-Clair
Georgian Bay
1× 102
APR
MAY
MAR
FEB
0.8
0.6
0.4
JUL
JAN
AUG
DEC
SEP
NOV
OCT
Figure 3.8. Seasonality in the precipitation over the Great Lakes
3.4.5 Seasonality assessment in Great Lakes runoff
The runoff time series of the Great lakes were analysed for seasonality assessment purpose and the results
are presented in Figure 3.9. Generally speaking, the maximum runoff occurs in late winter and early
spring for the whole Great Lakes system and there is no or minimal runoff for summer and early fall. It
can be seen that the values of precipitation in the lake St-Clair and the Lake Ontario are higher compared
to the other lakes. This is more remarkable for the Lake St-Clair where runoff observations are of several
orders of magnitude compared to other lakes for the whole year period and more particularly for April.
40
Superior
Michigan
Huron
Erie
Ontario
MCG-HUR
HUR-GB
St-Clair
Georgian Bay
APR
MAY
MAR
JUN
8× 102
FEB
6
4
JUL
2
JAN
AUG
DEC
SEP
NOV
OCT
Figure 3.9. Seasonality in runoff in different Lakes
3.4.6 Seasonality assessment in Great Lakes NBS
After seasonality assessment in the NBS components and explanatory variables, Seasonality in the Great
Lakes NBS was performed. Figure 3.10 shows a polar representation of seasonality in the Great Lakes
NBS. It can be seen from this figure that the NBS values are lowest in the cold portion of the year and
they are highest in warm portion of the year for most of the lakes. While the NBS is in highest level in
spring (APR, MAY, and JUN) and also in fall (OCT, NOV, DEC), it is in the lowest levels in winter
(JAN, FEB, MAR) and in summer (JUL, AUG, SEP). The high levels of the NBS in spring can be
attributed to the snowmelt season which causes higher magnitudes of runoff into the Great Lakes. The
second period of the maximum NBS values corresponds to heavy rainfall periods in the fall which causes
a significant increase in precipitation component as well as the runoff component of the NBS. However,
this is not the case for the lakes Sainte-Clair and Ontario where NBS is low in the summer and it reaches
its maximum values in the winter and early spring. One should recall that this difference in the behaviour
was observed for other variables such as temperature, evaporation, and runoff in theses two lakes. A
comparison between Figures 3.9 and 3.10 reveals that the timing of maximum NBS values in the St-Clair
and Ontario lakes correspond to the timing of the maximum runoff observed in theses lakes. Moreover, a
review of Figure 3.7 shows that evaporation, as the output component of the NBS, is minimal in this
period of the year in the St-Clair and the Ontario lakes.
41
Superior
Michigan
Erie
Ontario
MCG-HUR
HUR-GB
JUN
St-Clair
APR
MAY
MAR
FEB
1× 103
0.6
0.2
JUL
JAN
AUG
DEC
SEP
NOV
OCT
Figure 3.11. Seasonality evaluation of the NBS for different lakes
3.5 Trend detection in Residual Net Basin Supply
Results of trend analysis in the residual NBS time series are presented in Table 3.16. According to this
table, residual NBS experienced significant upward as well as downward trends (considering 10%
significance level) in Lake Superior in some periods of the year but not in annual scale. There are only
two significant downward trends (at 10% level) for Lake Michigan-Huron in May and September (similar
to Lake Superior). For Lakes St-Clair and Erie, however, the residual NBS experienced significant upward
trends in most periods of the year and in an annual scale as well. Lake Ontario RNBS show significant
trends in the cold portion of the year and also in annual scale.
Table 3.16. Trend detection in the Great Lakes residual NBS
Lake
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
DEC
Annual
a) under STP assumption (TFPW_MK)
SUP
0
0
∆
∆
∇
0
0
0
∇
0
0
0
0
MIC-HUR
0
0
0
0
∇
0
0
0
∇
0
0
0
0
STC
∆
∆
∆
0
0
∆
0
0
0
0
∆
∆
∆
ERI
∆
∆
∆
∆
ONT
∆
∆
0
0
0
∆
∆
∆
∆
∆
∆
∆
∆
∆
0
0
0
0
∆
∇
∆
0
∆
0
0
∇
0
b) under LTP assumption (FARIMA_MK)
SUP
0
0
∆
∆
0
0
0
0
42
0
0
0
0
0
0
0
0
∇
0
STC
∆
∆
∆
0
0
0
0
∆
0
0
0
0
∆
∆
∆
∆
ERI
∆
∆
ONT
0
∆
0
0
0
0
∇
0
MIC_HUR
0
∆
0
0
0
0
0
∆
∆
∆
∆
∆
∆
∆
∆
0
43
Chapter 4 Statistical Analysis of Lake Level
and Flow Estimates
4.1 Statistical Analysis of Water Levels
4.1.1 Trend detection in the Great Lakes average water levels
Great lakes mean water levels were analysed in order to detect any monotonic trends in monthly and
annual scales. Table 4.1 compares detected significant trends in different lakes for monthly and annual
time series. It can be seen from that water levels in the Lakes STC and ERI experienced significant
upward trends for all monthly time series and also annual time series. This table also shows that no
significant trends were detected in monthly or annual water levels for Lake SUP. Lake MHG monthly
water levels experienced significant upward trends in JAN, FEB, and MAR at a 10% significance level.
Annual and monthly water levels experienced significant upward trends in Lake ONT except for the fall
portion of the year (OCT, NOV, and DEC).
Table 4.1. Trend detection results for water levels in the Great Lakes
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
Lake
Annual
a) (Original MK test, 1918-2007).
SUP
0
0
0
0
0
0
0
0
0
0
0
0
0
MHG
∆
∆
∆
0
0
0
0
0
0
0
0
0
0
∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ 0 0 0
STC
ERI
ONT
∆
∆
∆
b) (Modified MK test- 1918-2007)
SUP
0
0
0
0
0
0
0
0
0
0
0
0
0
MHG
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
∆
∆
∆
∆
∆
∆
∆
0
0
0
0
0
0
0
STC
ERI
ONT
∆ ∆ ∆ ∆ ∆ 0
∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆
c) (using FARIMA_MK under LTP assumption, 1918-2007)
SUP
MHG
STC
∆
ERI
∆
∆
ONT
44
In order to evaluate the magnitude of autocorrelation and its likely impact on detected trends in water level
time series, lag-1 autocorrelation for monthly and annual water levels were estimated. The analysis
showed water levels in the Great Lakes are extremely correlated. Annual and monthly water levels are
dominated by positive serial correlations for all lakes for the whole period of year. In order to investigate
the impact of observed autocorrelations on the number of detected significant trends, autocorrelations
were removed from the time series and the MK test was applied to prewhitened water level time series. It
can be seen from the same table that the number of significant trends experienced a considerable drop
after modifications. According to this table, Water levels experienced significant upward trends in lakes
St-Clair, Erie, and Ontario in monthly scale in the first half of the calendar year. However, water levels in
Lake Erie experienced significant upward trends in almost all periods of the year. There was no evidence
of significant trends in Lakes SUP and MHG in monthly or annual scale. No significant trend (at a 5%
significance level) was observed in annual water level time series for any of the lakes.
