Simple equations to represent the volume–area–depth ... · that are dependent on aquatic plants...

Simple equations to represent the volume–area–depth relations ofshallow wetlands in small topographic depressions

M. Hayashia,* , G. van der Kampb

aDepartment of Geology and Geophysics, University of Calgary, Calgary, Alberta, Canada T2N 1N4bNational 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 ofshallow lakes, wetlands or ephemeral ponds. The relations between the areaA, the volumeV, and the depthh of water indepressions are important for evaluating water and dissolved-mass balances of the system. TheA–h andV–h relations areusually determined from fine-resolution elevation maps based on detailed survey data. Simple equations are presented in thispaper, which can be used to: (1) interpolateA–h andV–h data points obtained by a detailed survey; (2) approximate unknownA–h andV–h relations of a depression from a minimal set of field data without a time-consuming elevation survey; and (3) serveas a geometric model of depressions in simulation studies. The equations are simple power functions having two constants. Thefirst constants is related to the size of the depression, and the second constantp is related to the geometry of the depression. Thepower functions adequately representA–h andV–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 othersimilar topographic depressions, an observer only needs to measureA and h twice to determine the two constants in theequation. The equations will be useful in field studies requiring approximateA–h andV–h relations and in theoretical andmodeling studies.q 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 theform of small lakes, wetlands or ephemeral pondshave important hydrological and ecological functions.They store snowmelt and storm water to attenuateflood peaks, and provide habitats for birds and animalsthat are dependent on aquatic plants and invertebrates.To study these functions hydrologists need to evaluate

water balance and dissolved-mass balance in thedepressions. For example, after a runoff event theflux of water and nutrients into a wetland can be esti-mated from the change of water volume in the wetlandand the change of concentration of dissolved species.

A practical approach for determining water volumeV and areaA is to measure the depth of water (h) andestimateA andV from predetermined area–depth (A–h) and volume–depth (V–h) relations. These relationsare specific to each depression, and are usuallyderived from a detailed bathymetry map. Because ofthis site-specific nature, most hydrological researcharticles reportA–h and V–h relations merely as a

Journal of Hydrology 237 (2000) 74–85www.elsevier.com/locate/jhydrol

0022-1694/00/$ - see front matterq 2000 Elsevier Science B.V. All rights reserved.PII: S0022-1694(00)00300-0

* Corresponding author. Tel:11-403-220-2794; fax:11-403-284-0074.

E-mail address:[email protected] (M. Hayashi).

tool for mass balance calculations, and rarely treatthem as a significant topic of study. However, it isuseful to look for common factors in theA–h andV–h relations for a variety of topographic depressionsbecause this can lead to a more general understandingof their shapes and storage characteristics.

Generalized forms ofV–h andA–h relations havebeen used by some investigators in the mathematicalmodeling of lakes. For example, Gates and Diessen-dorf (1977) assumedV is proportional toA to modellake level fluctuation in response to stochastic forcing,Bengtsson and Malm (1997) assumedA is propor-tional to h2 to study the sensitivity of lake level toclimatic condition, and O’Connor (1989) assumedVis proportional tohm andA is proportional tohm21 tosimulate the variation of dissolved solids in lakes andreservoirs.

The scope of this paper is limited to ephemeralponds and wetlands in small natural depressions, forexample, “pothole” wetlands in glaciated plains.Simple power functions with two parameters will beproposed to representA–h and V–h relations, andtested using data for 27 shallow lakes, wetlands andponds in the northern prairie region of North America.The power functions presented in this study areexpected to be widely applicable to small wetlandsand ponds in isolated and smoothly sloped depres-sions, even though they have only been tested in thenorthern prairie environment.

