Volume 58, Issue 12 e2022WR032694
Method
Open Access

The Numerical Formulation of Simple Hysteretic Models to Simulate the Large-Scale Hydrological Impacts of Prairie Depressions

Martyn P. Clark

Corresponding Author

Martyn P. Clark

Centre for Hydrology, University of Saskatchewan Coldwater Laboratory, Canmore, AB, Canada

Correspondence to:

M. P. Clark,

[email protected]

Contribution: Conceptualization, Methodology, Software, Validation, Writing - original draft, Writing - review & editing, Visualization, Project administration, Funding acquisition

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Kevin R. Shook

Kevin R. Shook

Centre for Hydrology, University of Saskatchewan, Saskatoon, SK, Canada

Contribution: Conceptualization, Methodology, Validation, Writing - review & editing

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First published: 29 November 2022
Citations: 2

Abstract

Topographic depressions have large impacts on the hydrology, ecology, and biogeochemistry in prairie environments. Topographic depressions control the hydrological flow regime through their role in storing water from precipitation and snow melt, and the storage dynamics of topographic depressions affect the retention, transformations, and transport of sediment, carbon, and nutrients. The water stored in topographic depressions supports a diversity of animal and plant species. More generally, depressional wetlands are an integral component of global water, energy, and biogeochemical cycles. In this technical note, we present simple models to simulate hysteretic relationships between storage and contributing area. We consider both spatially distributed and spatially integrated models. The spatially distributed models (the ensemble-depression model) explicitly simulate storage dynamics for a distribution of discrete depressions. In these spatially distributed models, the large-scale storage-contributing fraction relationships are an emergent property of the modeled system. The spatially integrated models (the meta-depression model) parameterize the large-scale storage-discharge relationships. The spatially integrated models are formulated in a way to represent hysteretic behavior. We present a suite of numerical experiments and demonstrate that the ensemble-depression and meta-depression models have similar large-scale behavior. The simple hysteretic models defined here are formulated in a way so that they can easily be incorporated in conventional hydrological models.

Key Points

  • We present simple models to simulate hysteretic relationships between storage and contributing area in Prairie landscapes

  • We demonstrate that the spatially distributed and spatially lumped models have similar large-scale behavior

  • The simple hysteretic models defined here are formulated in a way so that they can easily be incorporated in conventional hydrologic models

1 Introduction

Topographic depressions have large impacts on the hydrology, ecology, and biogeochemistry in prairie environments. In many prairie landscapes, there has not been sufficient energy, water, or time to carve conventional drainage networks, and, as a result, much of the landscape drains internally to millions of small topographic depressions. Topographic depressions control the hydrological flow regime through their role in storing water from precipitation and snow melt and the subsequent impacts of depression storage dynamics on flood peaks and inundation patterns (e.g., Ehsanzadeh et al., 2012; Evenson, Golden, et al., 2018; Huang et al., 2011; Le & Kumar, 2014; Shook & Pomeroy, 2011; Spence & Mengistu, 2019). The storage dynamics of topographic depressions affect the retention, transformations, and transport of sediment, carbon, and nutrients (e.g., Baulch et al., 2021; Cohen et al., 2016; Costa et al., 2020; Euliss et al., 2006; Gleason et al., 2008; Lane et al., 2018; Marton et al., 2015; Pennock et al., 2010) and the water stored in topographic depressions supports a diversity of animal and plant species (e.g., Hansen & Loesch, 2017; Niemuth & Solberg, 2003). More generally, depressional wetlands are an integral component of global water, energy, and biogeochemical cycles (e.g., Bridgham et al., 2013; Fan & Miguez-Macho, 2011; Melton et al., 2013; Wik et al., 2016).

In this note, we focus on the general problem of simulating storage-discharge dynamics in complexes of depressional wetlands (Evenson, Golden, et al., 2018; Fan & Miguez-Macho, 2011; Golden et al., 2014). We are interested in depressional wetland complexes because the storage dynamics of wetland complexes and the outflow from the wetlands affects the catchment-scale hydrological connectivity (e.g., Shook & Pomeroy, 2011; M. Mekonnen et al., 2014; Shook et al., 2021; Zeng & Chu, 2021). In framing our work in terms of depressional wetlands, we recognize that the ponds of water in prairie depressions are ephemeral, and hence may not satisfy the strict definition that wetlands must have hydric soils (e.g., van der Kamp et al., 2016). Nevertheless, the ponds of water that form in prairie depressions are defined as wetlands in the global Earth System modeling literature, where wetlands are defined in general terms as locations that are inundated with water for some time period during the growing season (e.g., Lehner & Döll, 2004). In particular, the topographic depressions we consider here can be classified as Type A and F wetlands defined by Fan and Miguez-Macho (2011) as “wetlands formed in local depressions with low permeability substrate, disconnected from a river network and perched above the regional water table” (Type A) and wetlands that “exist largely because of the frozen soil in the shallow depths which prevents drainage” (Type F). Fan and Miguez-Macho (2011) define Type F wetlands in regions with patchy permafrost and in regions where there is substantial seasonal frost penetration (>0.5 m). Moreover, prairie potholes are given as an example as geographically isolated wetlands, defined by Tiner (2003) as “wetlands that are completely surrounded by upland at the local scale.” These overlaps between the definitions of topographic depressions and wetlands are useful in order to connect our work to model developments in the wetlands literature.

Many different approaches have been used to simulate the storage dynamics of depressional wetlands (e.g., Golden et al., 2014). Liu et al. (2008) and Wang et al. (2008) developed a scheme to simulate the storage dynamics of wetland complexes. In this approach, it was considered infeasible to simulate each wetland separately, so Liu et al. (2008) and Wang et al. (2008) lumped together isolated wetlands in a watershed to create a Hydrologically Equivalent Wetland (HEW) for simulation. The area-volume relationships for the HEW were defined using a power-law as A = βVα, where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0001 is the wetland area, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0002 is the wetland volume, and β and α are coefficients in the area-volume relationships. More recent model development efforts focused on improving the representation of wetlands by improving the spatial delineation of wetlands and their contributing areas, improving simulations of groundwater-wetland interactions (subsurface inflows and wetland seepage), and improving simulations of evapotranspiration (Evenson et al., 2016; Evenson, Jones, et al., 2018). Recent large-domain applications of the depression-integrated model for the Upper Mississippi River basin demonstrate that including depressions in the model substantially improves large-domain model simulations (Rajib et al., 2020).