A study of change points in the Great Lakes water level time series (next section) was performed using a
Bayesian method of change point detection (Seidou & Ouarda, 2007). This was done in order to verify
whether or not the water level time series experienced any change in trends during the observation period.
Since some change points where identified in annual water level time series, the MK test was performed
on each segment of the time series before and after detected change points. A detailed description of the
change point detection results is presented in section 4.2 of this report; however, for the sake of
comparison, the results of MK trend test for segmented water level time series are presented here.
The results obtained from the Bayesian change point detection method showed that there were single
change points in the Lakes St-Clair, Erie, and Ontario in 1969, 1969, and 1975, respectively. However, no
change point was detected in the Lake Superior whereas two change points in 1970 and 1989 were
detected for the MHG Lake. This was confirmed by an exploratory data analysis on the monthly and
annual water level time series where a change point around 1970 was detectable in the majority of the
time series. For the sake of simplicity, a common change point in 1972 (as a compromise for detected
change points for different lakes) was set for all lakes and monthly and annual time series were tested for
trends before and after determined change point. Table 4.2 presents original MK test results, and modified
MK test results, respectively, for the period 1918-1972 (before common change point). It can be seen
from Table 4.2 a, based on original MK test, that water level time series experienced significant upward
trends in the Lakes St-Clair and Erie in monthly and annual scales, inclusively. However, no significant
trend was detected in the Lakes Superior and MHG. Water levels in the Lake Ontario experienced
significant trends (at a 10% significance level) only in summer period (JUN, JUL, AUG, and SEP).
Table 4.2. Trend detection results for water levels in the Great Lakes (1918-1972)
Lake
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
Ann.
a) (original MK test)
SUP
0
0
0
0
0
0
0
0
∆
∆
0
0
0
MHG
0
0
0
0
0
0
0
0
0
0
0
0
0
STC
ERI
ONT
∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆
0
0
0
0
0
∆
∆
∆
∆
∆
∆
0
0
0
0
b) Modified TFPW_MK test
SUP
0
0
0
0
0
0
0
0
0
0
0
0
0
MHG
0
0
0
0
0
0
0
0
0
0
0
0
0
45
∆ ∆
STC
0
ERI
0
∆
ONT
0
0
∆
∆
0
0
0
∆
0
∆
∆
∆
∆
∆
0
0
0
0
0
∆
∆
∆
∆
0
0
0
0
0
0
0
0
0
0
0
Analysis of the autocorrelation showed that water level time series in this period were highly correlated.
The impact of high serial correlation is reflected in the results of TFPW_MK test (Table 4.2(b)) where
trends are significant in some seasons and also annual water levels (at 10% significance level) in Lakes StClair and Erie. No significant trends were detected in Lake Ontario in either monthly or annual scales after
modifications to account for autocorrelations.
The results of trend detection in water levels after change point (1973-2007) are presented in Table 4.3.
According to Table 4.3(a), based on original MK test, water levels experienced significant downward
trends in all lakes in both monthly and annual scales at a 5% significance level except in Lake Ontario in
some periods of the year.
Lake
Table 4.3. Trend detection results for water levels in the Great Lakes (1973-2007)
JAN
FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
Ann.
a) original MK test
ERI
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
ONT
0
0
0
SUP
MHG
STC
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
0
0
0
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
0
0
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
0
0
0
b) Modified MK test
SUP
MHG
STC
∇
∇
∇
ERI
∇
ONT
0
∇ ∇ ∇ ∇
∇ ∇ ∇ ∇
∇ ∇ ∇ ∇
0
∇
∇ ∇
0
0
0
0
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
∇
0
0
Although autocorrelations were still significantly high, their values for this period of observations were
lower compared to the previous period. TFPW_MK test results presented in Table 4.3(b) show that 51 out
of 65 tested time series (77%) experienced significant downward trends (at a 5% significance level) even
after autocorrelations were removed. The difference in the MK test results before and after prewhitening is
limited to Lakes Erie and Ontario where water levels did not show significant trends in some periods of
the year based on TFPW_MK test.
A comparison of the direction, significance level, and the number of detected trends in annual water levels
before and after segmentation of time series (based on detected change points) is performed in Tables 4.4
and 4.5 for original and TFPW_ MK test, respectively.
46
Table 4.4. Comparison of annual trends in water levels for different record periods (original MK test)
Period
1918-2007
1918-1972
1973-1988
1973-2007
1989-2007
Lake
(1)
(2)
(3)
(4)
(5)
SUP4
0
0
---------
MHG
0
0
0
∆
∆
∆
∆
∆
STC
ERI
ONT
0
-------------------------
∇
∇
∇
∇
∇
--------
∇
----------------------
A comparison between columns (1) and (2) of Table 4.4 reveals that there is a close agreement between
the number and direction of trends in annual water levels for the whole record period (1918-2007) and the
period before the common change point (1918-1972). Interestingly, however, the detected trends for the
second period of the observations (column (4)) have opposite directions compared to those for the whole
period of observations (column (1)). This is in full agreement with the observations of the responsibilities
reporting decreases in the Great Lakes water levels in few past years. This also underlines the importance
of simultaneous evaluation of trends and change points in hydro-climatic time series in order to have more
realistic understanding of the process under study. The results of TFPW_MK test, compared in Table 4.5,
show that no significant trends were detectable (at a 5% significance level) in annual water levels for the
whole period of observations and also for the period before the change point (1918-1972). However,
annual water levels show significant downward trends after the change point (1972) in the Great Lakes
(except Lake Ontario) even after removing autocorrelations. It is noteworthy that although shift analysis
identified three segments in Lake MHG water levels, there is no evidence of significant trends in the first
(1918-1972) and the second (1973-1988) segments; however, there is a downward trend for the third
segment (1989-2007) regardless of the trend detection method.
Table 4.5. Comparison of trends in annual water levels for different record periods (TFPW_MK test)
Period
1918-2007
1918-1972
1973-1988
1973-2007
1989-2007
Lake
SUP
0
0
---------
MHG
0
0
0
STC
0
∆
---------
ERI
∆
∆
---------
ONT
0
0
---------
∇
∇
∇
∇
0
--------
∇
----------------------
4.1.2 Trend detection on Multi-station Water Levels
Trend analysis under STP assumption
4
Although no change points was detected for Lake Superior, the observations before and after the common change
point were tested for comparison purpose.
