Journal of Hydrology 237 (2000) 74–85
www.elsevier.com/locate/jhydrol
Simple equations to represent the volume–area–depth relations of
shallow wetlands in small topographic depressions
M. Hayashi a,*, G. van der Kamp b
a
Department of Geology and Geophysics, University of Calgary, Calgary, Alberta, Canada T2N 1N4
b
National Water Research Institute, Saskatoon, Saskatchewan, Canada S7N 3H5
Received 28 September 1999; revised 5 April 2000; accepted 14 July 2000
Abstract
Small topographic depressions have important functions in hydrology and ecology because they store water in the form of
shallow lakes, wetlands or ephemeral ponds. The relations between the area A, the volume V, and the depth h of water in
depressions are important for evaluating water and dissolved-mass balances of the system. The A–h and V–h relations are
usually determined from fine-resolution elevation maps based on detailed survey data. Simple equations are presented in this
paper, which can be used to: (1) interpolate A–h and V–h data points obtained by a detailed survey; (2) approximate unknown
A–h and V–h relations of a depression from a minimal set of field data without a time-consuming elevation survey; and (3) serve
as a geometric model of depressions in simulation studies. The equations are simple power functions having two constants. The
first constant s is related to the size of the depression, and the second constant p is related to the geometry of the depression. The
power functions adequately represent A–h and V–h relations of all 27 wetlands and ephemeral ponds examined in this paper,
which are situated in the northern prairie region of North America. Assuming that the power functions are applicable for other
similar topographic depressions, an observer only needs to measure A and h twice to determine the two constants in the
equation. The equations will be useful in field studies requiring approximate A–h and V–h relations and in theoretical and
modeling studies. 䉷 2000 Elsevier Science B.V. All rights reserved.
Keywords: Wetlands; Ephemeral lakes; Ponds; Water balance; Bathymetry; Water storage
1. Introduction
Topographic depressions that hold water in the
form of small lakes, wetlands or ephemeral ponds
have important hydrological and ecological functions.
They store snowmelt and storm water to attenuate
flood peaks, and provide habitats for birds and animals
that are dependent on aquatic plants and invertebrates.
To study these functions hydrologists need to evaluate
* Corresponding author. Tel: ⫹1-403-220-2794; fax: ⫹1-403284-0074.
E-mail address: [email protected] (M. Hayashi).
water balance and dissolved-mass balance in the
depressions. For example, after a runoff event the
flux of water and nutrients into a wetland can be estimated from the change of water volume in the wetland
and the change of concentration of dissolved species.
A practical approach for determining water volume
V and area A is to measure the depth of water (h) and
estimate A and V from predetermined area–depth (A–
h) and volume–depth (V–h) relations. These relations
are specific to each depression, and are usually
derived from a detailed bathymetry map. Because of
this site-specific nature, most hydrological research
articles report A–h and V–h relations merely as a
0022-1694/00/$ - see front matter 䉷 2000 Elsevier Science B.V. All rights reserved.
PII: S0022-1694(00 )00 300-0
M. Hayashi, G. van der Kamp / Journal of Hydrology 237 (2000) 74–85
75
and one can be derived from another. For example,
suppose that the water level in a lake rises by a small
amount Dh. The resulting volume change DV in the
lake is equal to ADh. Therefore, V at any h is given by
V…h† ˆ
Fig. 1. Slope profile of symmetric basins with y=y0 ˆ …r=r0 †p : For
example, p ˆ 2 indicates a parabolic slope.
tool for mass balance calculations, and rarely treat
them as a significant topic of study. However, it is
useful to look for common factors in the A–h and
V–h relations for a variety of topographic depressions
because this can lead to a more general understanding
of their shapes and storage characteristics.
Generalized forms of V–h and A–h relations have
been used by some investigators in the mathematical
modeling of lakes. For example, Gates and Diessendorf (1977) assumed V is proportional to A to model
lake level fluctuation in response to stochastic forcing,
Bengtsson and Malm (1997) assumed A is proportional to h 2 to study the sensitivity of lake level to
climatic condition, and O’Connor (1989) assumed V
is proportional to h m and A is proportional to hm⫺1 to
simulate the variation of dissolved solids in lakes and
reservoirs.
The scope of this paper is limited to ephemeral
ponds and wetlands in small natural depressions, for
example, “pothole” wetlands in glaciated plains.