Other model development efforts focus on how the threshold-driven behavior of topographic depressions affects the temporal variability in the contributing fraction of catchments (e.g., Ehsanzadeh et al., 2012; Le & Kumar, 2014; M. Mekonnen et al., 2014; B. A. Mekonnen et al., 2016; Shook & Pomeroy, 2011; Spence & Mengistu, 2019; Zeng & Chu, 2021, see Figure 1). A key process is that topographic depressions act as switches, where the land areas that drain into topographic depressions only produce outflow when the storage capacity of the depression is satisfied (e.g., Shook & Pomeroy, 2011; Shook et al., 2013; Zeng & Chu, 2021). This is the fill-and-spill process (Leibowitz et al., 2016; Shaw et al., 2012; Shook & Pomeroy, 2011; Spence, 2007; Spence & Woo, 2003), which has important controls on hydrological processes across many different hydrological regimes (e.g., Arnold et al., 2014; McDonnell et al., 2021; Nyquist et al., 2018; Tromp-van Meerveld & McDonnell, 2006a2006b).

Details are in the caption following the image

Illustration of the hysteretic relationship between storage and contributing area. The four images denote different points on the hysteresis diagram in Figure 2. The blue arrows are water flows. The dashed lines represent the depression catchment divides. The brown areas are the unfilled regions of the depressions.

A major hydrological modeling challenge is simulating the strong hysteretic relationships between storage and the contributing fractions of basins (see Figures 1 and 2). Initially (Figures 1a and 2 point a), only the smallest depressions are filled. As water is added to the depressions by precipitation and/or runoff, there is a gradual increase in the catchment's contributing fraction as more depressions are filled (e.g., Chu et al., 2013; Le & Kumar, 2014; Shaw et al., 2012; Shook & Pomeroy, 2011; Shook et al., 2013; Figures 1b and 2 point b). This occurs because there is large spatial variability in the storage capacity of depressions, and the areas that contribute flow to them, meaning that individual depressions reach their maximum storage capacities at different times during a runoff event (e.g., Chu et al., 2013; Leibowitz et al., 2016; Shook et al., 2013). When all depressions are filled (Figures 1c and 2 point c), all depressions are connected to the drainage system, so the contributing fraction of the basin is 1. As water is removed from a topographic depression by evapotranspiration and/or infiltration, the depression immediately stops contributing outflow because the storage of water in the depression is less than its maximum capacity (e.g., Shook & Pomeroy, 2011; Shook et al., 2013). The water losses due to evapotranspiration have limited spatial variability, meaning that the water storage across all depressions drops below the storage capacity at approximately the same time (Shook & Pomeroy, 2011; Shook et al., 20132021). Thus the fraction of depressions connected to the outlet is zero (Figures 1d and 2 point d). As no water is flowing, the contributing fraction of the basin is also zero. The distribution of filled depressions is changed by the removal of water because water removal, unlike addition, is not affected by the catchment areas of the depressions. Therefore subsequent refilling of the depressions will result in contributing-fraction curves which differ from the original filling curve shown in Figure 2 (Shook et al., 2013).

Details are in the caption following the image

Illustration of the hysteretic relationship between storage and contributing area. The points on the plot relate to the four images in Figure 1. Such large-scale hysteretic impacts of prairie depressions are poorly represented (or missing) from conventional hydrological and land models.

The purpose of this technical note is to present simple models to simulate hysteretic relationships between storage and contributing area. We have two primary objectives:
  1. To parameterize the morphology of prairie depressions at local and catchment scales using physically meaningful quantities. At the scale of individual depressions and multidepression complexes, the depression bathymetry is defined directly from estimates of depression area and volume, which can be derived from terrain analysis. At the catchment scale, the bivariate area-volume distributions and the relationships between depression area and catchment area are defined using stochastic models that sample from bivariate lognormal distributions and hence rely on parameters that have a straightforward physical interpretation.

  2. To formulate and evaluate simple mechanistic models to simulate the hysteretic storage dynamics of prairie depressions. We consider both spatially distributed and spatially integrated models. The spatially distributed models (the ensemble-depression model) explicitly simulate storage dynamics for a distribution of discrete depressions. In these spatially distributed models the large-scale storage-contributing fraction relationships are an emergent property of the modeled system. The spatially integrated models (the meta-depression model) parameterize the large-scale storage-discharge relationships. The spatially integrated models are formulated in a way to represent hysteretic behavior. For both models, we define the state equations, the flux parameterizations, and the numerical solution.

The intent in writing this note is to define simple hysteretic models of prairie depressions that can easily be incorporated in conventional hydrological and land models.

The remainder of this paper is organized as follows. Section 2 summarizes methods to parameterize the morphology of prairie depressions, including the simple methods to parameterize the bathymetry of individual depressions and multidepression complexes (Hayashi & Van der Kamp, 2000), as well as methods to characterize the spatial distribution of depressions across the landscape in terms of their bivariate area-volume distributions and the relationships between depression area and catchment area (Shook et al., 20132021). Section 3 presents the ordinary differential equations and flux parameterizations for the simple hysteretic storage-discharge models and Section 4 presents numerical experiments that illustrate the hysteretic behavior for both spatially distributed and spatially integrated models. Finally, Section 5 summarizes our primary contributions.

