### The Hurst Phenomenon in Hydrology

```Aplinkos tyrimai, inžinerija ir vadyba, 2003.Nr.3(25), P.16-20
Environmental research, engineering and management, 2003.No.3(25), P.16-20
ISSN 1392-1649
The Hurst Phenomenon in Hydrology
Gaudenta Sakalauskienė
Institute of Mathematics and Informatics
(received in July, 2003; accepted in October, 2003)
Recent studies have shown that rescaled adjusted range analysis (R/S analysis) and fractional Gaussian
noise (FGN) are useful in characterizing subsurface heterogeneities in addition to geophysical time series.
Although those studies have led to a fairly good understanding of some aspects of R/S analysis and FGN, a
comprehensive introduction to these stochastic, fractal functions is still lacking in the subsurface hydrology
literature. In this paper we look for a possibility to use R/S analysis and fractional Gaussian noise in
prediction of the Nemunas river flow using data on the average annual flow over the period of January 1812 December 2002.
Key words: long-range dependence, R/S analysis, and hydrology.
1.
Introduction
The methods of time series analysis have been
recognized as important tools for assisting in solving
problems related to the management of water
resources.
R/S analysis has it roots in early work of the
British hydrologist H. E. Hurst [1], who investigated
dependence properties of such phenomena as levels of
the river Nile. The Hurst constant H , as the index of
dependence is often called, always lies between 0 and
1, and equals to 0.5 for processes that have
independent increments. Particular interest focuses on
the hypothesis that H > 0.5 indicates relatively longrange dependence. For example, Hurst observed that
H = 0.91 in the case of the Nile data. This indicates a
stretch of dependence that is well beyond what could
increments.
Today the R/S analysis is mostly used for the
hydrological studies, where H value is variously
interpreted either as an indicator of dependence range
or irregularity. Several authors have used R/S analysis
to analyze data on hydrological parameters such as
river flow, precipitation, temperature, etc [2-4]. R/S
analysis was adapted to estimate natural annual flow
for the Logan river for a 70-year period. The measure
of long-term persistence (the Hurst coefficient), was
approximately 0.8 and the lag-one correlation
coefficient (measure of shorter-term persistence), was
0.4 [5]. The R/S analysis of Adibi and Collins [6] has
been used to develop and evaluate long-term forecasts
for the dry and wet period phenomenon.
The most important statistical parameters of
average annual flow in Lithuania were collected and
researched by J. Jablonskis at the observatory of the
Nemunas river near Smalininkai during the period
from 1812 to 1991 [7].
This paper attempts to show that the Hurst
phenomenon is essentially very simple to formulate,
understand and reproduce in transformed time series.
A mathematical formulation based on the relationship
of the process variance with the temporal scale of the
process is offered. In addition, it is demonstrated that
the Hurst phenomenon could be used for analysis of
fluctuations of hydrological processes at different
time scales. This approach differs fundamentally from
more traditional methods currently used for
simulation, where the model is taken to be a relatively
conventional discrete time series such as an
autoregression or moving average, or autoregressive
moving average. In this paper we look for a
possibility to use R/S analysis and fractional Gaussian
noise in prediction of the Nemunas river flow using
data on the average annual flow over the period of
January 1812 – December 2002.
The Hurst Phenomenon in Hydrology
2.
Hurst exponent power function, and is shown in
Equation 5 [11].
Methods
 R* 
E  n  = cn H as n → ∞ .
 σˆ n 
2.1. R/S analysis
Hydrological processes such as rainfall, runoff,
evaporation, etc are often modeled as stationary
stochastic processes in discrete time. Let such a
process be denoted as Z i with i = 1,2,..., denoting
discrete time (e.g. years).
The R/S analysis is the range of partial sums of
deviations of a time series from its mean, rescaled by
its standard deviation.
For an annual time series z1 , z 2 ,..., z n define the
The expected value will be described by a power
function with an exponent of 0.5 if the data set is a
random walk.
 R* 
E  n  = cn 0.5 as n → ∞ .
 σˆ n 
(1)
where
z n is the mean of the n observations in the time series
and s 0* = s n* = 0 .
The adjusted range Rn* is defined as
Rn* = M n* − mn* ,
(6)
With this definition the Hurst exponent of 0.5
will correspond to a time series that is truly random
(Normal distribution, Brown Noise, or Brownian
motion).
When Hurst exponent is within an interval of
0.5 < H ≤ 1 , it describes a dynamically persistent, or
trend reinforcing series. A straight line with non-zero
gradient will have a Hurst exponent of 1.
Alternatively, the series have memory that increases
with H . Importantly, the values at the beginning of
the time series are as important to the dynamics as the
most recently observed.
When Hurst exponent is within an interval of
0 ≤ H < 0.5 , it describes an anti-persistent, or a mean
reverting system. At the limit of zero the time series
must change the direction of every sample as in the
case of white noise.
kth adjusted partial sum s k* by
s k* = s k* −1 + z k − z n ,
(5)
(2)
where
M n* = max(0, s1* , s 2* ,..., s n* ) is the adjusted surplus,
mn* = min(0, s1* , s 2* ,..., s n* ) is the adjusted deficit.