Simple power functions with two parameters will be
proposed to represent A–h and V–h relations, and
tested using data for 27 shallow lakes, wetlands and
ponds in the northern prairie region of North America.
The power functions presented in this study are
expected to be widely applicable to small wetlands
and ponds in isolated and smoothly sloped depressions, even though they have only been tested in the
northern prairie environment.
2. Theory
A–h and V–h relations are dependent on each other,
Zh
0
A…h† dh
…1†
where h is a dummy variable of integration and h is
the depth measured at the deepest point of the lake.
This relationship between V and A is fundamental and
applies to all lakes and wetlands that have a horizontal
water surface. Experimentally determined V–h and
A–h relations must satisfy Eq. (1), a failure of
which indicates that the V–h and A–h relations are
inconsistent.
The A–h and V–h relations of a basin can often be
approximated by simple analytical expressions such
as polynomials or power functions. In this paper
power functions are proposed that are based on the
shape of simple symmetric basins formed by rotating
a slope profile around the central axis (Fig. 1). The
slope profiles are given by
y=y0 ˆ …r=r0 †p
…2†
where y [L] is the relative elevation of the land surface
at a distance r [L] from the center, y0 [L] is the unit
elevation, for example 1 m in SI units, r0 is the radius
corresponding to y0, and p is a dimensionless constant.
It follows from Eq. (2) that the area of the water
surface corresponding to a depth of water h measured
at the center of the basin …r ˆ 0† is given by
Aˆ
pr02
h
h0
2=p
2=p
h
ˆs
h0
…3†
where h0 [L] is the unit depth, s [L 2] is a scaling
constant, which is equal to the area of water surface
when h ˆ h0 : The constant p provides the link
between the shape of the basin (Fig. 1) and A–h relation. A small value, for example p ˆ 2; corresponds to
a paraboloid basin that has smooth slopes extending
from the center to the edge, and a large value corresponds to a basin that has a flat bottom. In an extreme
case, we can set p ! ∞: This corresponds to a cylinder, for which A ˆ s regardless of h. It follows from
Eq. (1) that the volume of water corresponding to h is
76
M. Hayashi, G. van der Kamp / Journal of Hydrology 237 (2000) 74–85
Fig. 2. Location of the present and previous study sites. (1) St.
Denis; (2) Fort Qu’Appelle; (3) Melfort; (4) Saskatoon; (5) Swift
Current; (6) Wilkie; (7) Ward; (8) Dickey; (9) Stutsman. Shaded
area indicates the extent of the prairie wetland region (Winter,
1989).
given by
Vˆ
s
h1⫹…2=p†
…1 ⫹ 2=p† h2=p
0
…4†
Unlike the hypothetical basins represented in Fig. 1,
real wetlands have more complex, asymmetric shapes,
and they commonly occur in the lowest part of catchments where slopes are concave. With suitable p and
r0, Eq. (2) represents the concave portion of most
slope profiles reasonably well. A natural depression
is made up by many slope profiles each having different values of p and r0, and no single slope profile
represents the entire depression. Therefore, one
might expect that Eqs. (3) and (4) cannot adequately
represent natural depressions. However, as will be
shown, the field data suggest that A–h relations of
all natural depressions examined in this study are
well approximated by Eq. (3). In this case p and s in
Eq. (3) represent the shape and the size of depressions
in some average sense.
3. Field sites and methods
3.1. Present study
The present field study was conducted in the St.
Denis National Wildlife Area (NWA) in
Saskatchewan, Canada as part of a multidisciplinary
research project to understand the hydrology and the
ecology of prairie wetlands. The site is located at
106⬚06 0 W and 52⬚02 0 N, which is approximately
40 km east of Saskatoon (Fig. 2). The topography of
the site is described as moderately rolling knob and
kettle moraine with slopes varying from 10 to 15%
(Miller et al., 1985). The area is underlain by glacial
tills, which have a high clay content of 20–30%. The
mean annual precipitation in Saskatoon is 360 mm, of
which 84 mm occurs as snow (Atmospheric Environment Service, 1997). Air temperature frequently
becomes lower than ⫺30⬚C in winter during which
the soil frost penetrates as deep as 2 m. The uplands
around the wetlands have been under cultivation for
50–100 years.
