Introduction
The volume of water stored at the surface of wetlands, ponds, and lakes (as a
function of stage) is of great concern to those responsible for assessing
risks and balancing water supplies to societal demands. Arriving at reliable
estimates of such storage is difficult without some knowledge of the
feature's morphometry, i.e. information that is often time consuming and
impractical to acquire, especially when the features are numerous and
transient through space and time (Milly et al.,
2008). This is particularly true for beaver ponds owing to their cyclic
creation and abandonment.
Beaver dams and their associated ponds are ubiquitous in streams and
wetlands in the Northern Hemisphere and southern South America
(Whitfield et al., 2015). Beaver dam densities have been
reported to exceed 40 dams per kilometre (Macfarlane et al.,
2017), making them one of the most frequent obstructions to flowing water
(Naiman et al., 1986; Pollock et al., 2003).
Beaver dams increase the open-water area within watersheds
(Hood and Bayley, 2008) and ponds bring numerous
ecosystem benefits (Johnston, 2012), but beaver ponds can also be
viewed as burdensome or even dangerous from an anthropomorphic perspective
(Butler and Malanson, 2005; Green and Westbrook,
2009). Such concerns, whether positive or negative, generally centre around
the pond's capacity to store water and sediment, highlighting the need for
quick and accurate surface-water storage estimation methods.
Numerous hydrological investigations have sought to estimate surface-water
storage in other types of wetlands (Trigg et al., 2014; Xu
and Singh, 2004). For hydrological modellers, an ideal approach is one that
overcomes the need for often time-intensive topographic surveys and that is
more practical for use in models at varying scales and locations. Previous
studies have set about this by defining statistical relationships between
surface area and volume for wetlands of specific physiographic regions
(Gleason et al., 2007; Hubbard, 1982;
Lane and D'Amico, 2010; Wiens, 2001). Such approaches have been found useful
for modelling entire watersheds (Gleason et al., 2007), but
limited for estimating storage in individual wetlands because depth and
basin morphometry (i.e. surface area, volume, depth) are not considered
(Huang et al., 2011; Lane and D'Amico, 2010; Wiens, 2001). Brooks and Hayashi (2002)
presented an equation that includes depth and basin morphometry, but to use
it, basin morphometry must be predefined and no such information yet exists
for beaver ponds.
Another approach, the simplified volume–area–depth (V–A–h) method
(Hayashi and van der Kamp, 2000), accounts
for depth and calculates basin morphometry for each individual wetland.
Requiring only two measurements of depth and surface area, it has been shown
to provide reliable estimates of surface-water storage in the pothole
wetlands of the North American prairies for which it was designed
(Minke et al., 2010). Prairie potholes
are depressional wetlands that have fairly regular shapes, i.e. concave
profiles with smooth slopes. Beaver ponds, by contrast, typically encompass
a bathymetry that is far more complex because their size and shape is
controlled by the dimensions of the dam and the land surface that becomes
flooded upon dam establishment (Johnston and Naiman, 1987).
Whether statistical or analytical approaches can reliably estimate water
storage in beaver ponds has yet to be determined. Our goal was thus to
explore tools useful for estimating surface-water storage in beaver ponds.
We studied beaver ponds across much of their habitat range and (i) evaluated
the utility of the simplified V–A–h method in estimating surface-water
storage, (ii) evaluated correlations between surface-water storage and beaver
pond morphometry, and (iii) described beaver pond morphometry in relation to
surface-water storage capacity.
Methods
The simplified V–A–h method
The simplified V–A–h method is based on a simple power equation (Hayashi and
van der Kamp, 2000), where the area of a pond (A), at a given height above
the pond bottom (h), is described as
A=shh02/p,
where h0 is the unit height of the water surface (e.g. 1 m for SI units),
s is a scaling coefficient that represents the area of a circle (m2)
with a radius that corresponds to h0, and p is a dimensionless
morphometry coefficient that represents the shape of the bathymetric curve
(i.e. the area–depth relationship of the pond). The volume of the pond is
then determined by integrating all the area profiles below h to give
V(h)=∫0hsh*h02/pdh*=s1+2/ph1+2/ph02p.
Using Eqs. (1) and (2) requires parameterizing the s and p coefficients.