2 Parameterizing Depressions in Prairie Landscapes

2.1 Definitions

In this note, we use the terminology defined in Shook et al. (2021). We use the term “depression” to refer to any topographic minimum that is capable of holding water, defined by its area, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0003, and volume, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0004. We use the term “pond” to refer to the water within the depression, which is also defined by the pond area, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0005, and pond volume, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0006. Depressions are also characterized by their maximum depth, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0007, also called the sill, which defines the relative elevation where the water in the depression will spill. Ponds are also characterized by the relative elevation of the water surface, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0008, which is the elevation difference between the water surface and the deepest point in the depression. The simplified depression bathymetry is illustrated in Figure 3.

Details are in the caption following the image

Schematic of simplified depression bathymetry. Here hw is the height of the water level, hd is the relative elevation of the depression sill, rw is the distance of the pond perimeter from the center of the depression, and rd is the distance of the depression sill from the center of the depression.

Scaling up from individual depressions to the catchment, it is necessary to distinguish between the contributing fraction and the connected fraction of a catchment. A depression is “connected” if it is full, that is, the volume of water stored in the depression, Vw, is equal to the volume of the depression, Vd, and there is a filled path to the drainage network. Local connectivity hence defines the state of an individual depression—a depression is classified as “connected” if it is capable of providing flow. In this sense, the local depression is classified as “active” (e.g., see Ali et al., 2018 and the extensive literature on percolation theory). A connected depression only contributes flow if there are inflows to the depression and/or if precipitation falls over the depression area. Local connectivity is a necessary but not sufficient condition for the depression to contribute flow.

We define the contributing fraction of a catchment as simply the sum of the basin areas from all depressions that contribute flow. In the analysis that follows, we assume that each individual depression is directly connected to the large-scale river network. This assumption means that we ignore the cascade of depressions examined in previous work (Shook et al., 2013). As was recently shown by Shook et al. (2021), ignoring depression cascades is a reasonable assumption when the number of depressions is large and when the area of the largest depression is a small fraction (<5%) of the total depression area. Our focus on the hysteretic relationships between depression storage and the contributing fraction recognizes that the contributing fraction of a catchment requires connectivity at the scale of individual depressions. Models of discrete depressions ignore the aggregation and disaggregation of ponds which occur as depressions fill and empty. However, models of discrete depressions have been shown to produce very similar estimates of connected/contributing fractions to those of models which can simulate the merging and splitting of ponds Shook et al. (2021).

2.2 Individual Depressions

Following Hayashi & Van der Kamp (2000), we parameterize individual depressions as circular depressions with power law slope profiles. The slope profiles can be defined as follows:
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0009(1)
where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0010 is the relative elevation of the land surface at a distance urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0011 from the center of the depression, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0012 is a reference elevation, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0013 is the distance of the reference elevation from the center of the depression, and p is a dimensionless constant. Since the choice of h0 and r0 is arbitrary, it is convenient to use physically meaningful quantities such as the height, hd, and the radius, rd, of the depression (i.e., the height and radius of the depression sill) in place of h0 and r0.
The area-depth relationships follow from Equation 1. If we let hw denote the elevation of the water surface (e.g., see Figure 3), and given urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0014, then the area of the pond, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0015, is as follows:
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0016(2)
where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0017 is the area of the depression and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0018 is the height of the depression sill. As we will show below, the depression height hd can be obtained given information on Ad and Vd.
The volume-area-depth relationships defined in Hayashi & Van der Kamp (2000) are given as follows:
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0019(3)
where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0020 is the volume of water stored in the depression and η is the dummy variable of integration. The integral in Equation 3 has the analytical solution (Hayashi & Van der Kamp, 2000)
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0021(4)
Since the choice of h0 and r0 is arbitrary, Equation 4 can be simplified by setting h0 = hw and r0 = rw. Applying the exponent rule xa/xb = xab, then urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0022, and.
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0023(5)
where Aw is calculated using Equation 2. Substituting Equation 2 into Equation 5, dividing both sides by hd, and given urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0024, then
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0025(6)
where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0026 is the maximum depth of the depression (i.e., the height of the depression sill), which can be obtained from Equation 5 as urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0027.
To simulate the storage dynamics of depressions it is convenient to use the pond volume as the state variable. Rearranging Equation 6 as
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0028(7)
and substituting Equation 7 into Equation 2, then
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0029(8)

Equation 8 enables converting per-unit-area fluxes from a host land model into volume fluxes based on time-varying changes in pond area.

The presentation in Equations 1-8 is a departure from Hayashi and Van der Kamp (2000), and papers that follow (e.g., Minke et al., 2010; Shook et al., 20132021), that use h0 = 1. The use of physically meaningful quantities such as hd and rd has been derived before (e.g., Brooks & Hayashi, 2002) and simplifies parameterizations of depression bathymetry when information is available on the depression area Ad and the depression volume Vd. The simplified bathymetry described in Equations 1-8 is illustrated in Figure 4.

Details are in the caption following the image

Simplified depression bathymetry calculated using Equations 1-8 for varying values of p.

2.3 The Spatial Distribution of Depressions Across the Prairie Landscape

Large-domain simulations of prairie hydrology entail simulating the storage dynamics for distributions of depressions. This requires spatial information on the depression area, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0030, the depression volume, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0031, the area upland of each depression, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0032, and the parameter p that defines the power law slope profile in Equation 1.

In this study, estimates of Ad, Vd, and Au were derived from a limited-domain 2-m Digital Elevation Model (DEM) based on LiDAR data from the Smith Creek Research basin. The 2-m DEM was processed to breach roads at locations where there were culverts and bridges (Annand, 2022). The processed DEM was then resampled to 10 m, and the Wetland Digital Elevation DEM Ponding Model (WDPM) described by Shook et al. (20132021) was run to define Ad and Vd. The upstream areas Au were also derived from the resampled 10-m DEM using the program “Depression Catchments.” The Fortran source code is available at https://doi.org/10.5281/zenodo.7126668. This terrain analysis delineated 48,295 depressions with information on Ad, Vd, and Au. This derived data set of prairie depressions is publicly available at https://doi.org/10.5281/zenodo.7008095.