Within the St. Denis NWA, four wetlands identified
as S92, S109, S120, and S125S were selected for
detailed elevation surveys. The extent of the wetlands
is loosely defined by the growth of aquatic vegetation
such as sedge and spike rush and by the presence of
soft organic-rich soil, but the water-covered areas of
the wetlands drastically change during a year. The
wetlands become entirely or partially inundated in
spring after snowmelt runoff. Runoff rarely occurs in
summer and water levels in the wetlands gradually
decline (Hayashi et al., 1998). Topographical maps
of the four wetlands are shown in Fig. 3. Surveyed
areas did not completely cover the catchments of S92,
S109, and S125S, and part of drainage divides are
drawn along the limit of the surveyed area. The missing area is small compared to the areas included in the
maps.
In addition to the four wetlands, a detailed elevation
survey was conducted for four small depressions on
the cultivated uplands. These small depressions hold
ephemeral ponds only for a week to a few weeks in
early spring, and are not considered wetlands in usual
sense. However, they are hydrologically important
because they store snowmelt water and recharge
local groundwater. Three such depressions (D1, D2,
and D3 in Fig. 3b) are located in and adjacent to the
catchment of S109. The fourth one (S104) is located
200 m northeast of S109.
The catchments of the wetlands and depressions
were surveyed in 1994, 1998, and 1999 using total
stations. For the wetlands, survey points were spaced
horizontally at 10–15 m intervals in the uplands, and
M. Hayashi, G. van der Kamp / Journal of Hydrology 237 (2000) 74–85
77
Fig. 3. Topographical maps showing the elevation above the mean sea level of the four catchments in the St. Denis NWA. Principal contour
interval is 1 m. Scale bars indicates 50 m. Wetlands are indicated by shades and drainage divides are indicated by thick lines. The location of
small depressions D1, D2, and D3 are indicated in (b).
5–10 m intervals in the wetlands. For the small
depressions, survey points were spaced horizontally
at 2–5 m intervals. Estimated measurement error is
within a few centimeters for elevation and within a
few tens of centimeters for horizontal location. The
software package Surfer (Golden Software, Golden,
CO, USA) was used to estimate the elevation on regularly spaced grids by interpolation and to construct
digital elevation models (DEMs). The maps shown
in Fig. 3 are based on the DEMs. From the DEMs,
V–h and A–h relations were calculated using the
volume and area integration tool of Surfer. The
depth of water h is defined as the elevation difference
between the water surface and the lowest point in the
depression, which means that h ˆ 0 when the wetland
becomes completely dry. The kriging method (Davis,
78
h (m)
A (m 2)
V (m 3)
A (m 2)
V (m 3)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
S92
170
520
750
970
1180
1380
1590
1810
2050
2350
2730
3200
5.2
41
100
190
300
430
570
740
940
1160
1410
1700
S109
190
500
830
1130
1410
1690
1970
2250
2630
3120
3660
4120
6.5
41
110
210
330
490
670
880
1120
1410
1750
2140
A (m 2)
V (m 3)
S120
520
900
1190
1430
1660
1880
2100
2300
2570
2830
3150
21
93
200
330
480
660
860
1080
1330
1600
1890
A (m 2)
V (m 3)
S125S
360
790
1210
1610
1990
2360
2740
3120
3510
3850
17
74
170
320
500
710
970
1260
1590
1970
A (m 2)
V (m 3)
S104
180
340
500
660
820
1010
1210
9.5
36
78
140
210
300
410
A (m 2)
V (m 3)
D1
150
360
610
7.2
32
81
A (m 2)
V (m 3)
D2
63
130
210
310
460
3.2
13
30
56
94
A (m 2)
V (m 3)