47
The results of trend analysis in Multi-station water level data are presented in Table 4.6. The first column
of this table presents the name of the time series that trend detection was performed on. The second
column presents the standard normal variate (Z) for the S statistic of the MK test. The third column in this
table presents the lag-1 autocorrelation of the time series, and the last column presents the p-values
associated with the estimated trends. The p-values equal or smaller than 0.05 (significant trends at a 5%
level of significance) are shown using bold numbers (grey cells).
Table 4.6. Trend detection results for Multi-station water level time series
Water level time series
Cleveland
Gibraltar
Harbour Beach
St-Clair Shore
GLB- CLV
GLB- Fermi
HB - SCS
HB- CLV
HB - GIB
SCS - CLV
SCS - GIB
(HB - SCS) - (SCS-CLV)
(HB - SCS) - (SCS-GLB)
Standard Normal
Variate (Z)
1.98
0.95
-0.13
0.98
1.01
4.17
-5.66
-5.33
-2.67
-2.27
-2.00
-5.23
-2.28
Autocorrelation
(Lag-1)
0.723
0.741
0.798
0.788
0.741
0.557
0.599
0.484
0.649
0.343
0.679
0.755
0.696
P-values
(TFPW_MK)
0.047
0.338
0.896
0.324
0.312
3E-05
1E-08
9E-08
0.008
0.023
0.045
2E-07
0.023
According to Table 4.6, water levels in Cleveland station experienced an upward trend. Autocorrelation
for this time series is relatively high (0.723); however, trend is still significant at 5% significance level
after autocorrelation is removed from the time series. Table 4.6 also shows that water levels experienced
upward trends in St-Clair Shore and Gibraltar stations whereas they showed downward trend in Harbour
Beach. However, observed trends were not significant in any of mentioned water level stations. It can be
seen that autocorrelation coefficient is larger than 0.7 for all of the water level time series. Table 4.6 also
shows that while the fall in water levels between Gibraltar and Cleveland stations (GLB - CLV) did not
experience significant trends, the water level fall between all other examined stations (GLB- Fermi, HBSCS, HB-GIB, SCS-CLV, and SCS-GIB) experienced significant trends, inclusively. Observed significant
trends in fall between different stations had downward directions except for the fall between Gibraltar and
Fermi (GLB-Fermi) where detected significant trend showed upward direction. As it can be seen from
Table 4.6, the difference between water level falls for different stations ((HB-SCS) - (SCS-CLV) and
(HB-SCS) - (SCS-GLB)) were also tested for trends and it was observed that theses time series were
characterized by significant downward trends under the LTP assumption.
The results of trend analysis on the water levels in the stations obtained from the TWG's SharePoint
website are presented in Table 4.7. It can be seen that water levels in Buffalo station experienced upward
trends. Observed trend, however, is not significant at a 5% significance level but it is significant at a 10%
significance level. Water level in Harbour Beach experienced a downward trend where, similar to Buffalo
station, it was only significant at a 10% significance level. There was no evidence of significant trends in
Lakeport water level at any significance level. According to Table 4.7, although fall between Lakeport and
Buffalo stations (Lakeport – Buffalo) experienced a significant downward trend at a 10% significance
level, it was not significant at a 5% significance level. According to this table, unlike Lakeport – Buffalo
48
fall, water levels between Harbour Beach and Buffalo stations experienced a very significant downward
trend.
Table 4.7. Water level time series obtained from the TWG's SharePoint website
Standard Normal
Autocorrelation
P-values
Water level time series
Variate (Z)
(Lag-1)
(TFPW_MK)
Buffalo
1.86
0.718
0.062
Harbour Beach
-1.75
0.816
0.08
Lakeport
-0.30
0.817
0.763
Lakeport - Buffalo
-1.7
0.498
0.076
Harbour Beach - Buffalo
-5.9
0.445
3E-09
Trend Analysis under LTP assumption
This section is devoted to the results of trend analysis under the Long Term Persistence (LTP) hypothesis.
Table 4.8 presents the results of trend analysis in water level time series in Multi-station dataset. In this
table, the second column provides the MK S statistic and the third column presents the p-values under
LTP assumption. As it can be seen from this table, none of the tested water level time series (Cleveland,
Gibraltar, Harbour Beach, and St-Clair Shore) show significant trends under the LTP assumption. It can
also be seen that GLB- CLV, HB- SCS, HB- CLV, HB- GIB, SCS - CLV, and SCS- GIB falls did not
experience significant trends. However, GLB- Fermi fall experienced a significant upward trend whereas
HB- SCS and HB- CLV falls experienced significant downward trends even after LTP was accounted for.
Table 4.8 also shows that while the difference between HB - SCS and SCS- CLV fall experienced a very
significant downward trend, no significant trend was observed in the difference between HB- SCS and
SCS- GLB falls.
Table 4.8. Results of trend analysis under LTP assumption for Multi-station water level dataset
Water level data
Cleveland
Gibraltar
Harbour Beach
St-Clair Shore
GLB- CLV
GLB- Fermi
HB - SCS
HB- CLV
HB - GIB
SCS - CLV
SCS - GIB
(HB - SCS) - (SCS-CLV)
(HB - SCS) - (SCS-GLB)
MK statistic
1764
537
56
1417
245
531
-3342
-2986
-797
-1211
-455
-3667
-438
Simulated P-value
0.198
0.401
0.964
0.326
0.696
0.001
0
8.00E-04
0.150
0.121
0.450
0
0.308
Table 4.9 presents the results of trend analysis for the water level sample data obtained from the TWG’s
SharePoint website. As it can be seen from this table, no significant trend was observed for Buffalo,
Harbour Beach, and Lakeport water level stations under LTP assumption. This table also shows that
49
Lakeport - Buffalo fall did not show significant trend for the observational period. However, Harbour
Beach- Buffalo fall experienced a very significant downward trend under LTP assumption.
Table 4.9. Trend in water level data obtained from the TWG’s SharePoint website under LTP assumption
Water level data
MK statistic
Simulated P-value
1684
-3170
22
-234
-3778
0.34
0.22
0.95
0.41
4.00E-04
Buffalo
Harbour Beach
Lakeport
Lakeport - Buffalo
Harbour Beach - Buffalo
4.1.3 Trend detection in Connecting Channel discharges
Table 4.10 presents the results of trend detection in connecting channel discharges. It can be seen that no
significant trend is observed in connecting channel discharges in an annual scale. In a monthly scale,
however, there is strong evidence of significant upward trends in winter for all connecting channels.