Figure 5 illustrates the variability in Ad, Vd, and Au. Figure 5 (left plot) illustrates the area-volume relationships obtained from WDPM for the Smith Creek Research Basin. Results show that Vd is consistently lower than Ad, indicating shallow depressions with mean depths less than one m. The terrain analysis with WDPM can be accomplished using other models such as the Prairie Region Inundation MApping (PRIMA) (Ahmed et al., 2020), or from GIS programs which can fill depressions using algorithms such as that of Planchon and Darboux (2001). Similar approaches are used in the dynamic models described by Chu et al. (2013) and Chu (2017) that explicitly simulate puddle-to-puddle filling, spilling, splitting, and merging processes through cell-to-cell and puddle-to-puddle flow routing.

Details are in the caption following the image

Relationships between Ad and Vd (left plot) and between Ad and Ab (right plot) obtained from analysis of a LiDAR-derived Digital Elevation Model for the Smith Creek Research Basin. The estimates of Ad and Vd in the left plot were obtained from Wetland Digital Elevation DEM Ponding Model.

Figure 5 (right plot) illustrates relationships between Ad and Ab for the Smith Creek Research Basin, where Ab = Au + Ad. It is apparent from Figure 5 that the slope of the relationship between urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0033 and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0034 is less than one, meaning that small depressions occupy smaller fractions of their local catchments than do large depressions. As relative upland area is larger for small depressions, a given runoff depth on small depressions translates to larger depths of water in them than in large depressions. These relationships were previously reported by Shook et al. (2013), who documented similar scaling exponents for several prairie basins. Defining Ad, Vd, and Ab from limited-domain LiDAR data enables explicitly simulating individual depressions.

The general relationships illustrated in Figure 5 are shaped by a myriad of factors. One key issue is the filling and drainage of small depressions. For example, Dumanski et al. (2015) documents that land management practices in the Smith Creek research basin has resulted in a 58% decrease in the extent of ponds and almost an 8x increase in channel length. Substantial losses of wetlands are reported throughout the Prairie Pothole region (e.g., Bartzen et al., 2010; Baulch et al., 2021; Breen et al., 2018; Johnston, 2013; Oslund et al., 2010; Van Meter & Basu, 2015). Drainage is often not uniform, with small depressions being drained or infilled preferentially by farmers (Annand, 2022), which changes their relationship between Ad and Vd. Another key issue is the control of terrain on depression storage, that is, how the terrain controls the frequency distributions of Ad and Vd. Shook et al. (2021) showed that filling curves of contributing fraction versus depressional storage varied greatly between two basins having very different topographic types. More generally, Wolfe et al. (2019) introduced a classification of Canadian Prairie basins based largely on their topographies and distributions of depressional storage, and Pavlovskii et al. (2020) demonstrated how different types of glacial deposits in the prairie pothole region affect the runoff retention and storage capacity of Prairie depressions. Pavlovskii et al. (2020) suggest that the association between the type of surficial deposits and the storage capacity of depressions could provide a way to provide information on the frequency distributions of Ad and Vd in regions where LiDAR-derived DEMs are not available. Recently, Spence et al. (2022) showed that basins within a given class could be modeled with a single generic “virtual” basin model forced with local meteorological variables, demonstrating the importance of local topography in controlling basin hydrological responses. Taken together, the human and physical impacts on topographic depressions must be considered to provide large-domain geospatial information on the topographic parameters that affect hydrologic processes in prairie environments.

In this paper, we use a simple stochastic model to parameterize the frequency distributions of depression area, Ad, depression volume, Vd, and catchment area, Ab. Our stochastic model samples from the bivariate lognormal distributions that define the relationships between Ad and Vd and between Ad and Ab (see Appendix A). There are many other stochastic models that could be implemented here, for example, using alternative parametric probability distributions, such as the Generalized Pareto distribution, or copulas, as described by Shook et al. (2021). It is also possible to use regionalized regression relationships to estimate the frequency distributions of Ad and Vd (e.g., Wiens, 2001). We use the bivariate lognormal model here both because of its simplicity and because the parameters of lognormal probability distributions are easy to interpret. Such stochastic models can be applied to different landscape types, and in areas with different land management (e.g., wetland filling/drainage and restoration), to provide the information necessary for global application of the models introduced in this paper. Appendix A provides parameter values for the lognormal stochastic model developed for the Smith Creek research basin to provide initial estimates of the frequency distributions of Ad, Vd, and Ab in regions where high-resolution DEM analyses are not available.

3 Simple Hysteretic Models of Prairie Depressions

3.1 Model Forcing

The large-scale effects of prairie depressions can be simulated using information from a host land model. The typical (and desired) approach to model coupling requires that the computational elements in the host model (grid cells or subbasins) are disaggregated into at least one upland landscape type (of area urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0035) and a depression landscape type (of area urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0036). This coupling approach uses the following information from the host land model: surface runoff from the upland land area, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0037, surface runoff over the depression land area, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0038, rain + melt over the pond surface, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0039, and the losses of water from the depression due to evapotranspiration, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0040. An overbar is used to describe these flux terms because the host land model approximates the average model fluxes over the discrete time interval Δt.

It is also possible to simplify the approach to model coupling and force the depression models with runoff, rainfall, and potential evapotranspiration from the host land model. In this simplified approach, it is assumed that runoff includes both the runoff from upland and depression areas. Conceptually, this means that runoff from the host land model represents the aggregate impacts of spatial heterogeneity in snow accumulation and melt processes, including the preferential snow accumulation in topographic depressions. This simplified model integration approach also assumes that potential evapotranspiration is a good proxy for free water evaporation from the pond surface and that the water in the pond does not contribute greatly to transpiration. It is possible that the assumptions underlying this simplified coupling approach may be reasonable for numerous host land models, such that the development here provides a general model-agnostic solution to simulate hysteretic storage-discharge dynamics. In the developments that follow, we focus on the “typical” approach to coupling (described in the paragraph above) because it is general—this typical approach to coupling can easily be simplified if not all of the fluxes are available from the host land model.