D3
88
160
270
370
470
600
760
4.6
18
40
72
110
170
230
M. Hayashi, G. van der Kamp / Journal of Hydrology 237 (2000) 74–85
Table 1
Area A and volume V of water corresponding to depth h in the wetlands and small depressions in the St. Denis NWA
M. Hayashi, G. van der Kamp / Journal of Hydrology 237 (2000) 74–85
79
Table 2
Upper depth limit hmax, scaling constant s, power constant p, root-mean-squared error of area Aerr and volume Verr of the wetlands, the relative
magnitude of Aerr with respect to A1m, and Verr with respect to V1m. Asterisks indicate that the relative magnitudes of Aerr and Verr are evaluated
against A and V at hmax
s (m 2)
p
Aerr (m 2)
hmax (m)
St. Denis
S92
S109
S120
S125S
S104
D1
D2
D3
1.2
1.2
1.1
1.0
0.7
0.3
0.5
0.7
2450
3180
2820
3840
1720
2880
1160
1130
1.80
1.61
2.66
2.10
1.95
1.55
1.45
1.66
Shjeflo (1968)
Pothole 1
Pothole 2
Pothole 4
Pothole 5
Pothole 5A
Pothole 6
Pothole 7
Pothole 8
Pothole C1
1.4
1.7
1.5
2.3
1.6
1.5
1.5
1.1
1.9
51 900
109 700
93 400
78 200
9600
33 300
86 800
123 100
162 900
5.12
3.64
4.31
5.48
2.49
6.19
3.52
3.33
5.33
1360
8110
4610
1390
239
200
1820
6260
4540
2.6
6.9
5.2
1.7
2.6
0.6
2.1
5.3
2.8
685
2010
814
961
100
359
801
1120
3870
1.7
2.4
1.5
1.5
2.1
1.5
1.7
1.3
3.4
Lakshman (1971)
Ft. Qu’Appelle 1
Ft. Qu’Appelle 2
Ft. Qu’Appelle 17
Ft. Qu’Appelle 19
Ft. Qu’Appelle 20
Melfort 7
Saskatoon 16
Swift Current 1
Wilkie 6
Wilkie 12
1.4
1.5
1.5
1.2
0.9
1.1
0.9
2.0
0.6
1.2
7160
8570
5790
6720
4310
8990
1960
34 100
11 800
2150
3.22
3.00
2.79
3.75
3.11
2.44
4.06
3.26
4.72
3.28
57
342
83
142
98
392
31
2610
242
46
0.8
4.7
1.6
1.5
1.4 ⴱ
4.1
1.7 ⴱ
7.4
2.6 ⴱ
2.1
101
125
37
90
54
236
44
1240
118
13
2.0
3.1
1.4
1.6
1.4 ⴱ
4.3
5.1 ⴱ
7.1
2.9 ⴱ
0.9
1986, p. 239) was used for interpolation. Preliminary
analysis showed that the calculated V–h and A–h relations were essentially independent of the choice of
semivariogram and grid spacing. The relations
presented in this paper were calculated using linear
semivariograms with no drift and 5 m grids for the
wetlands and 1 m grids for the small depressions.
3.2. Previous studies
Two sets of published data are used in this paper;
the first data set from North Dakota, USA (Shjeflo,
1968) and the second data set from Saskatchewan,
Canada (Lakshman, 1971). The Shjeflo wetlands are
located in three locations; Potholes 1, 2, and 4 in Ward
97
101
55
20
17
2.4
11
15
Aerr/A1m (%)
Verr (m 3)
Wetland ID
4.1
3.2
1.9
0.5
1.4 ⴱ
0.4 ⴱ
2.5 ⴱ
2.0 ⴱ
17
20
8.6
5.3
2.6
0.9
1.6
2.6
Verr/V1m (%)
1.5
1.4
0.5
0.3
0.6 ⴱ
1.1 ⴱ
1.7 ⴱ
1.1 ⴱ
County, Potholes 5, 5A, 6, 7, and 8 in Dickey County,
and Pothole C1 in Stutsman County (Fig. 2). The
Lakshman wetlands are located near Fort Qu’Appelle,
Melfort, Saskatoon, Swift Current and Wilkie (Fig. 2).
Survey methods and the density of surveyed points
were not clearly described in the original articles,
which only included tables of the elevation of water
surface with respect to an arbitrary datum and the area
and the volume corresponding to each elevation. Shjeflo (1968) provided the elevation of the lowest point in
each basin so that we could calculate the depth of
water corresponding to each elevation and determine
A–h and V–h relations. Lakshman (1971) did not
provide such data, and we needed to estimate the
lowest elevation by inspecting the water level record.
80
M. Hayashi, G. van der Kamp / Journal of Hydrology 237 (2000) 74–85
Fig. 4. A–h relation of S92 and S109. Solid circles indicate data points and curves indicate the power function (Eq. (3)) with the values of s and
p listed in Table 2.