Table 4.10. Trend detection in connecting channels (1900-2007)
Channel
St_Mary
St-Clair
Detroit
Niagara
falls
JAN
0
∆
∆
∆
FEB
MAR
∆
∆
∆
∆
∆
∆
∆
∆
APR MAY JUN
JUL
AUG
SEP
a) Under STP assumption (TFPW_MK)
OCT
NOV
DEC
Annual
∆
∆
∆
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
∆
0
0
0
0
0
0
0
0
0
b) under LTP assumption (FARIMA_MK)
St-Marry
0
0
0
0
0
0
0
0
0
0
0
0
0
St-Clair
0
∆
0
0
0
0
0
0
0
0
0
0
0
Detroit
∆
∆
0
0
0
0
0
0
0
0
0
0
0
Niagara
0
0
0
0
0
0
0
0
0
0
0
0
0
4.1.4 Seasonality assessment in connecting channels discharges
Figure 4.1 shows the results of seasonality analysis in the connecting channels discharges. It can be seen
that no seasonality was observed in the connecting channels discharges. This figure also shows that
discharges in the St-Clair and Detroit River have the same magnitudes; however, St-Marys discharge is
several orders of magnitude smaller compared to the other rivers. Niagara River discharge measurements
were not supplied by the GLERL and as such are not included in seasonality assessments.
50
St-Marys
St-Clair river
Detroit
APR
MAY
MAR
JUN
6× 1035
FEB
4
JUL
3
2
JAN
AUG
DEC
SEP
NOV
OCT
Figure 4.1. Seasonality in the connecting channel discharges
4.1.5 Trend detection in Change in Fall (MH and Erie) and
Residual/Component NBS differences in the Lake Superior.
Annual discharges of fall between Huron and Erie Lakes and also the difference between residual and
component annual NBS time series for the Lake Superior were analysed for trends. The former has 147
years of record (1860-2006) whereas the latter has 59 annual observations (1948-2005). Table 1 presents
the results of trend analysis performed on the two time series. According to this table the time series of
Huron-Erie fall experienced a downward significant trend at 5% significance level. Although the time
series are highly correlated, modifications on the MK test to account for autocorrelations did not have any
impact on the test results. As for the difference between residual and component NBS time series in Lake
Superior, based on original MK test, the time series experienced a downward trend at a 5% significance
level; however, this time series is characterized by a relatively high autocorrelation, and therefore no
evidence of trend was found after modification was performed on the MK test to account for estimated
autocorrelation
Table 4.11. Trend detection results for two additional time series
Estimated
Fall
Lake Superior NBS
51
Variable
Z
Pvalue1 (original MK)
Pvalue2 (modified MK)
Lag-1 Autocorrelation
Downward trend at 5% SL (MK)
Downward trend at 5% SL (modified MK)
(Huron-Erie) (residual NBS-component NBS)
-11.9382
-2.2108
0
0.027
3.66E-04
0.1972
0.8423
0.5046
Yes
Yes
Yes
No
The results of trend analysis on the difference between residual and component NBS in Lake Superior and
also the difference between annual water levels in Lakes Huron and Erie under LTP assumption are
presented in Table 4.12. It can be seen that no significant trend was observed for the difference between
residual and component NBS in Lake Superior. The annual average fall between Lakes Huron and Erie,
however, experienced a very significant downward trend under LTP assumption.
Table 4.12. Trend analysis under LTP assumption for the difference between residual and component NBS in
Lake Superior and fall between Lakes Michigan-Huron and Erie
Variable
RNBS-CNBS (Lake Superior)
Huron-Erie Fall
4.2
MK statistic
-517
-7130
Simulated P-value
0.109
0
Change point detection in water levels and flows
This part is devoted to detection of change in the parameters (trend or mean) of water level time series. To
find abrupt shifts that might eventually be linked to climate variability, water level data series are modeled
as a linear combination of a set of relevant hydro-meteorological explanatory variables. The choice of
‘good’ predictors is mainly based on the statistical significance of the correlations between water levels
data series and climatic variables. As shown by the tables presented in Appendix A.2, the estimated
correlations between water level time series and available meteorological data are quite trivial. Hence, the
linear combination used to model water levels data may lead to lose a signal when the Bayesian method of
change point detection is applied. For these reasons, another analysis is performed in order to detect
sudden changes in the trend of water levels data series without any use of hydro-meteorological
covariates. The period of study goes from 1948 to 2005 for the first analysis (due to the availability of
meteorological data) and from 1918 to 2007 for the second. Let us mention that water level data are
provided for five sites namely: Lake Superior (SUP), Lake Michigan – Huron (MHG), Lake St. Clair
(STC), Lake Erie (ERI) and Lake Ontario (ONT).
4.2.1 Change detection in the linear combination of hydro-climatic variables
Correlations between water levels and climatic factors such as precipitation and temperature were first
computed and tested using Student t-test (α = 5%). Then, meteorological variables that demonstrated high
correlations with water levels data series were selected in the set of explanatory variables. For annual data
series, the correlations between water levels and precipitation were statistically significant except for Lake
Michigan-Huron. For all lakes, these correlations increased widely (more than 50% in general) when
considering the mean cumulative precipitations that recorded during the three past years. On the other
hand, for monthly data, the analysis of correlations between water levels and precipitation did not reveal
52
any significant linear dependence. Hence, the choice of this variable as covariate (case of monthly data)
appears to be less appropriate. Moreover, the correlations between water levels and temperatures for both
monthly and annual data were not high enough (except for Lake Superior and barely for Lake MichiganHuron) to be considered as significant. Nevertheless, the regression model that integrates precipitation and
temperature as covariates is validated using the method of maximum errors.
Case study: Lake Michigan-Huron (annual average)
Figure 4.2 shows the existence of one change with a probability of 14%, and three changes with nearly
20% chance; however, the probability of detecting two changes is more than 66%. Thus, for Lake
Michigan-Huron, two change-points (1970 and 1989) were detected. Sudden trend changes are observed:
note that the trend is slightly downward after 1989.
Number of changepoints (Michigan-Huron, annual average)
0.7
Trend analysis - Water levels (Michigan-Huron, annual average)
177.6
0.6
177.4
0.5
0.4
Pr(m)
177.2
0.3
177
Water levels(m)
0.2
0.1
0
0
1
2
m
3
4
Change-point 1- (Michigan-Huron, annual average)
0.8
176.8
176.6
176.4
Pr(τc )
0.6
176.2
0.4
0.2
176
0
1950
1955
1960
1965
1970
1975
1980
Years
1985
1990
1995
2000
2005
Change-point 2- (Michigan-Huron, annual average)
175.8
0.8
Pr(τc )
0.6
0.4
1940
0
1950
1960
1970
1980
1990
2000
2010
Years
0.2
1950
1955
1960
1965
1970
1975
1980
Years
1985
1990
1995
2000
2005
Figure 4.2. Change detection in water level data series: Lake Michigan-Huron (annual average)
As for the other annual average data series, no change was detected for Lake Superior where trend was
downward for the whole period of observation (1948-2005). A common change-point located in 1969 was
detected for the Lakes St. Clair, Erie and Ontario.