The simplified V–A–h method arrives at these values by rearranging Eq. (1)
to give (Minke et al., 2010)
s=A1h1h2-2/p,
and
p=2logh1/h2logA1/A2,
where A1 and A2 are surface areas of the pond at corresponding
depths of h1 and h2, respectively, and h1 < h2. With
only two measurements of area and depth in time, Eqs. (3) and (4) can be
used to calculate s and p coefficients that are then reinserted into Eqs. (1)
and (2) to define the entire area–depth and volume–depth relationship of the pond.
Beaver pond morphometry
Metrics for surface-water volume estimations
A beaver pond's capacity to store surface water is defined simply by its
bathymetry, and can be directly calculated if an accurate topographic survey
is available. The problem here relates to how well we can approximate that
volume given some simple measures of the dam and pond dimensions. To
discover if metrics exist, a series of morphometric variables were generated
in addition to the p coefficient described in Eq. (1). They include the
maximum dam height (hmax) defined as the difference in elevation (m)
between the dam crest and the lowest point in the pond, the maximum surface
area (m2) of the pond (Amax) at hmax, and the length (m) of
the dam (Dlen) measured along its crest. Regression analysis was then
used to determine if any of the variables are correlated to the maximum
volume of the pond (Vmax).
Perceptual diagram of the relationship between morphometric variables.
The area (a) at a given stage of the pond (h) is a point on the bathymetric
curve (thick black line), where RA is the relative area and RD is
the relative depth. The bathymetric integral (BI) is the integration
of everything below the bathymetric curve and the pond's capacity to store
water (BWC) is the integration of everything above the bathymetric
curve. The morphometry (p) coefficient represents the shape of the bathymetric
curve in the power function equation (red-dashed line; Eq. 7). The reference
solid is the box created by multiplying the maximum height of the dam (hmax)
by the maximum surface area created by the pond (Amax), and is entirely
comprised of land (Vland) and/or water (Vmax) proportional
to BI and BWC.
![]()
Morphometric analysis
Understanding the underlying mechanics of the simplified V–A–h method and
how morphometry relates to a pond's capacity to store water requires a
deeper analysis of the bathymetric curve. The bathymetric curve is
equivalent to the hypsometric curve defined by Strahler (1952) as the ground surface area of
a land mass with respect to elevation. To compare curves for ponds of
different size and relief, it is necessary to express the variables as
relative depth (RD) and relative area (RA) as
RD=hhmax,
and
RA=aAmax,
where h is the stage (m) elevation of the pond and a is the corresponding
surface area (m2) at any given h. For ease of visual interpretation, we
express the bathymetric curve as RD vs. 1 - RA (Fig. 1). Power
functions described by Eq. (1) can then be fit to a bathymetric curve with
the following equation:
RD=1-RAp/2,
where the p coefficient here is equal to the p coefficient in Eq. (1). This
allows for a visual aid in the analysis of error by superimposing estimated
curves produced via either Eq. (1) or Eq. (4) to the pond's actual
bathymetric curve. It also eliminates issues of scale between different
ponds so that bathymetric curves can be visually compared to one another.
From the relative bathymetric curve, it is possible to compute the
bathymetric integral (BI), a modified form of the hypsometric integral
defined as the measure of land mass volume with respect the entire reference
solid created by the maximum dimensions of the pond (Fig. 1; Strahler, 1952):
BI=VlandhmaxAmax=∫01RAdRD.
Equation (10) produces values between 0 and 1, with 1 representing a reference
solid entirely composed of land mass. Using the BI, we introduce a new
metric that represents the pond's bathymetric capacity to store water (BWC).
Since the total volume of the reference solid is comprised of either land or
water, the BWC, relative to the reference solid, is expressed as
BWC=1-BI=VmaxhmaxAmax.
The BI and BWC are quantitative measurements of the pond's morphometry
and capacity to store water, respectively. The value in using these metrics
is that they facilitate the comparison of surface-water storage capacity
among beaver ponds and other wetland types.