3.2 General Model Formulation

The prairie depressions we consider here, and illustrated in Figures 1 and 2, are a particular class of depressional wetlands that are not fed or supported by groundwater (Fan & Miguez-Macho, 2011; Golden et al., 2014). Much of the landscape of the Canadian Prairies is underlain by deep deposits of glacial till (Christiansen, 1979) which generally has very low hydraulic conductivities and therefore very low rates of infiltration (van der Kamp & Hayashi, 2009). Moreover, the freezing of soils to depths of ∼1.5 m controls the rate of infiltration in uplands and restricts the groundwater infiltration rates under depressions by the downward progression of the thawing front (Hayashi et al., 2003). Hayashi et al. (1998) found the annual recharge of a small depression to be between 1 and 3 mm, which is less than 1% of the mean annual precipitation of 360 mm. When conditions are sufficiently wet, shallow subsurface flows may occur among depressions (Brannen et al., 2015). However, deep groundwater inflows to or discharges from depressions are uncommon in most Prairie basins (Hayashi et al., 2016), and, consequently, base flows are absent for many prairie streams. Consequently, coupled surface-subsurface flow models (e.g., Restrepo et al., 1998; Wilsnack et al., 2001) or more conceptual models of groundwater-wetland interactions (e.g., Evenson et al., 2016; Evenson, Jones, et al., 2018; Fan & Miguez-Macho, 2011; Liu et al., 2008) are less relevant in the Canadian Prairies. In this paper, we use very simple models of focused infiltration at the bottom of topographic depressions and instead focus on fill-and-spill processes. The model representations of fill-and-spill dynamics we introduce here will still be useful for groundwater-supported wetlands, in which case these fill-and-spill models must be coupled with a more detailed representation of surface-subsurface exchange.

The cold climate of the Prairies strongly influences the region's hydrology. About one third of the region's annual precipitation (300–550 mm) has historically fallen as snow, with most of the surface runoff due to the spring melt of the accumulated snowpacks (Shook & Pomeroy, 2012). The depressions are filled with water by direct precipitation, runoff from the adjacent uplands, and by trapping blowing snow.

Given the forcing from the host land model defined in Section 3.1, along with the limited magnitude of groundwater inflows, a general state equation to simulate the time evolution of volume storage in an individual depression is as follows:
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0041(9)
with
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0042(10)
where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0043 is the volume of water in the pond, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0044 is the sum of water inputs to the pond, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0045 is the losses of water from the pond due to evapotranspiration, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0046 is the infiltration losses from the bottom of the pond, and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0047 is the pond outflow.
The volume fluxes from the host land model are defined as follows:
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0048(11)
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0049(12)
where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0050 is the surface runoff from the upland land area, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0051 is the surface runoff over the depression land area, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0052 is the rain + melt over the pond surface, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0053 is the losses of water from the depression due to evapotranspiration, and Aw is the pond area. Note that the time-varying pond area Aw, as defined in Equation 8, is computed as a function of pond volume. The volume flux Qdi defined in Equation 11 is the water that is available to spill.
The infiltration losses from the bottom of the depression, Qdix, are parameterized here as a linear reservoir:
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0054(13)
where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0055 is the time constant for the linear reservoir.
The outflow from a depression can be parameterized as follows:
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0056(14)
where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0057 is the fraction of inflow that spills and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0058 is the large-scale contributing fraction of a catchment. For spatially distributed models the depression outflow is simulated explicitly for individual depressions, and ϕd is formulated as a step function at the volume threshold Vd. In this case, ϕd describes the fill-and-spill process, where the depression outflow is simply equal to the net forcing if the depression is full. For spatially integrated models, Fc is a smooth function that parameterizes the large-scale contributing fraction of a catchment. We will consider both of these cases in Section 3.3.

3.3 Modeling Alternatives

3.3.1 Spatially Distributed Dynamics: The Ensemble-Depression Model

We first consider a spatially distributed model that explicitly simulates the storage dynamics for a distribution of discrete depressions (the ensemble-depression model). The ensemble-depression model is similar to the Pothole Cascade Model (PCM) described by Shook et al. (2013), where each individual depression is simulated using the volume-area-depth relationships of Hayashi & Van der Kamp (2000). The difference from PCM is that the ensemble-depression model assumes that each depression, once full, is connected to the large-scale river network (i.e., the ensemble-depression model does not consider the cascade of water through a sequence of depressions). As noted earlier, we ignore depression cascades because recent work shows that flow through a sequence of depressions has a limited impact on storage-discharge relationships at larger spatial scales (Shook et al., 2021), provided that the largest depression is only a small fraction of the total depressional area. Therefore, the ensemble-depression model is restricted to this case. In these spatially distributed models, the large-scale storage-contributing fraction relationships are an emergent property of the modeled system (Shook et al., 2015).

The key difference between the spatially distributed ensemble-depression model (described here) and the spatially integrated meta-depression model (described in Section 3.3.2) is the parameterization of the depression outflow. In the ensemble-depression model, the outflow from an individual depression, that is, Qdo in Equation 10, can be parameterized as a fill-and-spill process, which is as follows:
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0059(15)

In Equation 15, the depression outflow is simply equal to the net forcing if the depression is full.

The ensemble-depression model relies on the morphology of depressions described in Section 2, including the simple methods to parameterize the bathymetry of individual depressions (Hayashi & Van der Kamp, 2000), as well as methods to characterize the spatial distribution of depressions across in a region in terms of their bivariate area-volume distributions and the relationships between depression area and catchment area (Shook et al., 20132021). The spatial distribution of depressions can be defined using either terrain analysis or stochastic models (see Appendix A).