2. Theory

A–h andV–h relations are dependent on each other,

and one can be derived from another. For example,suppose that the water level in a lake rises by a smallamountDh. The resulting volume changeDV in thelake is equal toADh. Therefore,V at anyh is given by

V�h� �Zh

0A�h� dh �1�

whereh is a dummy variable of integration andh isthe depth measured at the deepest point of the lake.This relationship betweenV andA is fundamental andapplies to all lakes and wetlands that have a horizontalwater surface. Experimentally determinedV–h andA–h relations must satisfy Eq. (1), a failure ofwhich indicates that theV–h and A–h relations areinconsistent.

TheA–h andV–h relations of a basin can often beapproximated by simple analytical expressions suchas polynomials or power functions. In this paperpower functions are proposed that are based on theshape of simple symmetric basins formed by rotatinga slope profile around the central axis (Fig. 1). Theslope profiles are given by

y=y0 � �r=r0�p �2�

wherey [L] is the relative elevation of the land surfaceat a distancer [L] from the center,y0 [L] is the unitelevation, for example 1 m in SI units,r0 is the radiuscorresponding toy0, andp is a dimensionless constant.It follows from Eq. (2) that the area of the watersurface corresponding to a depth of waterh measuredat the center of the basin�r � 0� is given by

A� pr20

hh0

� �2=p

� shh0

� �2=p

�3�

where h0 [L] is the unit depth,s [L 2] is a scalingconstant, which is equal to the area of water surfacewhen h� h0: The constantp provides the linkbetween the shape of the basin (Fig. 1) andA–h rela-tion. A small value, for examplep� 2; corresponds toa paraboloid basin that has smooth slopes extendingfrom the center to the edge, and a large value corre-sponds to a basin that has a flat bottom. In an extremecase, we can setp! ∞: This corresponds to a cylin-der, for whichA� s regardless ofh. It follows fromEq. (1) that the volume of water corresponding toh is

M. Hayashi, G. van der Kamp / Journal of Hydrology 237 (2000) 74–85 75

Fig. 1. Slope profile of symmetric basins withy=y0 � �r=r0�p: Forexample,p� 2 indicates a parabolic slope.

given by

V � s�1 1 2=p�

h11�2=p�

h2=p0

�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 catch-ments where slopes are concave. With suitablep andr0, Eq. (2) represents the concave portion of mostslope profiles reasonably well. A natural depressionis made up by many slope profiles each having differ-ent values ofp and r0, and no single slope profilerepresents the entire depression. Therefore, onemight expect that Eqs. (3) and (4) cannot adequatelyrepresent natural depressions. However, as will beshown, the field data suggest thatA–h relations ofall natural depressions examined in this study arewell approximated by Eq. (3). In this casep ands inEq. (3) represent the shape and the size of depressionsin 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 multidisciplinaryresearch project to understand the hydrology and theecology of prairie wetlands. The site is located at1068060W and 528020N, which is approximately40 km east of Saskatoon (Fig. 2). The topography ofthe site is described as moderately rolling knob andkettle moraine with slopes varying from 10 to 15%(Miller et al., 1985). The area is underlain by glacialtills, which have a high clay content of 20–30%. Themean annual precipitation in Saskatoon is 360 mm, ofwhich 84 mm occurs as snow (Atmospheric Environ-ment Service, 1997). Air temperature frequentlybecomes lower than2308C in winter during whichthe soil frost penetrates as deep as 2 m. The uplandsaround the wetlands have been under cultivation for50–100 years.

Within the St. Denis NWA, four wetlands identifiedas S92, S109, S120, and S125S were selected fordetailed elevation surveys. The extent of the wetlandsis loosely defined by the growth of aquatic vegetationsuch as sedge and spike rush and by the presence ofsoft organic-rich soil, but the water-covered areas ofthe wetlands drastically change during a year. Thewetlands become entirely or partially inundated inspring after snowmelt runoff. Runoff rarely occurs insummer and water levels in the wetlands graduallydecline (Hayashi et al., 1998). Topographical mapsof the four wetlands are shown in Fig. 3. Surveyedareas did not completely cover the catchments of S92,S109, and S125S, and part of drainage divides aredrawn along the limit of the surveyed area. The miss-ing area is small compared to the areas included in themaps.