Finally, we described the shape of the beaver pond surface using a
dimensionless shape index (SI), which is essentially the ratio of the
pond perimeter to the circumference of a circle with the same area (Hutchinson, 1957):
SI=P2πAmax,
where P is the perimeter of the pond (m). Ponds with SI = 1 have shapes
that are perfectly circular, whereas ponds with SI > 1 are
increasingly complex. Pond shape is an important metric as much of the
interaction between surface water and groundwater happens at the shoreline
(Shaw and Prepas, 1990). We chose SI as it is easy to
interpret and enables a relative comparison between the shapes of beaver
ponds and other types of wetlands (Minke et al., 2010).
V–A–h models for surface-water storage estimation in beaver ponds
Three versions of the power function model described by Eq. (1) were tested
in this study. They are referred to as the full, simplified, and optimized models.
The simplified model is the actual test of the simplified V–A–h method and
the other two models were included to aid in the analysis of this approach.
The full model is a power function fitted to the complete data set of each
pond's bathymetry (i.e. empirical fit). We arrive at values for s and p by
fitting a simple power function, y = axb, to the pond's bathymetric
curve, and assume a = s and b = 2/p in accordance with Eq. (1).
Non-linear least-squares regression was used to determine the best fit; the ability of this
model to make accurate area and volume estimates depends on its “goodness of
fit” to the data set. Analysis of the full model was included to (i) identify
the p coefficient that best describes each beaver pond's morphometry and
(ii) assess the overall suitability of power functions to describe beaver
pond bathymetry.
The simplified model is a power function using s and p coefficients created
from the same two relative measurements of depth (i.e. h1 and h2 as
a percentage of hmax) in each pond. Minke et al. (2010) evaluated the
simplified V–A–h method by applying it to two scenarios: a dry one where
h1 and h2 are taken at 0.1 m and 25 % of hmax, and a wet one
where h1 and h2 are taken at 50 and 75 % of hmax. They
found that estimation errors were lowest using the wet scenario; therefore,
we chose this scenario to simulate the application of the simplified V–A–h
method as it may be practically used in the field.
Site locations, characteristics, and details of topographic pond surveys.
“n” is the number of ponds studied at each site.
Site
Latitude and
n
Soil
Terrain
Survey
DEM
longitude (degree, ′)
substrate
method
resolution
type
(m)
Kananaskis
51∘3.553′ N,
10
Organic
Mountainous
rtkGPS
1
Provincial
114∘52.009′ W
Park, AB,
Canada
Escondido,
54∘36.908′ S,
3
Organic
Mountainous
rtkGPS
1
Tierra del
67∘44.540′ W
Fuego,
Argentina
Logan River
41∘50.327′ N,
14
Mineral
Mountainous
Total station
0.1
Watershed,
111∘33.668′ W
UT, USA
41∘49.568′ N,
2
Mineral
Mountainous
Total station
0.1
111∘34.516′ W
41∘48.868′ N,
5
Mineral
Mountainous
Total station
0.1
111∘35.553′ W
Voyageurs
48∘32.773′ N
1
Organic
Lowland
Lidar
1
National Park,
93∘4.328′ W
MN, USA
48∘27.975′ N
3
Mineral
Lowland
Lidar
1
92∘53.864′ W
48∘30.405′ N
1
Mineral
Lowland
Lidar
1
92∘40.331′ W
48∘31.262′ N
1
Mineral
Lowland
Lidar
1
92∘52.794′ W
The optimized model differs from the simplified model through parameterizing
coefficients via the optimum combination of h1 and h2 for each pond.
This required calculating s and p coefficients at every possible combination
of h1 and h2 along the bathymetric curve (note that h1 and h2 are
expressed as a percentage of hmax from 1 to 100; therefore, the total
number of combinations where h1 < h2 is 5000 for each pond).
Each set of s and p coefficients was then reinserted into Eqs. (1) and (2)
to estimate area and volume, respectively, and the set that produced the
least combined area and volume error was selected as the optimum. The
optimum model was included in this study to discover how best to use the
simplified V–A–h method with regards to differences in pond morphometry.
Error for all three models was evaluated using root mean square error (ERMS),
defined as
ERMS=1m∑i=1mDACT-DEST2,
where m is the number of data points, DACT is the point on the actual
bathymetric curve calculated from the pond itself, and DEST is the point
on the estimated bathymetric curve derived from the s and p coefficients at a
given combination of h1 and h2. Finally, to allow for coherent
comparisons of error among the different beaver ponds, the magnitude of
error, referred to as AERR (%) for area and VERR (%) for
volume, was calculated by dividing the ERMS by the actual area and
volume of the pond at 80 % of hmax. This particular depth was chosen
to avoid inconsistencies in error magnitudes that arise when the evaluation
depth is set too close to the minimum and maximum (Minke et al., 2010).