3.3.2 Spatially Integrated Dynamics: The Meta-Depression Model

Our methods to simulate the storage-discharge dynamics for a meta depression follows the development of large-scale flux parameterizations in many conventional hydrological and land models. The challenge of defining large-scale flux parameterizations is often referred to as the problem of “closure” (e.g., Beven, 2006) where large-scale flux parameterizations are needed to close catchment-scale water budgets. Here, we are interested in developing spatially integrated models where we directly parameterize the large-scale hysteretic relationships between storage and contributing fraction.

A previous attempt to parameterize the large-scale hydrological impacts of prairie depressions, PDMROF (M. Mekonnen et al., 2014), provides a convenient starting point for the spatially integrated model. PDMROF uses the Probability Distributed Model (PDM) described by Moore (2007) to describe the storage dynamics of a probability distribution of depressions with different storage capacities. PDMROF uses catchment-averaged storage to determine the fraction of depressions where depression storage is at capacity, and hence parameterize the contributing fraction of the catchment (M. Mekonnen et al., 2014). PDMROF has several limitations (see Section 5), most notably, that the probability distribution of interacting storages in PDMROF provides a poor representation of the spatial heterogeneity of isolated depressions, and that PDMROF does not represent hysteresis in the relationships between storage and contributing fraction. Many of the limitations with PDMROF can be addressed using the methods described here.

We resolve the conceptual limitations of PDMROF by simply parameterizing the connected/contributing fraction of the catchment as an increasing function of total depression storage (i.e., we directly parameterize the large-scale emergent behavior). We use the same equations as in Moore (2007) but we do not rely on their probabilistic interpretation. Specifically, we parameterize the total basin storage as a “meta” depression, where the bathymetry of the meta depression is defined based on the spatial mean depression area, Ad, and depression volume, Vd, and the simple volume-area-depth relationships of Hayashi & Van der Kamp (2000) that are described in Equations 1-8. In this meta-depression model, the outflow from the depression Qdo is computed using the contributing fraction parameterizations of Moore (2007) as follows:
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0060(16)
where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0061 is the contributing fraction, defined as follows:
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0062(17)
and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0063 is the volume of water in the meta depression at the start of a fill period, and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0064 defines the shape of the contributing fraction curve.

Values of Vd can be estimated from filling DEMs as described in Section 2.3. Values of b can be estimated by filling DEMs and calculating their connected/contributing fractions using programs such as WDPM and PRIMA. Alternatively, b can be estimated by filling sets of discrete depressions, as will be demonstrated in Section 4. Drainage or restoration of depressions will obviously affect the value of Vd. Because the drainage (or restoration) of depressions is often nonuniform, as described in Section 2.3, the value of b may also be affected.

Hysteretic effects are simulated by treating urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0065 as a dynamic (time-varying) parameter:
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0066(18)
where the superscripts n and n + 1 denote the start and end of the time interval (tn+1 = tn + Δt).

Equation 18 sets urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0067 at every time step during depletion periods, and, as such, the contributing fraction parameterization in Equation 17 simulates the rapid decrease in connectivity that occurs when evapotranspiration and/or infiltration losses decrease the storage of depressions below their capacity. Moreover, in Equation 18 the urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0068 parameter is not updated during runoff events, and hence Equation 17 simulates the gradual increase in connectivity during filling periods. Hysteretic functions similar to Equation 18 have been used in other hydrological models (e.g., see Tritz et al., 2011).

3.4 Numerical Solution

A general solution for Equation 9 can be obtained using implicit Euler methods (also known as backward Euler methods). Using the superscripts n and n + 1 denote the start and end of the time interval (i.e., tn+1 = tn + Δt), the implicit Euler method finds the value of urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0069 such that urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0070. The Newton-Raphson solution of the implicit Euler method can be formulated as follows:
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0071(19)
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0072(20)
Where ζ is a trial value of Vw and the superscript m denotes the iteration index (not to be confused with the time index n). The Newton-Raphson solution is initialized as urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0073 and iterations continue until the absolute value of the residual, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0074, is less than the criteria for convergence (here the criteria for convergence is defined as 10−6 m3). The flux derivative dg/ can be computed either numerically using finite differences, i.e., urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0075), or analytically. In the model simulations presented here, we obtain dg/ analytically. The iteration increments Δζm+1 in Equation 19 are constrained with bracketing and bisection to handle overshoot.

The implementations of Equations 19 and 20 differ slightly for the spatially distributed and spatially integrated models because of the differences in the methods used to estimate Qdo. We will address the specific implementation details for these model variants in the next sections.

3.4.1 The Ensemble-Depression Model

It is desirable that the numerical solution of Equation 9 provides a good representation of the threshold fill-and-spill behavior defined in Equation 15. While it is possible to define the step function in Equation 15 using a continuously differentiable logistic smoother over a very small range of Vw at Vd, for example, see the logistic smoothers defined by Kavetski and Kuczera (2007) and implemented by Clark et al. (2008), these smoothing approaches have two limitations: First, the logistic smoothers allow for a small amount of outflow when the depression is lower than its capacity, complicating calculations of contributing fraction; and second, the logistic smoother increases the nonlinearity of the flux equations, meaning that the solver may take longer to converge.

Here, we use an operator splitting approach, where we provide an intermediate solution with outflow excluded. The depression spills if the intermediate solution (without outflow) exceeds storage capacity. Excluding the depression outflow Qdo from Equation 10 provides the modified flux parameterization:
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0076(21)
Given Equation 21, the implicit Euler method in Equations 19 and 20 is used to estimate the intermediate solution where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0077. Depression outflow is then estimated as follows:
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0078(22)
and the state update obtained as follows:
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0079(23)

Equations 21-23 hence preserve the threshold behavior in Equation 15.