4.2.2 Change detection in water levels without hydro-meteorological
covariates
Water levels data series were analysed on the same period of observation (from 1918 to 2007). Two time
scales were investigated: month and year (annual average). The Bayesian method developed by Seidou
and Ouarda (2007) was applied. To detect changes in the trend of water level data series, simulated
random vectors (uniform or normal distributed) were used as predictors.
The results obtained for annual average data series give a good survey of the findings derived from the
analysis of monthly data series (cf. Table 4.13 below). In all investigated cases, 3 or 4 changes were
detected. As a reminder, for the method of analysis, the rate of false detection is negligible when the
number of detected change points does not exceed 3 (large samples with more than 75 data).Thus, to limit
the number of false detections, the requirement for the highest probability associated to the number of
53
detected changes is 50% or more. On the other hand, concerning the posterior probability distribution of
detected change-points, many histograms show that the most probable date of change has a low
probability of occurrence or this probability is very close to the one associated to many other neighbour
dates (grouped change-points; e.g. Figure 4.3(b) (N°4)). This indicates that a range of probable dates of
change (instead of a unique date) should be taken into consideration. It, therefore, appears that for all the
lakes, the most probable periods of trend changes in water level data follow one another during 19291932, 1942-1943 (except for Lake Superior), 1955-1957 and 1967-1971. For Lake Superior, a last period
of trend change was detected during the time interval 1987-1988. Note that in the considered cases, a clear
downward trend is observed after the last detected change-point.
Figure 4.3. Detection of trend changes in water level data series: Lake Ontario (annual average data series)
Table 4.13 Change points for water level annual data series.
Lake Superior
Lake Michigan – Huron
Lake St. Clair
Lake Érié
Lake Ontario
1929 - 1954 – 1968 -1988
1931 – 1942 – 1957 - 1969
1943 - 1957 - 1969
1943 - 1957 - 1969
1931 - 1943 - 1956 - 1971
4.3 Seasonality assessment in water levels in the Great Lakes
4.3.1
Seasonality assessment considering the whole record length
54
The seasonality in the Great Lakes water levels was investigated using a polar plot of the water level time
series and the results are shown in Figure 4.4. This figure shows the difference in the water levels and also
proximity of water levels for different lakes except the Lake Ontario whose water levels are significantly
lower than other lakes (mainly due to Niagara Falls). However, the seasonality can not be observed in this
figure due to the scale problem. Figure 4.5 provides a more clear presentation of seasonality in water
levels by using polar plot presentation of individual lake water levels.
Superior
Erie
Ontario
MCG-HUR
St-Clair
APR
MAY
MAR
JUN
2× 102
FEB
1.6
1.2
JUL
0.8
JAN
AUG
DEC
SEP
NOV
OCT
Figure 4.4. Seasonality in water levels in the Great Lakes
Figure 4.5 shows that the maximum water levels in the Superior Lake are observed in summer and early
fall and minimum water levels are observed in spring. Michigan-Huron lakes water levels show a small
shift in the timing of maximum/minimum water levels where maximum water levels in this lake occur in
summer (more particularly in August). It can be seen that this shift in timing of maximum/minimum water
levels is repeated in downstream lakes as well. For example, in the Lake Ontario as the most downstream
lake, maximum water levels occur in July and June which compared to the Lake Superior (the most
upstream lake) timing of maximum water levels shows a one season shift toward earlier dates.
55
Lake Superior
Lake Michigan-Huron
APR
MAY
MAR
JUN
FEB
102
1.8355×1.8345
JUL
JAN
JUL
DEC
SEP
MAR
JUN
FEB
1.7665×1.7655
102
1.8335 1.8325
AUG
APR
MAY
1.7645 1.7635
AUG
NOV
DEC
SEP
NOV
OCT
Lake St-Claire
OCT
Lake Erie
APR
MAY
MAR
JUN
JAN
DEC
SEP
MAR
JUN
102
1.744×1.743
1.75 1.749
1.748
AUG
APR
MAY
FEB
102
1.752×1.751
JUL
JAN
NOV
JUL
FEB
1.742 1.741
1.74
JAN
AUG
DEC
SEP
OCT
NOV
OCT
Lake Ontario
APR
MAY
MAR
JUN
75.1
FEB
74.9
JUL
74.7
JAN
AUG
DEC
SEP
NOV
OCT
Figure 4.5. Seasonality in water levels for different lakes
56
4.3.2
Seasonality assessment considering change points
A study of shifts was performed for the Great Lakes water level time series using a Bayesian method of
change point detection (Seidou and Ouarda, 2007). The methodology uses a multivariate linear regression
model to detect any change in mean or trend in the relationship between the response variable and
explanatory variables. The response variable in this case is the annual water level whereas the explanatory
variables are Mean air temperature (Co) over basin and Precipitation (mm) over lakes. It was observed that
some annual water levels experienced significant shifts in some of the Great Lakes; however, no shifts
were detected in mean water levels in the Lake Superior. Annual water levels in Lake Michigan-Huron
and Georgian Bay experienced 2 shifts in mean in 1970 and 1989. Figure 4.6 shows a polar presentation
of water levels before and after change points in this lake. It can be seen that the water levels experienced
significant changes for the corresponding seasons. Interestingly, for the first and third segment of
observations, (1918 to 1969) and (1989-2007), respectively, the values of water levels were in a close
agreement in different portions of the year. However, annual water levels experienced an increasing shift
for a period starting in 1970 and ending in 1988 compared to the first and last period of records. This shift
amounts to about 50 centimetres in average for different periods of the year. There is a similarity in the
fluctuations of water levels in different seasons for the three periods of observations. That is, water levels
are in their lowest levels in late winter (FEB, MAR, and APR) whereas they are in their highest levels in
summer (JUL, AUG, and SEP). It can be seen that the difference in water levels in low and high seasons
reaches a maximum of 25 centimetres in MHG Lake for all three periods.
From 1989 to 2007
From 1970 to 1988
From 1918 to 1969
APR
MAY
MAR
JUN
1.771× 102
1.769
FEB
1.767
JUL
1.765
1.763
JAN
AUG
DEC
SEP
NOV
OCT
Figure 4.6. Seasonality assessment in Lake MHG for different periods of observations
Statistical analysis showed that annual water levels in the Lake St-Clair experienced an increasing shift in
1969. Figure 4.7 depicts the seasonality of water levels for the periods before and after the change point in
this lake. This figure shows that mean water levels experienced both an increasing shift and a shift in
seasonality behaviour for the second period of observations (1969-2007) compared to the previous period
(1918-1968). The maximum difference in water levels for the two consecutive periods occurs in March
57
where it amounts to 65 centimetres whereas it is close to 40 centimetres for other periods of the year. For
the second period, similar to Lake MHG, water levels are at highest levels in the summer (JUL, AUG, and
SEP); however, they experience a sharp decrease of about 20 centimetres where they reach their lowest
levels in early winter (DEC,JAN, and FEB). For the first period (1918-1968) minimum water levels
occurred in March whereas it occurred in early winter in the second period and this shows a change in
seasonality behaviour with time in this lake.