Test sites
Forty beaver ponds were selected for this study and simulated in digital
elevation models (DEMs). Our sample design captured the range of structures
built by beaver along streams with mineral and organic substrates in both
mountainous and lowland terrain. Beaver ponds were thus analyzed from
multiple locations where bathymetric data existed, which included
Kananaskis Provincial Park, Alberta, Canada; Escondido, Tierra del Fuego,
Argentina; the Logan River watershed, Utah, USA; and Voyageurs National
Park, Minnesota, USA. Details of the location, terrain, number of ponds,
survey methods, and survey resolution for each site are provided in Table 1.
DEM creation and manipulation for variable calculations
Sites selected for this study were former beaver ponds that had drained
sufficiently to either reveal pond bottom bathymetry or allow it to be
surveyed. Beaver ponds extracted from lidar, when available, were fully
drained with visible relic dams, whereas some ponds surveyed by total
station and real-time kinetic geographical positioning system (rtkGPS) often were still full with water up to their crest
elevations, but not enough to impede point collection by wading. DEMs that
relied on total station and rtkGPS surveys were created with
Surfer®v10 (Golden Software, Colorado) using ordinary
kriging. The beaver ponds were then isolated from the unneeded areas of the
DEM by extracting all the points in the raster below and upstream of the dam
crest (i.e. hmax). This was done in ArcGIS v10.2 (ESRI, 2015)
as was the calculation of the morphometric variables. The V–h relationship,
as well as
bathymetric curve of each pond, was calculated at 5 cm increments using a
script written in PythonTM that utilizes the “volume” feature of
ArcGIS Toolbox. The V–h relationship and bathymetric curve of each pond were
the primary inputs for the three models, which were built and run in
RStudio (RStudio Team, 2015). Finally, the bathymetric curve for
each pond was established using linear interpolation to create 100 points,
i.e. 1–100 % of hmax.
Results
Beaver pond morphometry
Pond morphometric characteristics are provided in Table 2 and examples of
the DEMs from each location are provided in Fig. 2. The 40 ponds well
represented the various types of beaver ponds that are created in riverine
and wetland habitats (Baker and Hill, 2003), with maximum dam
heights (hmax) ranging from 0.25 to 2 m and dam lengths (Dlen)
spanning 3–308 m, with medians of 0.83 and 40 m, respectively. Pond
volumes (Vmax) ranged between 1 and 9001 m3 and showed strong power
correlations to Dlen, hmax, and Amax (Fig. 3). Among the ponds,
there was considerable variability in shape as SI values ranged
from 1.5 to 5.3 (mean = 2.6). No strong correlations (i.e. -0.10 > R2 < 0.10)
were found between SI and the other morphometric variables used in
this study (i.e. p, BI, BWC, Dlen, hmax).
The p coefficients for the beaver ponds followed a log-normal distribution,
and ranged from 0.45 to 2.58 (median of 0.91) (Fig. 4). Of the 40 beaver ponds
analyzed, 70 % (28) had p coefficients that were < 1, indicating
that beaver ponds tend to have convex bathymetries. Most beaver ponds tended
to be more convex than they are concave, given the shape of the bathymetric
curves (Fig. 5) and the range of BI (0.45–0.85; median of 0.69). In all
but one case, Vland was greater than 50 % of the total volume of space,
indicating that most beaver ponds are shallow, which limits the volume of
surface water they can store. This phenomenon is well described by the
strong exponential relationship between the p coefficient (R2 = 0.96)
and BI and BWC (Fig. 6). Soil substrate type (Table 1; organic
vs. mineral) did not affect the value of the p coefficient, as evidenced by a
t test (P = 0.97).
Four examples of detrended beaver pond DEMs used for this study, one
from each study area (SI = shape index, BI = bathymetric
integral, BWC = bathymetric water capacity, p = morphometry
coefficient (full model), s = scaling coefficient, Dlen = dam length,
hmax = maximum height of the dam, Amax = maximum
surface area of the pond, and Vmax = maximum volume of the pond).