The rescaled adjusted range (RAR) is the given
as
Rn* =
Rn*
,
σˆ n
2.2. The fractional Gaussian noise
(3)
The Fractional Gaussian Noise (FGN) was
developed specifically to account for the Hurst
phenomenon [9]. The connection with Hurst’s law is
the parameter H in FGN, which is often estimated by
the Hurst coefficient H in (4).
To derive FGN first consider a Brownian motion
or Wiener process that is denoted by B (t ) in
continuous time where its increments B (t + u ) − B (t )
are Gaussian with mean zero and variance u and are
independent for non-overlapping time intervals.
Fractional Brownian motion is defined as
where
σˆ n is the sample standard deviation.
By the studies of long term storage requirements
on the Nile river Hurst stimulated in the RAR statistic
in (3). On the basis of a study of 690 annual time
series comprising stream flow, river or lake levels,
precipitation, temperature, pressure and other
geophysical processes, Hurst developed the empirical
relationship
Rn*
t
H
n
=  ,
2
∫
B H (t ) − B H (0) = (t − s ) H −1 / 2 dB( s ) ,
(4)
(7)
0
where
H is the Hurst coefficient. Hurst found H to have
an average value of 0.73 with a standard deviation of
0.09 from the historical records [8-10].
where
t > 0 , 0 < H < 1 and B H (0) is the level of time
t = 0.
Discrete time fractional Gaussian noise (i.e.,
FGN) is then calculated as
The Hurst exponent is estimated by calculating
the average rescaled range over multiple regions of
the data. In statistics, the average (mean) data set Z
is sometimes written as the expected value E [Z ] .
Using this notation, the expected value of R/S, is
calculated over a set of regions converges on the
z t = B H (t + 1) − B H (t ) ,
where
t = 1,2,... .
17
(8)
G. Sakalauskienė
The sample mean and variance of FGN are
consistent estimators of the true mean and variance
and FGN is covariance stationary. The theoretical
autocorrelation function (ACF) of FGN at lag k is
given by
[
1
(k + 1) 2 H − 2k 2 H + (k − 1) 2 H
2
ρk =
]
The data set used in this study consists of
average time series of the Nemunas river flows near
Smalininkai from January 1812 to December 2002.
The data source is the Lithuanian Hydro
meteorological Service at the Ministry of
Environment.
Annual data was used for the empirical study, 5
and 10 years moving average data on the Nemunas
river flow. Fig. 1 shows estimated natural annual flow
as well as the 5 and 10 years moving average flows of
the Nemunas river near Smalininkai for the 190-year
period.
The proposed method for generating R/S
analysis is demonstrated by transforming records with
Hurst exponent equal to those of the historical
standardized flow series at the Nemunas river.
Fig. 2 shows R/S analysis for the Nemunas river
near Smalininkai site. The measure of long-term
persistence (the Hurst coefficient) is approximately
0.67 and the lag-one correlation coefficient (a
measure of shorter-term persistence) is 0.29 (record
length 190 years from 1812 to 2002). Moving average
for the 5 and 10 years time series, Hurst coefficient
are 0.80 and 0.87, lag-one correlation coefficient –
0.51 and 0.63, respectively.
The first step of our empirical analysis was to
test whether the Hurst coefficient of an asset was
significantly different from 0.5 or not. A significant
difference from 0.5 would indicate that Z i did not
follow a Brownian motion. In order to test the null
hypothesis that H = 0.5 against the alternative
H ≠ 0.5 , i.e.
(9)
for 0 < H < 1 and k ≥ 1 . When a theoretical ACF is
the able sum it must satisfy [12]
∞
∑ ρk
<∞.
(10)
k = −∞
The theoretical ACF for FGN in (9) is not the
sum able for 0.5 < H ≤ 1 and therefore FGN is called
a long memory process. The theoretical ACF of
ARMA processes satisfy (10) and are referred to as
short memory processes. When the parameter H lies
in the interval 0 ≤ H < 0.5 , FGN also constitutes a
short memory process.
For many geophysical time series the estimates
for H lie in the range 0.5 < H ≤ 1 . The statistical
effect of past events on present behavior attenuates
very slowly because FGN models possess long
memory for H contained in this interval.
3.
Results and discussion
The Nemunas river is a major source of surface
water for all Lithuanian territory. The Nemunas river
basin accounts for 72% of all Lithuanian surface
water. The total surface of the Nemunas river basin is
96100 km2 (97900 km2 with the river delta). The
Nemunas river near Smalininkai hydrology
monitoring station was established in October 1811.
The monitoring station is 111 km away from the river
mouth and it measures water flow from an area of
81190 km2. So, the Nemunas river flow in this
monitoring station is the best indicator of changes in
Lithuanian hydrology.
m /s
h1 : H ≠ 0.5 ,
we approximated the distribution of Hˆ − H
conditional on the null hypothesis, and calculated the
p-values.
We studied the assets for which the estimated
Hurst coefficient H was significantly different from
0.5. Hence the Nemunas river flow near Smalininkai
time series (both annual and 5 and 10 moving
average) is a long memory process.