In addition to the four wetlands, a detailed elevationsurvey was conducted for four small depressions onthe cultivated uplands. These small depressions holdephemeral ponds only for a week to a few weeks inearly spring, and are not considered wetlands in usualsense. However, they are hydrologically importantbecause they store snowmelt water and rechargelocal groundwater. Three such depressions (D1, D2,and D3 in Fig. 3b) are located in and adjacent to thecatchment of S109. The fourth one (S104) is located200 m northeast of S109.

The catchments of the wetlands and depressionswere surveyed in 1994, 1998, and 1999 using totalstations. For the wetlands, survey points were spacedhorizontally at 10–15 m intervals in the uplands, and

M. Hayashi, G. van der Kamp / Journal of Hydrology 237 (2000) 74–8576

Fig. 2. Location of the present and previous study sites. (1) St.Denis; (2) Fort Qu’Appelle; (3) Melfort; (4) Saskatoon; (5) SwiftCurrent; (6) Wilkie; (7) Ward; (8) Dickey; (9) Stutsman. Shadedarea indicates the extent of the prairie wetland region (Winter,1989).

5–10 m intervals in the wetlands. For the smalldepressions, survey points were spaced horizontallyat 2–5 m intervals. Estimated measurement error iswithin a few centimeters for elevation and within afew tens of centimeters for horizontal location. Thesoftware packageSurfer (Golden Software, Golden,CO, USA) was used to estimate the elevation on regu-larly spaced grids by interpolation and to construct

digital elevation models (DEMs). The maps shownin Fig. 3 are based on the DEMs. From the DEMs,V–h and A–h relations were calculated using thevolume and area integration tool ofSurfer. Thedepth of waterh is defined as the elevation differencebetween the water surface and the lowest point in thedepression, which means thath� 0 when the wetlandbecomes completely dry. The kriging method (Davis,


Fig. 3. Topographical maps showing the elevation above the mean sea level of the four catchments in the St. Denis NWA. Principal contourinterval is 1 m. Scale bars indicates 50 m. Wetlands are indicated by shades and drainage divides are indicated by thick lines. The location ofsmall depressions D1, D2, and D3 are indicated in (b).

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Table 1AreaA and volumeV of water corresponding to depthh in the wetlands and small depressions in the St. Denis NWA

h (m) A (m2) V (m3) A (m2) V (m3) A (m2) V (m3) A (m2) V (m3) A (m2) V (m3) A (m2) V (m3) A (m2) V (m3) A (m2) V (m3)

S92 S109 S120 S125S S104 D1 D2 D30.1 170 5.2 190 6.5 520 21 360 17 180 9.5 150 7.2 63 3.2 88 4.60.2 520 41 500 41 900 93 790 74 340 36 360 32 130 13 160 180.3 750 100 830 110 1190 200 1210 170 500 78 610 81 210 30 270 400.4 970 190 1130 210 1430 330 1610 320 660 140 310 56 370 720.5 1180 300 1410 330 1660 480 1990 500 820 210 460 94 470 1100.6 1380 430 1690 490 1880 660 2360 710 1010 300 600 1700.7 1590 570 1970 670 2100 860 2740 970 1210 410 760 2300.8 1810 740 2250 880 2300 1080 3120 12600.9 2050 940 2630 1120 2570 1330 3510 15901.0 2350 1160 3120 1410 2830 1600 3850 19701.1 2730 1410 3660 1750 3150 18901.2 3200 1700 4120 2140

1986, p. 239) was used for interpolation. Preliminaryanalysis showed that the calculatedV–h andA–h rela-tions were essentially independent of the choice ofsemivariogram and grid spacing. The relationspresented in this paper were calculated using linearsemivariograms with no drift and 5 m grids for thewetlands 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 arelocated in three locations; Potholes 1, 2, and 4 in Ward