![]()
Power regression relationships between the maximum volume of the beaver
ponds (Vmax) and (a) the length of the beaver
dams (Dlen),
(b) the product of the maximum depth of the ponds (hmax) and
the length of the beaver dams, (c) the maximum surface area (Amax)
of the ponds, and (d) the product of the maximum surface area and
maximum depth of the pond.
![]()
Distribution of morphometry (p) coefficients (full model) for all beaver ponds
sampled (n = 40).
![]()
Bathymetric curves for ponds shown in Fig. 1. RD is relative
depth, RA is relative area, BI is the bathymetric integral,
BWC is the bathymetric water capacity, and p is the optimum morphometry coefficient.
![]()
surface-water storage estimations
The full model had the least AERR, and the optimized model had the
least VERR (Fig. 7; Table 3). The highest AERR and VERR was
associated with simplified model estimates, which also produced the greatest
variability of error among the different ponds. With regards to study
locations, full VERR ranked as Escondido < Voyageurs < Logan < Kananaskis,
whereas full AERR ranked Logan < Escondido < Kananaskis < Voyageurs. Overall,
the beaver ponds in Kananaskis proved most difficult to model (i.e. highest VERR
and AERR overall); however, mean error for the full model remained
below 5 % for both area and volume estimates.
Compared to the full model (Fig. 7), the simplified model had higher VERR
in 65 % of cases (26 ponds) and higher AERR in 98 % of
cases (39 ponds), whereas the optimized model had lower VERR in 100 %
of cases but slightly (< 1 %) higher AERR in 100 % of cases.
The optimum p coefficients for volume tended to be slightly different than
the optimum p coefficients for area, which are the coefficients derived from
the empirical fit of the Full model. The optimum model proved useful for
revealing the two points on the bathymetric curve that can be used to obtain
the optimum p coefficient for volume estimates. Pond 7 had the largest
AERR and VERR (Fig. 7), and therefore was selected for more detailed study
(Fig. 8). The optimum points were found at the approximate location of where
the empirical fit intersects with the bathymetric curve. Thus, using the
optimum points in Eq. (4) computes a p coefficient that is closest to the
same coefficient generated by the curve fitted by non-linear least-squares
regression. The points used by the simplified model for Pond 7 fall on
segments of the bathymetric curve that are farther away in distance from the
empirical fit; hence, the p coefficient generated by these points creates a
curve that is farther away from the bathymetric curve, which ultimately
leads to a less accurate estimate of volume.
In a number of ponds, the empirical fit nearly overlapped the entire
bathymetric curve, and in such cases, there were many combinations of h1
and h2 that produced reasonable estimates of volume. For example,
Pond 10 had the lowest full AERR and VERR of all the beaver
ponds. In this case, there were 1899 combinations of h1 and h2 that
produced estimates with total error below 5 %, and the distance between
the points ranged from 1 to 84 % of hmax. Overall, the error was
not sensitive to distance between h1 and h2if the points were on or
near the full fitted curve. That said, the average minimum and maximum for h1
(for all the optimum combinations for each pond) was 18–74 %,
and for h2 it was 42–98 %.
Discussion
The simplified V–A–h method estimated surface-water storage in the beaver
ponds with high accuracy. Also, strong statistical relationships were found
between surface-water storage capacity in beaver ponds and the dimensions of
the dam and pond. The beaver ponds studied have a convex shape that permits
less water storage than do other open-water wetland types. surface-water
storage estimates can be made in beaver ponds without the need for
topographic surveys if pond morphology is used instead.
Relationship between the morphometry (p) coefficient (full model) and the bathymetric
water capacity (BWC).
![]()
Volume (VERR) and area error (AERR) from each beaver
pond using the three different approaches (a–f). Plots on the bottom
show the difference in volume (g) and area (h) error of the
simplified (solid circles) and optimized (open circles) models relative to the
full model (the full model is represented by the solid black line at zero on
the y axis). Bars and solid circles are colour coded by location as per the
legend at the top of the figure.