The estimation was only possible for those wetlands
that dried up frequently in the reported study period of
1964–1970. In addition the elevation-area-volume
data were incomplete for some wetlands. Therefore,
out of 25 wetlands in Lakshman (1971), V–h and A–h
relations were determined for only 15 of them. The
accuracy of estimating h is expected to be in the order
of 0.05 m.
4. Results
4.1. St. Denis wetlands
The A–h and V–h relations of the wetlands and
small depressions are listed in Table 1. The data are
listed for a depth range between 0 and hmax. The hmax is
defined by the overflow point for S120, S125 and all
micro-depressions, and by the highest water level
recorded in 1968–1997 for S92 and S109 (van der
Kamp et al., 1999). The values of hmax are listed in
Table 2.
Fig. 4 shows the A–h relation of S92, which represents an irregularly-shaped end member, and for
S109, which represents a reasonably regularly-shaped
end member. Solid circles indicate data points calculated from the DEM and curves show Eq. (3) with the
best-fit values of s and p determined by the leastsquares method. Table 2 lists the values of s and p.
The power function (Eq. (3)) approximates the A–h
relation of all wetlands reasonably well. To evaluate
the goodness of fit between the data points and the
power function, root-mean-squared (RMS) error Aerr
is defined by
v
u X
u1 m
…A
⫺ APF †2
…5†
Aerr ˆ t
m iˆ1 DEM
Fig. 5. V–h relation of S92 and S109. Solid circles indicate data points and curves indicate the power function (Eq. (4)) with the values of s and
p listed in Table 2.
M. Hayashi, G. van der Kamp / Journal of Hydrology 237 (2000) 74–85
Fig. 6. Relationship between s and p.
where ADEM is the area calculated from DEM, and APF
is the area given by the power function, and m is the
number of data points. The magnitude of Aerr is
comparable to the values of A in a range 0 ⬍ h ⱕ
0:1 m; and APF does not give a meaningful estimate
in this depth range. The relative magnitude of Aerr
81
becomes less significant at greater depth range as A
increases. The ratio Aerr =A is generally smaller than
10% in a range 0:3 m ⬍ h ⱕ hmax : Table 2 lists Aerr as
well as Aerr =A1m ; where A1m is the data point closest to
h ˆ 1 m:
Once s and p are determined by fitting Eq. (3) to A–
h data points, the same s and p can be used in Eq. (4)
to approximate V–h relation. Fig. 5 shows the V–h
relation of S92 and S109, in which solid circles indicate data points calculated from DEM and curves
show Eq. (4). The goodness of fit between the data
points and the power function (Eq. (4)) is expressed
by RMS error, Verr, defined similarly to Eq. (5). The
relative magnitude of Verr with respect to V is generally smaller than 10% in a range 0:3 m ⬍ h ⱕ hmax :
Table 2 lists Verr and Verr =V1m ; where V1m is the data
point closest to h ˆ 1 m:
Eqs. (3) and (4) were similarly applied to the A–h
and V–h relations of wetlands S120 and S125, and
small depressions D1, D2, D3, and S104. The leastsquares-fit values of s and p, and RMS errors Aerr and
Verr are listed in Table 2. The match between the
power functions and the data points for the wetlands
and small depressions is similar to that S92 and S109.
The ratio Aerr =A and Verr =V are generally smaller than
10% in a depth range 0:3 m ⬍ h ⱕ hmax for wetlands
and 0:1 m ⬍ h ⱕ hmax for small depressions.
The relationship between s and p is shown in Fig. 6.
St. Denis wetlands and depressions have relatively
small sizes, which is reflected in the range of the
scaling constant s. The values of p fall in a relatively
narrow region around 2, which indicates that the
depressions have a reasonably smooth shape that
resembles a paraboloid (Fig. 1).
4.2. Previously studied wetlands
Fig. 7. A–h and V–h relations of Pothole 4 (Shjeflo, 1968). Solid
circles indicate data points and curves indicate the power functions
with values of s and p listed in Table 2.
A–h and V–h data were available for 10 wetlands in
Shjeflo (1968) and 15 wetlands in Lakshman (1971).