County, Potholes 5, 5A, 6, 7, and 8 in Dickey County,and Pothole C1 in Stutsman County (Fig. 2). TheLakshman wetlands are located near Fort Qu’Appelle,Melfort, Saskatoon, Swift Current and Wilkie (Fig. 2).Survey methods and the density of surveyed pointswere not clearly described in the original articles,which only included tables of the elevation of watersurface with respect to an arbitrary datum and the areaand the volume corresponding to each elevation. Shje-flo (1968) provided the elevation of the lowest point ineach basin so that we could calculate the depth ofwater corresponding to each elevation and determineA–h and V–h relations. Lakshman (1971) did notprovide such data, and we needed to estimate thelowest elevation by inspecting the water level record.


Table 2Upper depth limithmax, scaling constants, power constantp, root-mean-squared error of areaAerr and volumeVerr of the wetlands, the relativemagnitude ofAerr with respect toA1m, andVerr with respect toV1m. Asterisks indicate that the relative magnitudes ofAerr andVerr are evaluatedagainstA andV at hmax

Wetland ID hmax (m) s (m2) p Aerr (m2) Aerr/A1m (%) Verr (m3) Verr/V1m (%)

St. DenisS92 1.2 2450 1.80 97 4.1 17 1.5S109 1.2 3180 1.61 101 3.2 20 1.4S120 1.1 2820 2.66 55 1.9 8.6 0.5S125S 1.0 3840 2.10 20 0.5 5.3 0.3S104 0.7 1720 1.95 17 1.4p 2.6 0.6p

D1 0.3 2880 1.55 2.4 0.4p 0.9 1.1p

D2 0.5 1160 1.45 11 2.5p 1.6 1.7p

D3 0.7 1130 1.66 15 2.0p 2.6 1.1p

Shjeflo (1968)Pothole 1 1.4 51 900 5.12 1360 2.6 685 1.7Pothole 2 1.7 109 700 3.64 8110 6.9 2010 2.4Pothole 4 1.5 93 400 4.31 4610 5.2 814 1.5Pothole 5 2.3 78 200 5.48 1390 1.7 961 1.5Pothole 5A 1.6 9600 2.49 239 2.6 100 2.1Pothole 6 1.5 33 300 6.19 200 0.6 359 1.5Pothole 7 1.5 86 800 3.52 1820 2.1 801 1.7Pothole 8 1.1 123 100 3.33 6260 5.3 1120 1.3Pothole C1 1.9 162 900 5.33 4540 2.8 3870 3.4

Lakshman (1971)Ft. Qu’Appelle 1 1.4 7160 3.22 57 0.8 101 2.0Ft. Qu’Appelle 2 1.5 8570 3.00 342 4.7 125 3.1Ft. Qu’Appelle 17 1.5 5790 2.79 83 1.6 37 1.4Ft. Qu’Appelle 19 1.2 6720 3.75 142 1.5 90 1.6Ft. Qu’Appelle 20 0.9 4310 3.11 98 1.4p 54 1.4p

Melfort 7 1.1 8990 2.44 392 4.1 236 4.3Saskatoon 16 0.9 1960 4.06 31 1.7p 44 5.1p

Swift Current 1 2.0 34 100 3.26 2610 7.4 1240 7.1Wilkie 6 0.6 11 800 4.72 242 2.6p 118 2.9p

Wilkie 12 1.2 2150 3.28 46 2.1 13 0.9

The estimation was only possible for those wetlandsthat dried up frequently in the reported study period of1964–1970. In addition the elevation-area-volumedata were incomplete for some wetlands. Therefore,out of 25 wetlands in Lakshman (1971),V–h andA–hrelations were determined for only 15 of them. Theaccuracy of estimatingh is expected to be in the orderof 0.05 m.