From 1969 to 2007
From 1918 to 1968
APR
MAY
MAR
JUN
1.755× 102
1.753
FEB
1.751
JUL
1.749
1.747
JAN
AUG
DEC
SEP
NOV
OCT
Figure 4.7. Seasonality assessment in the Lake St-Clair for different periods
Chang point detection analysis showed that an upward shift in water levels occurred in 1969 in the Lake
Erie. The seasonality behaviour of water levels in this lake for the periods 1918-1968 and 1969-2007 is
compared in Figure 4.8. This figure shows that water levels experienced an increase of approximately 40
centimetres for the second period of observations compared to the previous one in average. For both
periods, maximum water levels occurred in summer (JUN, JUL, AUG) and minimum water levels
occurred in late fall through early spring (DEC, JAN, FEB, MAR). However, the difference between
maximum and minimum water levels in the first period (1918-1968) amounts to almost 40 centimetres
whereas it is approximately 20 centimetres for the second period (1969-2007). This implies a change,
though not overwhelming, in the seasonality behaviour of water levels in the Lake Erie.
58
From 1969 to 2007
From 1918 to 1968
APR
MAY
MAR
JUN
1.747× 102
1.745
FEB
1.743
JUL
1.741
1.739
JAN
AUG
DEC
SEP
NOV
OCT
Figure 4.8. Seasonality assessment in the Lake Erie for different periods
Bayesian change point detection performed on water levels in the Lake Ontario showed an upward shift in
1975. A polar presentation of seasonal water levels in this lake for the periods before and after change
point is depicted in Figure 4.9. According to this figure, the maximum increase in water levels occurred in
FEB and MAR where it amounts to almost 20 centimetres. This increase was as low as 5 centimetres
approximately in OCT and NOV. An inspection of figure 4.8 reveals that, overall, the observed increasing
shift in water levels was maximal in the first portion of the calendar year (FEB- AUG) and it was minimal
in the second portion of the year (AUG-FEB). The timing of minimum water levels show one month shift
toward earlier dates when it shifts from JAN-FEB for the first period (9118-1974) to DEC-JAN for the
second period (1975-2007). There is no evidence of shifts in timing of maximum water levels for the
studied periods. Furthermore, the difference between maximum and minimum water levels did not
experience significant changes for studied periods in this lake.
59
From 1975 to 2007
From 1918 to 1974
APR
MAY
MAR
JUN
75.2
FEB
75
74.8
JUL
74.6
JAN
AUG
DEC
SEP
NOV
OCT
Figure 4.9. Seasonality assessment in the Lake Ontario for different periods
60
4.3.3 Change point detection in Water Level stations
A change point analysis was performed on the station water level data to investigate the presence of any
abrupt change in mean or change in the direction of trends for the observational periods. The results of
Bayesian change point detection for studied variables are presented in the following subsections.
4.2.1 Change points in the Multi-station water level dataset
Table 44 present the results of change point analysis for the time series in Multi-station dataset.
Table 4.14. Trend detection results for Multi-station water level time series
Water level time series
Cleveland
Gibraltar
Harbour Beach
St-Clair Shore
GLB- CLV
GLB- Fermi
HB – SCS
HB- CLV
HB – GIB
SCS - CLV
SCS – GIB
(HB – SCS) - (SCS-CLV)
(HB – SCS) - (SCS-GLB)
Starting
Year
1900
1937
1900
1900
1937
1964
1900
1900
1938
1900
1938
1900
1948
Abrupt changes observed during the observational period
1923
1943
1923
1923
1942
1943
1924
1924
1957
1957
1957
1957
1957
1969
1970
1969
1969
1987
1988
1989
1989
1989
1912
1923
1935
1988
It can be seen that water levels in the four water level stations in this dataset (Cleveland, Gibraltar,
Harbour Beach, and St-Clair Shore) experienced similar variability in terms of change points. That is, the
Bayesian change point detection technique identified four change points in Cleveland, Harbour Beach, and
St-Clair Shore stations located in 1923, 1942-43, 1957, and 1969. For Gibraltar station, however, two
change points in 1957 and 1970 were identified. Figure 4.10 provides more details on the specifications of
the abrupt changes in water levels in mentioned stations.
61
cleveland
gibralter
175
175.4
174.8
175.2
174.6
175
174.4
174.8
174.2
174.6
174
174.4
173.8
174.2
173.6
174
173.4
1900
1920
1940
1960
Year
1980
2000
2020
173.8
1930
1940
1950
1960
1970
Year
1980
1990
2000
2010
st. clair shores
176
harbor beach
177.6
175.8
177.4
177.2
175.6
177
175.4
176.8
175.2
176.6
176.4
175
176.2
174.8
176
174.6
175.8
1900
1920
1940
1960
Year
1980
2000
2020
174.4
1900
1920
1940
1960
Year
1980
2000
2020
Figure 4.10. Change detection in water levels in Multi-station dataset
Figure 410 (top-left) shows that water levels in Cleveland experienced a decreasing shift in 1923, an
increasing shift in 1943, a decreasing shift in 1957, and finally an increasing shift in 1969. This is also true
for Harbour Beach and St-Clair shore stations (bottom-left and bottom-right, respectively). A visual
inspection of these figures shows that a significant change in trend direction can be located in 1969.
According to Figure 4.10 (top-right), water levels in Gibraltar station experienced a downward as well as
an upward shift in 1957 and 1970, respectively. A change in trend direction after 1970 in this station is
obvious too.
Change point detection results for the falls between water level stations used in this part of the study are
also presented in Table 4. It can be seen from this table that the most common change point is located in
1987-1989 time period. Another potential common change point can be located in 1923-1924 time period.
Figure 41 illustrates the location and the nature (increasing/decreasing) of detected change points in the
fall between the stations discussed above. This figure shows that observed abrupt changes in the fall
between different stations/lakes are decreasing shifts except for the fall between Gibraltar and Cleveland
where observed shifts in 1957 and 1987 are, respectively, downward and upward. This figure shows that
62
there is no significant change in trend direction for the fall time series, except for the Gibraltar- Cleveland
fall where a downward trend is followed by an upward trend after the change point in 1987. A visual
inspection of this time series reveals that the change in trend direction can be located even before the
change point, probably in 1970s.