The data were examined with respect to the consistency condition given by Eq. (1). At each reported
value of h, a value of V was calculated from A–h
relation using Eq. (1), and compared to the reported
value of V. If the relative difference was greater than
15% at any h greater than 0.2 m, the data set was
considered inconsistent. The source of inconsistency
may be inappropriate methods used to calculate A and
V from the survey data. Nine wetlands in Shjeflo
82
M. Hayashi, G. van der Kamp / Journal of Hydrology 237 (2000) 74–85
(1968) and ten in Lakshman (1971) satisfied the
condition, and will be used in the following analysis.
The upper depth limit hmax, listed in Table 2, is arbitrarily set to be the highest water level recorded in the
study period (1960–1964 in Shjeflo (1968) and 1964–
1970 in Lakshman (1971)) plus 0.5 m. If A–h and V–h
data do not cover hmax, the highest value of h within
the data set is considered to be hmax.
Eq. (3) was fitted to the A–h relation of the nine
Shjeflo wetlands to determine s and p (Table 2). The
least-squares method was applied to each data set
within a range 0:1 m ⱕ h ⬍ hmax : The data points
having h less than 0.1 m were excluded from the
analysis because the accuracy of h at such a small
value is questionable. As an example Fig. 7a shows
the A–h relation of a wetland identified as Pothole 4,
and Fig. 7b shows the V–h relation of the same
wetland. Solid circles indicate data points and curves
show the power functions (Eqs. (3) and (4)). The
power functions agree with the data points reasonably
well for both A–h and V–h relations. Similarly the
power functions adequately represent A–h and V–h
relations of all other Shjeflo wetlands. The RMS
errors Aerr and Verr and the relative magnitudes
Aerr =A1m and Verr =V1m are listed in Table 2. For all
nine Shjeflo wetlands, Aerr =A and Verr =V are generally
smaller than 10% in a range 0:3 m ⬍ h ⱕ hmax :
The power function (Eq. (3)) was also applied to the
Lakshman wetlands to determine the least-squares-fit
values of s and p (Table 2). The agreement between
data points and the power function is reasonably good
for all ten wetlands. The RMS errors and their relative
magnitude with respect to A1m and V1m are listed in
Table 2. For all ten Lakshman wetlands, Aerr =A and
Verr =V are generally smaller than 10% in a range
0:3 m ⬍ h ⱕ hmax :
The relationship between s and p is shown in Fig. 6.
The positive correlation indicates that Shjeflo and
Lakshman wetlands, which are much larger in size
than St. Denis wetlands, have higher values of p.
The implication of the size-shape relationship will
be discussed later.
5. Simple field methods for determining
approximate A–h and V–h relations
In general, accurate determination of A–h and V–h
relations requires a detailed elevation survey over an
entire wetland, which is labor intensive and time
consuming. Based on the above results, it is likely
that Eqs. (3) and (4) approximately represent A–h
and V–h relations of most small topographic depressions. Therefore, at least for the first approximation,
one only needs to measure A and h at a few different
times to determine s and p.
For a given wetland, let z be the elevation of the
water surface with respect to an arbitrary datum, for
example a staff gauge, and zmin be the elevation of the
lowest point in the wetland. Eq. (3) can be written as
A ˆ s‰…z ⫺ zmin †=h0 Š2=p
…6†
In principle, three unknown constants s, p, and zmin can
be
determined
from
three
independent
measurements of A and z. However, it is easy
to measure zmin for most wetlands in small
depressions. For example, an observer can locate
the lowest point in the wetland just before it
becomes completely dry, or an observer on a
boat can probe around the central part of the
wetland to find the deepest point. Therefore, in
most cases only s and p need to be determined
from two independent measurements of A and z.
There are a number of ground-based and airborne
methods to estimate A of many wetlands relatively
easily.
If time and resources are limited, an observer may
chose to estimate p from the size–shape relationship
shown in Fig. 6. For example, Fig. 6 indicates that p is
likely close to 2 in small seasonal wetlands and
ephemeral ponds. In this case, an observer may
assume p ˆ 2 and determine s from a single measurement of A and h. The accuracy of A–h and V–h relations determined this way may not be high, but some
field studies can benefit from the simplicity and the
practicality of the method.
We used this method in the St. Denis NWA in
1998 to estimate the volume of snowmelt runoff
collected in several depressions that did not have
detailed survey data. This measurement showed
that significant portion of snowmelt runoff is
stored in depressions without draining to the
main wetland in the catchment, and gave us an
important step forward in understanding the
hydrology of prairie wetlands.