![]()
Comparison of the bathymetric curve for Pond 7 with the full and
simplified curve. The top shows the area (AERR) and volume (VERR)
error associated with the simplified curve that was calculated using simplified
depths h1 and h2 and the bottom shows the error associated with the
full curve and the optimum location for depths h1 and h2. RD is
relative depth, RA is relative area, BI is the bathymetric
integral, and p is the morphometry coefficient.
![]()
V–A–h model performance in beaver ponds
The low full AERR and VERR overall indicates that beaver pond
morphometry is adequately described by power functions. This is because the
bathymetric curve proved resilient to fluctuations in “elevation” inherent
to the impounded land surface. Also, the dams, intricate canals and holes
that beavers create in the areas they inhabit (Hood and Larson,
2015) do not warp the shape of the bathymetric curve enough that a power
function becomes inappropriate to sufficiently describe it. However, it
appears that volume estimations are more resilient to aberrations in the
bathymetric curve than are area estimates. The power functions in the full
model are fitted to pond bathymetry. When the power curve moves up and down,
AERR will increase, but sometimes the VERR can decrease because
volume is the integration of everything above the bathymetric curve. When
the curve moves slightly up or down from the empirical fit, irregularities
on the bathymetric curve are captured, which improves volume estimations at
the sacrifice of area estimations. This explains why the optimum p coefficients
for volume are different than they are for area. It also explains why, in
many cases, the simplified model had VERR that was less than 10 %,
while AERR was greater than 25 %. Without a complete set of pond
bathymetry, it is unlikely that users of the simplified V–A–h method would
be able to discern the optimum points for h1 and h2; however, as
long the chosen values for h1 and h2 are selected within the range
identified here (i.e. 18–74 % of hmax for h1 and 42–98 %
of hmax for h2), fairly accurate estimates of surface-water storage
should be expected. Overall, the simplified model performed reasonably,
exceeding 10 % VERR in only three cases. Given that the simplified
V–A–h method appears to work well across the broad range of beaver pond
bathymetry reported here, and across a wide range of prairie potholes
(e.g. Minke et al., 2010), it should be a robust enough approach to be used other
open-water wetlands.
Pond morphometric characteristics, including the full model morphometry (p) and
scaling (s) coefficients, shape index (SI), bathymetric integral (BI),
bathymetric water capacity (BWC), length of the dam (Dlen),
and maximum depth (hmax), area (Amax), and volume (Vmax)
of the ponds.
Location
Pond
SI
BI
BWC
p
s
Dlen
hmax
Amax
Vmax
no.
(m2)
(m)
(m)
(m2)
(m3)
Kananaskis
1
2.05
0.75
0.25
0.69
889
164
1.50
2974
1135
2
2.37
0.77
0.23
0.61
356
152
1.75
2006
867
3
2.57
0.69
0.31
0.97
959
127
0.85
686
186
4
1.79
0.77
0.23
0.61
123
27
1.50
446
163
5
3.71
0.77
0.23
0.62
705
226
1.95
5496
2503
6
3.47
0.74
0.26
0.67
1845
199
2.00
16357
9001
7
1.76
0.76
0.24
0.56
1334
308
1.85
12912
5734
8
2.55
0.75
0.25
0.63
701
159
1.80
3787
1757
9
1.71
0.68
0.32
0.92
290
39
1.25
448
184
10
1.51