HB−SCS
gib−clev
0.18
1.9
1.8
0.16
1.7
0.14
1.6
0.12
1.5
1.4
0.1
1.3
0.08
1.2
0.06
1930
1940
1950
1960
1970
Year
1980
1990
2000
2010
1.1
1900
1920
1940
1960
Year
1980
2000
2020
HB−GIB
hb−clev
2.6
3
2.5
2.8
2.4
2.3
2.6
2.2
2.4
2.1
2
2.2
1.9
2
1.8
1.8
1900
1920
1940
1960
Year
1980
2000
2020
1.7
1930
1940
1950
1960
1
1.05
0.95
1
0.9
0.95
0.85
0.9
0.8
0.85
0.75
0.8
0.7
0.75
0.65
1920
1940
1960
Year
1980
1990
2000
2010
SCS−GIB
SCS−CLEV
1.1
0.7
1900
1970
Year
1980
2000
2020
1930
1940
1950
1960
1970
Year
1980
1990
2000
2010
63
Figure 4.11. Change detection in fall between different water level stations
Figure 4.12 present the results of change point detection for the difference between HB-SCS and SCSCLV falls as well as the difference between HB-SCS and SCS-GLB falls. According to this table, the
former experienced abrupt changes in 1923, 1935, and 1988 whereas the latter did not show any shift
during the observational period. Figure 4.12 depicts the details of change detection results in these two
time series. It can be seen from this figure that the time series of the former difference ((HB - SCS) (SCS-CLV)) experienced three change points; however, no change in trend direction occurred for the
observational period. Figure 4.12 also shows that there is no evidence of abrupt changes in the time series
of the latter difference ((HB - SCS) - (SCS-GLB)).
scs−gib
1
scs−clev
1
0.95
0.9
0.9
0.8
0.85
0.7
0.8
0.6
0.75
0.5
0.7
0.4
1900
0.65
1920
1940
1960
Year
1980
2000
2020
1940
1950
1960
1970
1980
1990
2000
2010
Year
Figure 4.12. Change detection in difference between falls
4.4 Analysis of trends in a selected number of segmented time
series
Considering the results obtained from abrupt change detection analysis and after a visual inspection of the
time series, a number of sample data with change in trend direction were selected for a trend analysis
before and after detected change points. The most identified significant changes in trend direction for the
studied variables were observed in a time period spanning from late 1960s to early 1970s. Therefore, a
common trend direction change point in 1972 was considered for the time series with change in trend
direction and a trend analysis was performed on the two separated segments (beginning -1972, 1973 present). The list of selected time series for further trend analysis after partitioning the time series and the
results of trend analysis on the segmented time series are presented in Table 4.16.
Table 4.16. Trend detection results for segmented time series
64
Variable
Buffalo
CLV
ERI_NTS
ERI-MIC_NTS
GLB_CLV
GLB
HB
Lakeport
RNBS_CNBS
SCS
Before common change point
Correlation
0.66
0.67
0.68
0.17
0.29
0.71
0.78
0.69
0.09
0.75
Z
0.74
1.00
0.86
0.75
5.12
0.85
-2.47
2.02
-1.56
0.46
P
0.454
0.314
0.389
0.451
2.91E-07
0.394
0.013
0.042
0.118
0.644
After Common change point
Correlation
0.518
0.502
0.395
0.064
0.051
0.509
0.590
0.624
0.462
0.587
Z
-2.58
-2.61
-2.58
-0.79
-4.14
-2.24
-2.66
-2.02
1.25
-2.63
P
0.009
0.008
0.009
0.426
3.3E-05
0.024
0.007
0.04
0.209
0.008
It can be seen, regardless of statistical significance, that trends are upward for selected time series before
the change point except for Harbour Beach water level and the difference between the residual and
component NBS in Lake Superior. Considering statistical significance at a 5% level, water level in
Harbour Beach experienced a significant downward trend whereas Lakeport water level and the fall
between Gibraltar and Cleveland stations (GLB-CLV) experienced significant upward trends for the first
segment of observations.
Unlike the first segment, the time series in the second segment experienced downward trends in all cases
except the difference between residual and component NBS in Lake Superior which experienced an
insignificant upward trend. Observed downward trends for the second segment of observations were
significant in all cases except the difference between the NTS in Lakes Michigan- Huron and Erie. In
order to have a clearer understanding of the implication of change point detection in a trend analysis, the
results of trend detection for the whole record periods as well as identified segments for the selected
variables are compared in Table 4.17.
Table 4.17. Comparison of trend detection results for the whole and segmented time series
Total record
period
Before common
change point
∆
----------
CLV
∆
----------
GLB
----------
----------
Variable
After Common
change point
Lakeport
----------
∇
∆
SCS
----------
----------
GLB – CLV
----------
∆
∇
∇
∇
∇
∇
∇
∇
RNBS – CNBS
----------
----------
----------
∆
----------
∆
∇
----------
----------
Buffalo
HB
ERI_NTS
ERI-MH (NTS)
∇
65
In this table, the upward small and large triangles represent significant increasing trends at 10 and 5
percent significance levels, respectively, whereas the downward small and large triangles show significant
decreasing trends at 10 and 5 percent significance levels, respectively. Table 4.17 shows that although
trend analysis based on whole record period for the Buffalo and Cleveland stations indicated upward
trends, theses time series are characterized by decreasing trends for the last few decades (after 1972).
While Gibraltar and St-Clair Shore water levels did not show significant trends for the whole record
period and also for the first segment of observations, significant downward trends were observed for the
second segment of sample data in these two stations. Lakeport water levels did not show significant trends
for the whole record period but they exhibit significant upward (downward) trends for the first (second)
segment of observations. The whole records for the fall between Gibraltar and Cleveland (GLB- CLV) did
not show significant trends; however, water levels for the first and the second segments of this fall
experienced significant upward and downward trends, respectively. The difference between the residual
and component NBS of Lake Superior did not show significant trends for the whole or segmented records.
While the NTS time series of Lake Erie showed upward trends (at 10% significance level) for the whole
record period, it showed no significant trend for the first segment but a significant downward trend for the
second segment of observational period.
The exceptional case in this study is the difference between NTS time series of Lakes Michigan – Huron
and Erie. As it can be seen from Table 4.17, the time series experienced a significant upward trend for the
whole observational period; however, trends were not significant when the time series was partitioned
based on observed change point. It can be hypothesized that the non-stationary behaviour identified as
trend by MK test is due to the shift in the time series rather than a monotonic trend. Therefore, when the
sample data is partitioned at detected change point (where the shift occurs) none of separated segments
show significant trends.
66
Chapter 5 Water balance evaluation of
detected trends in the Great Lakes
I. Lake Superior
The results of trend detection for different variables in each lake are summarized and compared in order to
have a general idea of changes in water balance in each lake. Table 5.1 presents the trend detection results
for different variables in Lake Superior in monthly and annual scales. It can be observed from that
although there are some upward and downward trends in water balance components in some periods of the
year, no water balance component shows any significant trend in an annual scale in Lake Superior. It
remains unknown yet why St-Marys discharge increased significantly in the first portion of the year while
water balance input components (PrcLd and Run) in this lake did not experience significant increases.