M. Hayashi, G. van der Kamp / Journal of Hydrology 237 (2000) 74–85
83
Fig. 8. Hypsometric curves of the catchments of S92 and S109.
Shaded regions indicate wetlands.
6. Discussion
6.1. Hypsometry: statistical meaning of the shape
parameter p
The field data indicate that the A–h relation of
many wetlands is represented by a well-defined
value of p that reflects the basin shape in some average sense. This finding is not trivial and warrants
some discussion. To this end it is informative to reexamine how the A–h relation is determined from
detailed survey data.
The A–h relation of a basin is determined from a
high-resolution DEM of the basin. On the DEM, the
basin is hypothetically filled with water to a series of
values of h, and for each h the grid points that are
under the water surface are counted. If there are n
grid points under the water surface located at h, then
the area of the water surface A is given by
A…h† ˆ n…h†Da
…7†
where Da is the area associated with each grid point.
The above procedure involves “counting” the number
of grid points, which suggests that A can also be
expressed in terms of the frequency distribution of
grid points within the basin. The cumulative
frequency distribution F…h† is defined as
F…h† ˆ n…h†=N
…8†
where N is the total number of grid points in the basin.
Since the total area of the basin Atot is given by
Atot ˆ NDa
…9†
Fig. 9. North–south (NS) and east–west (EW) slope profiles of
Pothole 4 and S109. Open squares indicate EW profiles and solid
circles indicate NS profiles. Curves show the best fit power functions having power p. (a) Pothole 4. (b) S109.
\tflt="PS6F00" \tfnm="PS6F00" A…h† and F…h† are
related by
A…h† ˆ Atot F…h†
…10†
Eq. (10) clearly demonstrates the equivalence
between A–h relation and frequency distribution.
Similar ideas have been used to study the distribution of landmass within a drainage basin by geomorphologists, who use the term hypsometric analysis for
studying how F…h† changes in relation to landform
evolution (Strahler, 1952; Willgoose and Hancock,
1998). The F…h† of the entire catchments of S92 and
S109 are shown in Fig. 8. The graphs, called hypsometric curves, have F…h† on the horizontal axis and h
on the vertical axis following the geomorphological
convention (Strahler, 1952). The values of A corresponding to F are also plotted on the top horizontal
axis. Each wetland occupies only a small portion of
the catchment, which is indicated by a shaded region
at the toe of the hypsometric curves. The wetland
84
M. Hayashi, G. van der Kamp / Journal of Hydrology 237 (2000) 74–85
portion of the hypsometric curves have relatively
simple shape that is adequately represented by the
power function (Eq. (3)) even though the overall
shape of the hypsometric curves is more complex.
It is clear from the above discussion that the A–h
relation of a depression should be regarded as the
frequency distribution of land elevation within the
depression. Therefore, two given depressions may
have an identical A–h relation even though their
actual shape is significantly different. The shape parameter p in Eq. (3) represents the slope profile of a
hypothetical basin (Fig. 1) that is hypsometrically
equivalent to the actual depression.
6.2. Size–shape relationship and landform evolution
Fig. 6 shows that larger wetlands tend to have
higher values of p. In general large prairie wetlands
have water for a long period of time in a given year, in
fact many of them are semi-permanent lakes, while
small prairie wetlands have water only for a few
months after snowmelt. As a result, large wetlands
have a flat bottom formed by sedimentation. For
example Fig. 9a shows north–south (solid circles)
and east–west (open squares) profiles of Pothole 4
in Shjeflo (1968) and Fig. 9b shows north–south and
east–west profiles of S109 in St. Denis. Markers in
Fig. 9 show the data points and lines show Eq. (2)
using the least-squares-fit values of p and r0. As indicated in Fig. 9, the profiles in Pothole 4 have large p
reflecting the flat bottom, while those in S109 have
small p reflecting relatively smooth slope from the
center to the edge.