0.66
0.34
1.05
247
30
1.10
297
113
Escondido
11
2.32
0.59
0.41
1.16
5352
162
0.55
1528
325
12
1.72
0.45
0.55
2.58
2181
59
0.30
748
130
13
1.99
0.54
0.46
1.61
3223
124
0.55
1342
344
Logan
14
2.19
0.66
0.34
1.06
438
7
0.30
54
6
15
2.03
0.72
0.29
0.83
464
3
0.25
15
1
16
1.89
0.56
0.44
1.51
87
4
0.60
41
11
17
2.63
0.75
0.25
0.67
112
17
0.75
52
10
18
2.14
0.70
0.30
0.91
91
19
0.80
63
15
19
2.17
0.67
0.33
0.97
138
10
0.65
53
11
20
1.95
0.67
0.33
0.94
352
16
0.45
50
8
21
2.47
0.64
0.36
1.11
179
11
0.50
45
8
22
2.70
0.67
0.33
0.98
96
7
0.45
17
2
23
1.90
0.64
0.36
1.20
56
10
0.55
23
5
24
1.97
0.69
0.31
0.80
430
27
0.60
82
15
25
2.37
0.59
0.41
1.36
154
6
0.30
22
3
26
2.83
0.75
0.25
0.68
124
21
0.90
90
19
27
2.79
0.73
0.27
0.75
114
5
0.60
36
6
28
1.73
0.67
0.33
0.96
278
13
1.00
265
87
29
4.32
0.81
0.19
0.45
975
87
1.00
980
189
30
3.43
0.71
0.29
0.79
620
21
0.85
374
94
31
5.31
0.69
0.31
0.90
551
43
0.85
432
115
32
2.61
0.69
0.31
0.83
1647
46
0.50
210
32
33
2.59
0.66
0.34
0.99
409
51
1.65
1123
621
34
2.40
0.58
0.42
1.41
470
12
0.45
130
26
Voyageurs
35
4.65
0.71
0.29
0.83
3683
144
1.10
4725
1517
36
3.82
0.70
0.31
0.94
4539
161
1.10
5928
1999
37
3.54
0.72
0.28
0.78
36105
57
0.40
2297
264
38
2.52
0.66
0.34
0.88
11836
58
1.10
12985
4740
39
2.72
0.61
0.39
1.11
18033
97
0.90
12482
4350
40
2.78
0.63
0.37
1.18
4867
41
0.55
1504
316
V–A–h model performance comparisons based on the mean (± standard
deviation) volume (VERR) and area (AERR) error magnitude.
“n” is the number of ponds studied at each site.
Site
n
Full
Simplified
Optimized
VERR (%)
AERR (%)
VERR (%)
AERR (%)
VERR (%)
AERR (%)
Kananaskis
10
4.3 ± 3.1
3.8 ± 2.1
7.2 ± 6.0
14.6 ± 12.3
2.3 ± 1.6
4.2 ± 2.5
Escondido
3
3.1 ± 1.4
3.8 ± 0.7
4.3 ± 2.5
6.7 ± 3.8
1.6 ± 0.7
4.0 ± 0.9
Logan
21
4.0 ± 2.6
3.6 ± 1.7
4.6 ± 3.5
9.9 ± 8.5
2.0 ± 1.2
3.9 ± 1.9
Voyageurs
6
3.8 ± 1.8
4.1 ± 1.6
3.8 ± 1.8
4.1 ± 1.6
1.9 ± 0.9
4.4 ± 1.7
All ponds
40
4.0 ± 2.5
3.8 ± 1.7
5.2 ± 4.1
11.0 ± 9.4
2.1 ± 1.2
4.0 ± 1.9
Beaver pond morphometry and surface-water storage capacity
Our results show that p coefficients in beaver ponds are lower overall than
those reported in prairie wetlands (Hayashi
and van der Kamp, 2000) and those reported in forest pools in New England
(Brooks and Hayashi, 2002). Because of the strong exponential
relationship between p coefficients and BWC, we can conclude that beaver
ponds typically store less water. For example, the prairie potholes studied
by Hayashi and van der Kamp (2000) had a median p coefficient of 3.22. Using
Fig. 6, this p coefficient is equivalent to a BWC of 0.61, which is almost
double the median beaver pond BWCequivalent of 0.32. The most likely
explanation for this is the ontogeny of beaver ponds compared to other open
water wetland types. Beaver ponds occur via inundation of an existing
channel and adjacent riparian area surface, whereas prairie potholes are
bowl shaped geomorphic depressions created by the deposition of glacial till
(Richardson et al., 1994). These different origins are
reflected in the shape of the bathymetric curves, and they also explain the
strong statistical relationships between surface-water storage capacity and
the dimensions of the dam and pond. The stream channel in Fig. 3, for
example, is represented on the far-right side of the bathymetric curve.