Annual value
Average, 10 years
3
900
h0 : H = 0.5
Average, 5 years
800
700
600
500
400
1999
1988
1977
1966
1955
1944
1933
1922
1911
1900
1889
1878
1867
1856
1845
1834
1823
1812
300
Year
Fig. 1.
The Nemunas river annual, 5 and 10 years moving average flow time series in 1812 - 2002
18
The Hurst Phenomenon in Hydrology
H=0.67
H=0.80
H=0.85
2000
1500
1000
500
Fig. 2.
1999
1988
1977
1966
1955
1944
1933
1922
1911
1900
1889
1878
1867
1856
1845
1834
1823
-500
1812
0
Year
Plot of a transformed time series generated using the statistics of standardized discharge at the
*
Nemunas river ( s k )
6.
As a result of the normalization process, every
realization curve contains at least on minimum or
one maximum global point (i.e. it has at least one
half cycle if a full cycle incorporates wet-dry-wetdry periods). It was possible to analyze the time
series of the Nemunas river flow near Smalininkai
by employing R/S analysis and predict that
durations of wet and dry periods are 27 and 5 years
respectively.
7.
8.
9.
10.
4.
References
1.
Hurst H. E. Long – term storage capacity of
reservoirs. // Transactions of the American Society
of Civil Engineering Nr. 116. 1951, P. 770-799.
Bellin A., Pannone M., Fiori A., Rinaldo A. On
transport in porous formations characterized by
heterogeneity of evolving scales. // Water Resource
Research. Vol 32. No. 12 3485.
Koutsoyiannis, D. The Hurst phenomenon and
fractional Gaussian noise made easy. //
Hydrological Sciences Journal. Nr. 47(4) 2002, P.
573-595.
Koutsoyiannis, D. Climate change, the Hurst
phenomenon, and hydrological statistics. //
Hydrological Sciences Journal. Nr. 48(1) 2003, P.
3-24.
Burgers Stephen J. Some aspects of Hydrologic
variability. // Managing water Resources in the
West Under Conditions of climate uncertainly.
1991, P. 275-280.
2.
3.
4.
5.
11.
12.
Adibi M. and Collins M. A. Anticipatory
conservation based drought management of water
supply
reservoirs.
P.
7.
http://www.eng.warwick.ac.uk.
Jablonskis J. Run-off of the Nemunas during the
180 years. // Power engineering. Nr. 4. ISSN 02357208. Vilnius: Academia, 1994. P. 19-32 (in
Lithuanian).
Kottegoda N. T. Stochastic water resources
technology. Halsted Press, New, 1980. P. 11-47.
Anderson O. D. Time series. 1980, P. 73-103.
Hall P. and et. all. Semiparametric bootstrap
approach to hypothesis tests and confidence
intervals for the Hurst coefficients. P. 22.
http://netec.mcc.ac.uk/WoPEc/data/Papers//wophu
mbsf1999-62.html.
Bras R.L. and Rodriguez-Iturbe I. Random function
and hydrology, Addison-Wesley, USA. 1985, P.
221.
Billinger D. R. Time series data analysis and
theory. 1975, P. 10-35.
Dr. Gaudenta Sakalauskienė, researcher at the
Institute of Mathematics and Informatics.
Research interests: water pollution, water quality
modeling, and time series analysis.
LT-2005 Vilnius, Lithuania
Tel.:
+370 5 2722554
Fax.:
+370 5 2729209
E - mail:
[email protected]
19
G. Sakalauskienė
Hursto reiškinys hidrologijoje
Gaudenta Sakalauskienė
Matematikos ir informatikos institutas
(gauta 2003 m. liepos mėn.; atiduota spaudai 2003 m. spalio mėn.)
Straipsnyje išsamiai išdėstyta R/S statistikos ir fraktalinio Brauno judėjimo teorija bei jų
pritaikymo galimybės hidrologijoje. Nemuno upės hidrologinėje stotyje ties Smalininkais
nepertraukiamai atliekami vandens lygio, vandens nuotėkio stebėjimai, matavimai ir kaupiami
duomenys nuo 1812 m. Gautų matavimų duomenys sudaro ilgalaikę laiko eilutę. Sukauptų
duomenų eilutės yra unikalios ir istorine, ir moksline prasme. Turimoms metinėms vandens
nuotėkio laiko eilutėms nuo 1812 iki 2002 metų apdoroti buvo panaudota procesų teorija, t. y. R/S
statistikos ir fraktalinio Brauno judėjimo teorija. Pritaikę minėtą procesų teoriją Nemuno ties
Smalininkais metiniams bei 5 ir 10 slankių vidurkių nuotėkio laiko eilutėms gauta, kad visos trys
nagrinėtos laiko eilutės atspindi ilgalaikę priklausomybę visam nagrinėtam laikotarpiui (Hursto
koeficientas: 0,67, 0,8 ir 0,85, atitinkamai su tikimybe 0,05). Be to, šią procesų teoriją labai
sėkmingai galima taikyti hidrologinėms laiko eilutėms, norint įvertinti sausringus ir labai drėgnus