Table 5.1. Water Balance evaluation of detected trends in Lake Superior (original MK test)5
VAR.
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
DEC
Ann.
NBS
∆
0
0
0
0
0
0
0
0
0
0
∆
PrcLd
0
0
0
0
∆
∆
0
0
0
0
∆
0
0
0
0
0
0
0
∆
0
0
∆
0
0
0
Run
∆
∆
∆
Evp
0
0
0
∆
0
0
0
∆
∆
∆
∆
∆
0
0
0
0
0
0
0
StMarys
StClair
CIS
Water
level
∆
+∆
+∆
-∆
-∆
0
∇
∆
∆
∆
-∆
+
+
∆
+
∆
+
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
∆
∆
∆
0
0
0
0
0
0
0
0
0
-
∆
0
0
II. Lake Michigan-Huron (with Georgian Bay)
Trend detection results in Lake Michigan-Huron (with Georgian Bay) are presented in Table 5.2. This
table shows that in Lake Michigan-Huron (with Georgian Bay) there is a significant increase in annual
precipitation and runoff observations (both act as input components in the lake water balance). However,
St-Clair river discharge which is an output component of water balance for this lake experienced an
upward significant trend as well. It can be concluded that the increase in the outflow compensates for the
increases in runoff and precipitation over this lake and therefore no significant trend in annual water levels
in this lake was observed.
5
For the connecting channels, a positive sign (+) represents an input whereas a negative sign (-) represents an output
in a water balance context.
67
Table 5.2. Water Balance evaluations of detected trends in Lake Michigan-Huron with Georgian Bay
(original MK test)
VAR.
JAN
FEB
MAR
APR
MAY
NBS
0
0
0
0
0
PrcLd
0
0
0
0
∆
0
0
0
Evp
∇
∇
∇
∆
∆
0
0
∇
Run
0
0
0
0
0
StClair
CIS
Water
level
JUL
∆ ∆
∆ +∆ +∆ 0 0
0
-∆
-∆
-∆
-∆
∆ 0 ∇ 0 ∇
∆ ∆ ∆ ∆ ∆
+
Detroit
JUN
∆ ∆
AUG
∆
∆
SEP
OCT
∆ ∆
∆ 0
0
∇
∆ ∆
NOV
DEC
Ann.
∆
0
0
0
0
0
0
0
0
∆
0
0
0
0
0
0
0
0
0
0
0
0
0
0
∇
0
∇ 0
∆ ∆ ∆
0
∆ ∆
∆
-∆
-∆
0
∆
∆
+
0
0
∆
∆
III. Lake St-Clair
Lake St-Clair trend detection results for different variables are compared in Table 5.3 to evaluate changes
in water balance in monthly and annual scales.
Table 5.3. Water Balance evaluation of detected trends in the St-Clair Lake (original MK test)
VAR.
JAN
FEB
MAR
APR
MAY
NBS
0
0
0
0
0
PrcLd
0
0
0
0
∆
0
0
0
Evp
∇
∇
∇
∆
∆
0
0
∇
Run
0
0
0
0
0
StClair
Detroit
CIS
Water
level
∆ +∆ +∆ 0 0
0
-∆
-∆
-∆
-∆
∆ 0 ∇ 0 ∇
∆ ∆ ∆ ∆ ∆
+
JUN
JUL
∆ ∆
∆ ∆
AUG
∆
∆
SEP
OCT
∆ ∆
∆ 0
0
∇
∆ ∆
NOV
DEC
Ann.
∆
0
0
0
0
0
0
0
0
∆
0
0
0
0
0
0
0
0
0
0
0
0
0
0
∇
0
∇ 0
∆ ∆ ∆
0
∆ ∆
∆
∆
∆
-∆
-∆
0
+
0
0
∆
∆
According to Table 5.3, although there are no significant trends in precipitation, runoff, and evaporation in
an annual scale in Lake St-Clair, there are increasing significant trends in St-Clair River and Detroit River
discharges which act, respectively, as input and output components of Lake St-Clair water balance. It
seems, however, that the increase in the St-Clair river discharge (input) is superior to the increase in the
Detroit river discharge and as such the water level experienced a significant upward trend in this lake.
This is the case for monthly water balance of this lake as well. When connecting channels discharge does
not show significant trends, this is the runoff that takes the turn to cause an increase in the water levels in
this lake. Therefore, there is an increase in the water level in all periods of the year for Lake St-Clair.
68
IV. Lake Erie
Lake Erie trend detection results for different variables are summarized in Table 5.4. It can be seen from
this table that there is an increase in water levels in monthly and annual scales in Lake Erie.
Table 5.4. Water Balance evaluation of detected trends in the Erie Lake (original MK test)
VAR.
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEP
OCT
NOV
DEC
Ann.
NBS
0
0
0
0
0
0
0
∆
∆
0
∆
0
∆
PrcLd
0
0
∇
0
∆
0
0
0
∆
0
0
0
0
Evp
0
0
0
∆
0
0
0
0
0
0
∇
0
0
Run
0
0
0
0
0
∆
0
Detroit
+
∆
+
∆
+
∆
+
∆ ∆
0
0
∆
0
∆ ∆
0
0
∆
0
∆
+∆ +∆
0
Niagara
CIS
Water
level
∆
∆
0
∆
∇
∆
∇
∆
∇ 0 ∇ 0 0 0
∆ ∆ ∆ ∆ ∆ ∆
0
∆
∆
∆
0
∆
Similar to the STC, there is an increase in Detroit river discharge in the winter and early spring which acts
as an input in Lake Erie’s water balance. Although Detroit River discharge does not show significant
trends in late spring and in the summer, this is a significant increase in the runoff in the summer that
results in an increase in water levels in this period of the year. In other words, in a water balance context,
annual and monthly increases in water levels in Lake Erie can be attributed to the combination of
increases in runoff and Detroit River discharge (both act as inputs) in different periods of the year.
V. Lake Ontario
Table 5.5 presents the results of trend detection in Lake Ontario. Since some of the water balance
components have not been analyzed in this lake, an evaluation of changes in water balance in this lake is
not fissile.
Table 5.5. Water Balance evaluations of detected trends in the Ontario Lake (original MK test)
VAR.
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
NBS
0
0
0
0
0
∆
0
0
PrcLd
0
0
0
0
0
0
0
0
Evp
0
0
0
0
0
0
0
0
0
Run
0
0
0
0
0
∆
0
0
∆
Ann.
SEP
OCT
NOV
DEC
∆
∆
∆
∆
0
∆
0
0
0
0
0
∆
∆
∆
∆
∆
∆
∆
Niagara
StLawr.
CIS
69
Water
level
∆ ∆
∆
∆
∆
∆ ∆
∆
∆
0
0
0
∆
70
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