Beaver ponds built on deeper and narrower stream channels tend to have lower
p coefficients than ponds built on wider, fewer constrained channels. This
happens because there is a rapid expansion of surface area inundated as the
dam exceeds the height and width of the stream channel; a phenomenon that is
well described by the “power” relationships between Dlen, hmax,
Amax and Vmax. Pond 12 is a good example of this; the p coefficient
was highest (2.58) and a distant outlier compared to the other ponds. The
uniqueness of this site is that the beaver built a small dam (0.3 m) with
excavated peat and impounded groundwater seepage rather than damming channel
flows. Even though the dam is relatively small, it has a large BWC (0.55)
relative to the other ponds because the dam is entirely dedicated to
impounding a mostly flat land surface. In contrast, Pond 6, which was also
built in a peatland, has a much lower BWC (0.26) because most of the dam
height (2 m) is dedicated to impounding water in an incised stream channel.
An advantage of using the BWC metric over pond volumes is that it allows
for a comparison of surface-water storage capability in a way that is
independent of pond size and shape.
Tools for surface-water storage estimation in beaver ponds
There are a variety of ways our results can be used to estimate surface-water
storage in beaver ponds under different data availability scenarios.
In situations where only aerial or remotely sensed imagery is available
(i.e. world wide), dam length and pond area can be approximated and used in
the generalized power regression relationships presented in Fig. 3. This is
a quick and easy way to incorporate beaver pond surface-water storage
capacity into land use planning decisions and watershed-scale hydrological
models. However, this approach is not suitable for detailed study in
individual beaver ponds as it does not account for pond morphometry (Huang et al., 2011; Wiens, 2001).
Including dam height should improve estimates. Measuring dam height in the
field is quick and straight forward, but it can also be reasonably
approximated with remotely sensed imagery alone using spectral-depth
correlation methods (e.g. Passalacqua et al., 2015). If dam heights are available, we recommend using
our median p coefficient (0.91) for beaver ponds in the equation presented by
Brooks and Hayashi (2002):
Vmax=Amax×hmax1+2/p.
This equation is a modified form of Eq. (2) used to estimate surface-water
storage capacity. It is easily incorporated into spatially distributed
hydrological models. Fang et al. (2010) had success in
using this approach, albeit for prairie potholes, in their Cold Regions Hydrological Model.
With a moderate amount of data, the simplified V–A–h method offers an
alternative that produces surface-water storage estimates with minimal
error. The advantage of this method over the others is that it is robust, it
is customized to each pond's basin morphometry, and it calculates a
coefficient of scale (i.e. s coefficient) for use in estimating surface-water
storage across the range of pond stages, unlike the generalized power
regression models and Eq. (12), which are limited to estimates of Vmax.
Combined with a few field visits and something as simple as
automated water level observations, the simplified V–A–h method can be a
powerful tool. But, it also has practical application in relatively data
rich environments. For example, many lidar data sets are collected when
beaver ponds are not fully drained. If the beaver pond is not entirely full,
the measurements for A2 and h2 can be measured within the vertical
distance between the crest of the dam and the surface of the water, thus
allowing for an appropriate p coefficient to be derived. Furthermore, the
simplified V–A–h method is increasingly practical with the advent of new
technologies. For example, structure from motion software facilitates the
creation of high resolution DEMs from ordinary photographs (Javernick et al.,
2014). Theoretically, with both tools, one field visit to collect a few
pictures and depths measurements should be all that is needed to make
reliable estimates of wetland surface-water storage.
Implications of study results
The results of our study provide some simple tools that enable surface-water
storage in beaver ponds to be estimated without the need for topographic
surveys. This allows environmental managers to better assess the risks and
benefits associated with beaver ponds that appear on landscapes, and allows
for the easy inclusion of the surface-water storage component of beaver ponds
into hydrological models at various scales. This study also demonstrates
that beaver pond morphometry is different than other types of wetlands,
which requires consideration. For example, based on this analysis we might
expect beaver ponds to reach their capacity faster during rainfall events,
while impounding larger surface areas than depressional wetlands. Although
we show that some beaver ponds store less surface-water than other wetland
types, their relevance to local and regional water balances should not be
underestimated. Beaver population recovery, post fur trade, has led to the
creation of between 9494 and 42 236 km2 of new beaver ponds globally
(Whitfield et al., 2015). Using the estimates of Whitfield et al. (2015) and our
median p coefficient (0.91) and median dam height (0.83 m) in Eq. (12), we
crudely estimate that between 2.5 and 11 km3 of water are stored in beaver ponds.