4. Results

4.1. St. Denis wetlands

The A–h and V–h relations of the wetlands andsmall depressions are listed in Table 1. The data arelisted for a depth range between 0 andhmax. Thehmax isdefined by the overflow point for S120, S125 and allmicro-depressions, and by the highest water level

recorded in 1968–1997 for S92 and S109 (van derKamp et al., 1999). The values ofhmax are listed inTable 2.

Fig. 4 shows theA–h relation of S92, which repre-sents an irregularly-shaped end member, and forS109, which represents a reasonably regularly-shapedend member. Solid circles indicate data points calcu-lated from the DEM and curves show Eq. (3) with thebest-fit values ofs and p determined by the least-squares method. Table 2 lists the values ofs and p.The power function (Eq. (3)) approximates theA–hrelation of all wetlands reasonably well. To evaluatethe goodness of fit between the data points and thepower function, root-mean-squared (RMS) errorAerr

is defined by

Aerr ��1m

Xmi�1

�ADEM 2 APF�2vuut �5�


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 ofsandp listed in Table 2.

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 ofsandp listed in Table 2.

whereADEM is the area calculated from DEM, andAPF

is the area given by the power function, andm is thenumber of data points. The magnitude ofAerr iscomparable to the values ofA in a range 0, h #0:1 m; and APF does not give a meaningful estimatein this depth range. The relative magnitude ofAerr

becomes less significant at greater depth range asAincreases. The ratioAerr=A is generally smaller than10% in a range 0:3 m , h # hmax: Table 2 listsAerr aswell asAerr=A1m; whereA1m is the data point closest toh� 1 m:

Oncesandp are determined by fitting Eq. (3) toA–h data points, the sames andp can be used in Eq. (4)to approximateV–h relation. Fig. 5 shows theV–hrelation of S92 and S109, in which solid circles indi-cate data points calculated from DEM and curvesshow Eq. (4). The goodness of fit between the datapoints and the power function (Eq. (4)) is expressedby RMS error,Verr, defined similarly to Eq. (5). Therelative magnitude ofVerr with respect toV is gener-ally smaller than 10% in a range 0:3 m , h # hmax:

Table 2 listsVerr andVerr=V1m; whereV1m is the datapoint closest toh� 1 m:

Eqs. (3) and (4) were similarly applied to theA–hand V–h relations of wetlands S120 and S125, andsmall depressions D1, D2, D3, and S104. The least-squares-fit values ofs andp, and RMS errorsAerr andVerr are listed in Table 2. The match between thepower functions and the data points for the wetlandsand small depressions is similar to that S92 and S109.The ratioAerr=A andVerr=V are generally smaller than10% in a depth range 0:3 m , h # hmax for wetlandsand 0:1 m , h # hmax for small depressions.

The relationship betweensandp is shown in Fig. 6.St. Denis wetlands and depressions have relativelysmall sizes, which is reflected in the range of thescaling constants. The values ofp fall in a relativelynarrow region around 2, which indicates that thedepressions have a reasonably smooth shape thatresembles a paraboloid (Fig. 1).

4.2. Previously studied wetlands

A–h andV–h data were available for 10 wetlands inShjeflo (1968) and 15 wetlands in Lakshman (1971).The data were examined with respect to the consis-tency condition given by Eq. (1). At each reportedvalue of h, a value ofV was calculated fromA–hrelation using Eq. (1), and compared to the reportedvalue ofV. If the relative difference was greater than15% at anyh greater than 0.2 m, the data set wasconsidered inconsistent. The source of inconsistencymay be inappropriate methods used to calculateA andV from the survey data. Nine wetlands in Shjeflo


Fig. 6. Relationship betweens andp.

Fig. 7. A–h andV–h relations of Pothole 4 (Shjeflo, 1968). Solidcircles indicate data points and curves indicate the power functionswith values ofs andp listed in Table 2.

(1968) and ten in Lakshman (1971) satisfied thecondition, and will be used in the following analysis.The upper depth limithmax, listed in Table 2, is arbi-trarily set to be the highest water level recorded in thestudy period (1960–1964 in Shjeflo (1968) and 1964–1970 in Lakshman (1971)) plus 0.5 m. IfA–h andV–hdata do not coverhmax, the highest value ofh withinthe data set is considered to behmax.