3.4.2 The Meta-Depression Model

The use of the implicit Euler solution for the meta-depression model requires smoothing the discontinuity in the fractional contributing area parameterization in Equation 17. To this end, we use the quadratic smoothers defined by Kavetski and Kuczera (2007) to smooth the discontinuity of Fc at urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0080, defining urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0081 as a smoothed value of the max function
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0082(24)
with ms a smoothing parameter (here ms = 10−4). The value of ms is selected such that the smoothed flux parameterization is visually indistinguishable from the original flux parameterization over the full range of the contributing fraction. The contributing fraction can then be given as
urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0083(25)

The smoothed contributing fraction parameterization is illustrated in Figure 6. By substituting Equation 25 in Equation 16, the direct application of the implicit Euler method in Equations 19 and 20 is possible because all of the flux equations in the meta-depression model are formulated as smooth continuously differentiable functions of pond volume.

Details are in the caption following the image

Illustration of the smoothed contributing fraction parameterization defined in Equation 25 using the parameters urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0084 = 0.2. and Vd = 1. The left plot shows the parameterization over the full range of storage and the right plot restricts attention to the shaded area in the left plot near the urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0085 threshold.

4 Simulations of Storage Dynamics and Fractional Contributing Area

We first evaluate the filling behavior for an ensemble of depressions in order to evaluate how depression complexes shape the contributing fraction of a catchment at large spatial scales. In this experiment, we add an arbitrary runoff depth to an ensemble of depressions, convert the runoff depth to runoff volume based on the catchment area of each depression, and compute the total contributing area of depressions that are at capacity (i.e., where Vw = Vd). These filling experiments are performed for a set of 48,295 real depressions as well as for 10,000 realizations of 1000-depression complexes generated using the stochastic methods defined in Appendix A. The results presented in Figure 7 reveal simple large-scale relationships between fractional pond volume and fractional contributing area, supporting the large-scale parameterizations of fractional contributing area defined in Equations 17 and 25. Similar large-scale relationships were documented previously by Shook et al. (2021).

Details are in the caption following the image

Illustration of the filling behavior for an ensemble of depressions, showing the filling behavior for the 48,295 real depressions in the Smith Creek research catchment (solid line) and the probability distribution of filling behavior obtained by using simulating 10,000 realizations of 1,000-depression complexes (shading). The left plot shows filling behavior where all depressions are initialized as empty and the right plot shows filling behavior when depressions are initialized at a water level 200 mm below the depression sill.

Next, we evaluate the storage dynamics and contributing area simulations from the ensemble- and meta-depression models. To this end, we use a single realization of the synthetic hydrological time series generated using the methods described in Appendix B, where the example time series (top plot of Figure 8) are characterized by a runoff pulse at the start of the time period, representing snowmelt runoff, and intermittent precipitation throughout the time series. Following Shook and Pomeroy (2011) and (Shook et al., 20132021), the ensemble-depression model (second and third plots in Figure 8) is run using the parameter p = 1.72 to define the shape of the depression bathymetry as defined in Equations 1-8. We use the parameter τdix = 0.01 to define the time constant for infiltration defined in Equation 13. The simulations from the ensemble-depression model in Figure 8 illustrate the hysteretic behavior defined by Shook et al. (2013) where there is a cessation in connectivity in depletion periods. The meta-depression model (fourth plot in Figure 8) is run using the same parameters in the ensemble-depression model (p = 1.72 and τdix = 0.01), and with the parameter b = 1.5 to define the shape of the contributing fraction curve defined in Equations 17 and 25 (and illustrated in Figure 7). Similar to the ensemble-depression model, the meta-depression model simulates the cessation of connectivity during depletion periods. An interesting difference between the ensemble-depression model and the meta-depression model is that the ensemble-depression model has a more gradual cessation of connectivity near the end of the snowmelt period—this occurs because differences in catchment area among the depressions in the ensemble can cause differences in timing in their switching between filling and depletion.

Details are in the caption following the image

Illustration of the behavior of the ensemble-depression model and the meta-depression model for a 90-day time period using a single realization of the synthetic hydrological time series generated using the methods described in Appendix B. The synthetic hydrological time series (shown in the top plot) are generated using snowmelt runoff qseas = 75 mm, total precipitation pseas = 250 mm, maximum water evaporation emax = 7.0 mm day−1, and the probability of precipitation occurrence p0 = 0.4. The second and third plots show simulations from the ensemble-depression model, illustrating the probability distribution of pond volume (shading) and the fractional contributing area (bars). The second plot shows simulations for the 48,295 real depressions and the third plot shows simulations for a single stochastic complex of 1,000 depressions. The fourth plot shows simulations for the meta-depression model, illustrating the fractional pond volume (line) and the fractional contributing area (bars).

Finally, we evaluate the hysteretic behavior of the ensemble-depression and meta-depression models. To this end, we run the ensemble-depression model using the same parameters defined above (with a complex of 1,000 depressions) but with 10,000 different depression complexes (generated using the methods in Appendix A) and 10,000 different realizations of synthetic forcing (generated using the methods defined in Appendix B). We also run the meta-depression model using the same parameters defined above and the same 10,000 realizations of synthetic forcing. In both the ensemble-depression and meta-depression models, the depressions are initialized at a range of initial pond volumes. The relationships between fractional pond volume and fractional contributing area presented in Figure 9 are similar for the ensemble-depression and the meta-depression model, where both models simulate a range of different fractional volume and fractional contributing area combinations. Note that nonhysteretic models such as PDMROF will always produce single valued functions of Fc will in their meta-depression simulations. Note also that the meta-depression model does not simulate pond volumes near capacity because the depression outflow has a greater compensation for depression inflows as the fractional pond volume approaches one.

Details are in the caption following the image

Relationships between fractional pond volume and fractional contributing area for the ensemble-depression model (left plot) and the meta-depression model (right plot). Note that nonhysteretic models such as PDMROF will always produce single-valued functions for Fc in their meta-depression simulations.

5 Summary and Discussion

The purpose of this technical note is to present simple models to simulate hysteretic relationships between storage and contributing area. We consider both spatially distributed and spatially integrated models. The spatially distributed models (the ensemble-depression model) explicitly simulate storage dynamics for a distribution of discrete depressions. In these spatially distributed models, the large-scale storage-contributing fraction relationships are an emergent property of the modeled system. The spatially integrated models (the meta-depression model) parameterize the large-scale storage-discharge relationships. The spatially integrated models are formulated in a way to represent hysteretic behavior. The intent in writing this note is to define simple hysteretic models of prairie depressions that can easily be incorporated in conventional hydrological and land models. In the simplest case, the models defined here can be forced by runoff and free-water evaporation fluxes from a conventional hydrological model.