In Fig. 9 p takes a range of values because the
wetlands are asymmetric and each individual slope
has different curvature. The A–h relation integrates
all slope profiles within the wetland and defines a
single value of p in an average sense. For example,
a representative p of Pothole 4 (Fig. 9a) may be given
by a harmonic average pav of the north, east, south,
and west profiles; pav ˆ 4:85: This is comparable to
p ˆ 4:31; which was obtained by fitting Eq. (3) to the
A–h data set. Similarly, a harmonic average of the
four profiles of S109 (Fig. 9b) is pav ˆ 1:78; which
is comparable to p ˆ 1:61; which was obtained by
fitting Eq. (3) to the A–h data set.
The wetlands examined in this paper occur in the
northern prairie region that is characterized by semi-
arid climate having warm summer and cold winter,
and by thick glacial tills and hummocky topography.
Many prairie wetlands become partially or completely
dry in late summer and fall, and the land surface is
subjected to gravity-driven soil creep induced by
freezing and thawing, drying and wetting (Kirkby,
1967), cultivation (de Jong et al., 1998), and animal
burrow activities (Black and Montgomery, 1991).
Geomorphological literatures suggest that soil creep
tends to dissipates the irregularity of the landform by
diffusion-like processes (Culling, 1960). It is reasonable to expect that the dissipation of irregularity at a
local scale results in the smooth frequency distribution of land elevation at a basin-wide scale, and hence
to simple A–h and V–h relations. If the landform
evolution is solely driven by soil creep, slope profiles
are expected to approach curves having a low value of
p (Fig. 1). In contrast, if the landform evolution is
strongly influenced by other mechanisms like underwater sedimentation, slope profiles may take a higher
value of p that reflects the balance between several
driving forces.
It is not clear if similar smoothing mechanisms
exist in different environments, for example deep
lakes on a rocky terrain or playas on tropical savanna.
Therefore, the applicability of Eqs. (3) and (4) is so far
limited to the prairie region where the equations have
been tested. However, it will be interesting to examine
whether the equations are useful for the lakes and
wetlands in different environments.
7. Conclusions
The main objective of this paper is to examine the
relation between the volume V, area A, and depth h of
wetlands in isolated depressions. For the 27 wetlands
and ephemeral ponds examined in this paper, A is
proportional to h2=p to a good approximation, and V
is proportional to h1⫹2=p ; where p is a dimensionless
constant. In other words, A–h and V–h relations are
expressed as power functions. The constant p is
related to the shape of the depression, more specifically the functional form of slope profiles. For example, a paraboloid-shaped depression has p ˆ 2; and a
cylinder-shaped depression has p ! ∞: Natural
depressions have more complex and asymmetric
shape. In general, low values of p occur in depressions
M. Hayashi, G. van der Kamp / Journal of Hydrology 237 (2000) 74–85
having smooth slopes from the center to the margin,
and high values occur in depressions having a flat
bottom. For the 27 depressions, p ranges approximately between 2 and 6. Large wetlands have high
p and small wetlands have low p reflecting the basin
shape. By definition the A–h relation is closely related
to the frequency distribution of land elevation within
the depression, and it reflects the shape of the depression in a statistical sense. Therefore, A–h curves are
similar to the hypsometric curves that are used by
geomorphologists to study landform evolution.
Assuming that the power functions approximate the
unknown A–h and V–h relations of a given wetland,
an observer only needs to determine two constants in
the function; the scaling constant s and the shape
constant p. This can be achieved by two independent
measurements of A and h, and offers a less laborintensive alternative to the standard method of determining A–h and V–h relations from detailed elevation
data. This simple geometric model can be used in field
studies requiring approximate values of A and V. It
also provides a valuable tool for theoretical and
modeling studies because the parameters s and p
sensitively reflect the size and the geometry of small
lakes and wetlands. The applicability of the equations
have only been tested in the northern prairie region of
North America, and future studies are required to
examine if the equations adequately represent lakes
and wetlands in other environments.
Acknowledgements
We gratefully acknowledge the Canadian Wildlife
Service for the use of the St. Denis NWA. We thank
Randy Schmidt, Vijay Tumber, Catheryne Staveley,
David Parsons, Trevor Dusik, Herman Wan, and
Geoff Webb for their assistance in the collection and
analysis of survey data; and Yvonne Martin for the
discussion on landform evolution. We also thank Lars
Bengtsson and an anonymous reviewer for constructive comments. The research was supported by Ducks
Unlimited’s Institute for Wetland and Waterfowl
Research and Natural Sciences and Engineering
Research Council of Canada.
85
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