Eq. (3) was fitted to theA–h relation of the nineShjeflo wetlands to determines andp (Table 2). Theleast-squares method was applied to each data setwithin a range 0:1 m # h , hmax: The data pointshaving h less than 0.1 m were excluded from theanalysis because the accuracy ofh at such a smallvalue is questionable. As an example Fig. 7a showstheA–h relation of a wetland identified as Pothole 4,and Fig. 7b shows theV–h relation of the samewetland. Solid circles indicate data points and curvesshow the power functions (Eqs. (3) and (4)). Thepower functions agree with the data points reasonablywell for both A–h and V–h relations. Similarly thepower functions adequately representA–h and V–hrelations of all other Shjeflo wetlands. The RMSerrors Aerr and Verr and the relative magnitudesAerr=A1m and Verr=V1m are listed in Table 2. For allnine Shjeflo wetlands,Aerr=A andVerr=V are generallysmaller than 10% in a range 0:3 m , h # hmax:

The power function (Eq. (3)) was also applied to theLakshman wetlands to determine the least-squares-fitvalues ofs andp (Table 2). The agreement betweendata points and the power function is reasonably goodfor all ten wetlands. The RMS errors and their relativemagnitude with respect toA1m and V1m are listed inTable 2. For all ten Lakshman wetlands,Aerr=A andVerr=V are generally smaller than 10% in a range0:3 m , h # hmax:

The relationship betweensandp is shown in Fig. 6.The positive correlation indicates that Shjeflo andLakshman wetlands, which are much larger in sizethan St. Denis wetlands, have higher values ofp.The implication of the size-shape relationship willbe discussed later.

5. Simple field methods for determiningapproximate A–h and V–h relations

In general, accurate determination ofA–h andV–h

relations requires a detailed elevation survey over anentire wetland, which is labor intensive and timeconsuming. Based on the above results, it is likelythat Eqs. (3) and (4) approximately representA–handV–h relations of most small topographic depres-sions. Therefore, at least for the first approximation,one only needs to measureA andh at a few differenttimes to determines andp.

For a given wetland, letz be the elevation of thewater surface with respect to an arbitrary datum, forexample a staff gauge, andzmin be the elevation of thelowest point in the wetland. Eq. (3) can be written as

A� s��z2 zmin�=h0�2=p �6�

In principle, three unknown constantss, p, andzmin canbe determined from three independentmeasurements ofA and z. However, it is easyto measure zmin for most wetlands in smalldepressions. For example, an observer can locatethe lowest point in the wetland just before itbecomes completely dry, or an observer on aboat can probe around the central part of thewetland to find the deepest point. Therefore, inmost cases onlys and p need to be determinedfrom two independent measurements ofA and z.There are a number of ground-based and airbornemethods to estimateA of many wetlands relativelyeasily.

If time and resources are limited, an observer maychose to estimatep from the size–shape relationshipshown in Fig. 6. For example, Fig. 6 indicates thatp islikely close to 2 in small seasonal wetlands andephemeral ponds. In this case, an observer mayassumep� 2 and determines from a single measure-ment ofA andh. The accuracy ofA–h andV–h rela-tions determined this way may not be high, but somefield studies can benefit from the simplicity and thepracticality of the method.

We used this method in the St. Denis NWA in1998 to estimate the volume of snowmelt runoffcollected in several depressions that did not havedetailed survey data. This measurement showedthat significant portion of snowmelt runoff isstored in depressions without draining to themain wetland in the catchment, and gave us animportant step forward in understanding thehydrology of prairie wetlands.


6. Discussion

6.1. Hypsometry: statistical meaning of the shapeparameter p

The field data indicate that theA–h relation ofmany wetlands is represented by a well-definedvalue ofp that reflects the basin shape in some aver-age sense. This finding is not trivial and warrantssome discussion. To this end it is informative to reex-amine how theA–h relation is determined fromdetailed survey data.

The A–h relation of a basin is determined from ahigh-resolution DEM of the basin. On the DEM, thebasin is hypothetically filled with water to a series ofvalues ofh, and for eachh the grid points that areunder the water surface are counted. If there arengrid points under the water surface located ath, thenthe area of the water surfaceA is given by

A�h� � n�h�Da �7�whereDa is the area associated with each grid point.The above procedure involves “counting” the numberof grid points, which suggests thatA can also beexpressed in terms of the frequency distribution ofgrid points within the basin. The cumulativefrequency distributionF�h� is defined as

F�h� � n�h�=N �8�whereN is the total number of grid points in the basin.Since the total area of the basinAtot is given by

Atot � NDa �9�

\tflt="PS6F00" \tfnm="PS6F00"A�h� and F�h� arerelated by

A�h� � AtotF�h� �10�Eq. (10) clearly demonstrates the equivalencebetweenA–h relation and frequency distribution.

Similar ideas have been used to study the distribu-tion of landmass within a drainage basin by geomor-phologists, who use the term hypsometric analysis forstudying howF�h� changes in relation to landformevolution (Strahler, 1952; Willgoose and Hancock,1998). TheF�h� of the entire catchments of S92 andS109 are shown in Fig. 8. The graphs, called hypso-metric curves, haveF�h� on the horizontal axis andhon the vertical axis following the geomorphologicalconvention (Strahler, 1952). The values ofA corre-sponding toF are also plotted on the top horizontalaxis. Each wetland occupies only a small portion ofthe catchment, which is indicated by a shaded regionat the toe of the hypsometric curves. The wetland


Fig. 8. Hypsometric curves of the catchments of S92 and S109.Shaded regions indicate wetlands.

Fig. 9. North–south (NS) and east–west (EW) slope profiles ofPothole 4 and S109. Open squares indicate EW profiles and solidcircles indicate NS profiles. Curves show the best fit power func-tions having powerp. (a) Pothole 4. (b) S109.

portion of the hypsometric curves have relativelysimple shape that is adequately represented by thepower function (Eq. (3)) even though the overallshape of the hypsometric curves is more complex.

It is clear from the above discussion that theA–hrelation of a depression should be regarded as thefrequency distribution of land elevation within thedepression. Therefore, two given depressions mayhave an identicalA–h relation even though theiractual shape is significantly different. The shape para-meter p in Eq. (3) represents the slope profile of ahypothetical basin (Fig. 1) that is hypsometricallyequivalent to the actual depression.

6.2. Size–shape relationship and landform evolution

Fig. 6 shows that larger wetlands tend to havehigher values ofp. In general large prairie wetlandshave water for a long period of time in a given year, infact many of them are semi-permanent lakes, whilesmall prairie wetlands have water only for a fewmonths after snowmelt. As a result, large wetlandshave a flat bottom formed by sedimentation. Forexample Fig. 9a shows north–south (solid circles)and east–west (open squares) profiles of Pothole 4in Shjeflo (1968) and Fig. 9b shows north–south andeast–west profiles of S109 in St. Denis. Markers inFig. 9 show the data points and lines show Eq. (2)using the least-squares-fit values ofp andr0. As indi-cated in Fig. 9, the profiles in Pothole 4 have largepreflecting the flat bottom, while those in S109 havesmall p reflecting relatively smooth slope from thecenter to the edge.

In Fig. 9 p takes a range of values because thewetlands are asymmetric and each individual slopehas different curvature. TheA–h relation integratesall slope profiles within the wetland and defines asingle value ofp in an average sense. For example,a representativep of Pothole 4 (Fig. 9a) may be givenby a harmonic averagepav of the north, east, south,and west profiles;pav � 4:85: This is comparable top� 4:31; which was obtained by fitting Eq. (3) to theA–h data set. Similarly, a harmonic average of thefour profiles of S109 (Fig. 9b) ispav � 1:78; whichis comparable top� 1:61; which was obtained byfitting Eq. (3) to theA–h data set.

The wetlands examined in this paper occur in thenorthern 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 completelydry in late summer and fall, and the land surface issubjected to gravity-driven soil creep induced byfreezing and thawing, drying and wetting (Kirkby,1967), cultivation (de Jong et al., 1998), and animalburrow activities (Black and Montgomery, 1991).Geomorphological literatures suggest that soil creeptends to dissipates the irregularity of the landform bydiffusion-like processes (Culling, 1960). It is reason-able to expect that the dissipation of irregularity at alocal scale results in the smooth frequency distribu-tion of land elevation at a basin-wide scale, and henceto simple A–h and V–h relations. If the landformevolution is solely driven by soil creep, slope profilesare expected to approach curves having a low value ofp (Fig. 1). In contrast, if the landform evolution isstrongly influenced by other mechanisms like under-water sedimentation, slope profiles may take a highervalue of p that reflects the balance between severaldriving forces.

It is not clear if similar smoothing mechanismsexist in different environments, for example deeplakes on a rocky terrain or playas on tropical savanna.Therefore, the applicability of Eqs. (3) and (4) is so farlimited to the prairie region where the equations havebeen tested. However, it will be interesting to examinewhether the equations are useful for the lakes andwetlands in different environments.

7. Conclusions

The main objective of this paper is to examine therelation between the volumeV, areaA, and depthh ofwetlands in isolated depressions. For the 27 wetlandsand ephemeral ponds examined in this paper,A isproportional toh2=p to a good approximation, andVis proportional toh112=p

; wherep is a dimensionlessconstant. In other words,A–h andV–h relations areexpressed as power functions. The constantp isrelated to the shape of the depression, more specifi-cally the functional form of slope profiles. For exam-ple, a paraboloid-shaped depression hasp� 2; and acylinder-shaped depression hasp! ∞: Naturaldepressions have more complex and asymmetricshape. In general, low values ofp occur in depressions


having smooth slopes from the center to the margin,and high values occur in depressions having a flatbottom. For the 27 depressions,p ranges approxi-mately between 2 and 6. Large wetlands have highp and small wetlands have lowp reflecting the basinshape. By definition theA–h relation is closely relatedto the frequency distribution of land elevation withinthe depression, and it reflects the shape of the depres-sion in a statistical sense. Therefore,A–h curves aresimilar to the hypsometric curves that are used bygeomorphologists to study landform evolution.

Assuming that the power functions approximate theunknownA–h andV–h relations of a given wetland,an observer only needs to determine two constants inthe function; the scaling constants and the shapeconstantp. This can be achieved by two independentmeasurements ofA and h, and offers a less labor-intensive alternative to the standard method of deter-miningA–h andV–h relations from detailed elevationdata. This simple geometric model can be used in fieldstudies requiring approximate values ofA and V. Italso provides a valuable tool for theoretical andmodeling studies because the parameterss and psensitively reflect the size and the geometry of smalllakes and wetlands. The applicability of the equationshave only been tested in the northern prairie region ofNorth America, and future studies are required toexamine if the equations adequately represent lakesand wetlands in other environments.

Acknowledgements

We gratefully acknowledge the Canadian WildlifeService for the use of the St. Denis NWA. We thankRandy Schmidt, Vijay Tumber, Catheryne Staveley,David Parsons, Trevor Dusik, Herman Wan, andGeoff Webb for their assistance in the collection andanalysis of survey data; and Yvonne Martin for thediscussion on landform evolution. We also thank LarsBengtsson and an anonymous reviewer for construc-tive comments. The research was supported by DucksUnlimited’s Institute for Wetland and WaterfowlResearch and Natural Sciences and EngineeringResearch Council of Canada.

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