In this note, we make two main contributions:
  1. We parameterize the morphology of prairie depressions at local and catchment scales using physically meaningful quantities. At the scale of individual depressions and multidepression complexes, we define the depressions based on the maximum depth of water in the depression (the height of the depression sill) and the radius of the depression. This approach simplifies parameterizations of depression bathymetry when information is available on the depression area and the depression volume. We also provided more complete derivations of the depth-area-volume relationships to simplify application of these relationships across a myriad of models. At the catchment scale, we present simple stochastic models to simulate the spatial distribution of prairie depressions. The simple stochastic models sample from bivariate lognormal distributions and hence rely on parameters that have a straightforward physical interpretation. The stochastic models complement the recent work by Shook et al. (2021) and can be used in other regions where detailed analysis of depressions is not available.

  2. We formulate simple mechanistic models to simulate the hysteretic storage dynamics of prairie depressions. We define an ensemble-depression model and a meta-depression model, specifying the state equations, the flux parameterizations, the numerical solution, and methods to couple these simple hysteretic models with conventional hydrological models. We demonstrate that, in general, the ensemble-depression and meta-depression models have similar large-scale behavior. An interesting difference between the ensemble-depression model and the meta-depression model is that the ensemble-depression model has a more gradual cessation of connectivity near the end of the snowmelt period—this occurs because differences in catchment area among the depressions in the ensemble can cause differences in timing in their switching between filling and depletion. The ordinary differential equations to describe the hysteretic storage dynamics of prairie depressions were never defined in previous papers (e.g., Shook et al., 20132021; Shook & Pomeroy, 2011) and the numerical formulations presented here provide a stronger basis for further model development.

The ensemble-depression and meta-depression models defined here resolve many of the conceptual limitations with previous attempts to parameterize the large-scale hydrological impacts of prairie depressions (i.e., the PDMROF model described by M. Mekonnen et al., 2014). First, PDMROF does not represent the relationships between the depression catchment areas and the depression areas that were described in Section 2.3, for example, see Figure 5. PDMROF fills all depressions equally, applying the same depth of water to each depression. Second, PDMROF uses a single probability distribution to describe the depression volumes, their areas, and their catchment areas. This is obviously incorrect, due to the nonlinear relationships among the variables defined in Sections 2.2 and 2.3. Third, during depletion periods, the PDM approach redistributes basin-average storage to storages of different size. Conceptually, this means that water is unrealistically redistributed among geographically isolated depressions, even when the depressions are below their storage capacity. It also means that the model is unable to reproduce the documented effects of hysteresis caused by the filling and emptying of depressions of varying sizes (see Figure 2). Finally, the contributing fraction is based on a single-valued function of catchment-average storage, meaning that PDMROF cannot represent the abrupt decrease to zero of the connected/contributing area fraction due to evaporation and/or infiltration. The limitations of PDMROF require that its parameters be obtained by calibration (M. Mekonnen et al., 2014), meaning that the scales of its parameter values are constrained to those of hydrometric gauges.

The deficiencies in existing meta-depression models such as PDMROF can be resolved using the developments in this paper. If information is required on the probability distribution of individual depressions, then this information can be obtained using the ensemble-depression model. If information is only required on the large-scale hydrological impacts of depressions, then this information can be obtained using the meta-depression model. Future work is needed to incorporate these simple hysteretic modeling approaches in different hydrological and land models and evaluate the extent to which these hysteretic models improve large-domain simulations of hydrological processes in prairie environments.

Acknowledgments

This work was completed as part of the Core Modeling theme in the Global Water Futures (GWF) program. We thank Holly Annand for providing the high-resolution LiDAR DEM for the Smith Creek Research Basin that was used to estimate Ad, Vd, and Ab. We also thank Louise Arnal for her help producing Figure 1.

    Appendix A: Stochastic Simulations of Ad, Vd, and Au

    The stochastic simulations of depression area, Ad, depression volume, Vd, and upslope area, Au, are obtained by sampling from the joint distribution of Ad and Vd and the joint distribution of Ad and Au. We do not explicitly simulate the dependence between Vd and Au.

    The variables Ad, Vd, and Au are parameterized using a three-parameter lognormal distribution (LN3). The probability density function urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0086 and the cumulative distribution function urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0087 for the LN3 distribution is defined as follows:
    urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0088(A1)
    urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0089(A2)
    where χmin is the minimum value of χ and μ and σ are the mean and standard deviation parameters of the lognormal probability distribution. We use the LN3 distribution in order to impose minimum values for Ad, Vd, and Au (here we set urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0090 50 m2, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0091 0.1 m3, and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0092 0.0001 m2). Setting a minimum value for Au is necessary for the rare case when the area of the depression is equal to the area of the catchment. The bivariate stochastic model requires the parameters urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0093, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0094, and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0095 to describe the probability distribution of depression area, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0096, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0097, and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0098 to describe the probability distribution of depression volume, and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0099, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0100, and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0101 to describe the probability distribution of upstream area, along with the parameters urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0102 and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0103 to define the correlation between urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0104 and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0105 and between urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0106 and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0107, respectively.
    In order to produce a probability distribution with the desired mean μχ and variance σχ, the parameters of the LN3 distribution are estimated as (Burges et al., 1975) follows:
    urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0108(A3)
    urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0109(A4)
    where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0110 and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0111 are the mean and standard deviation of the original (untransformed) data.
    The bivariate stochastic simulations are generated as follows. First, we represent the normalized depression area as follows:
    urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0112(A5)
    where wa,i defines the ith independent random draw from a standard-normal distribution. Next, we obtain a correlated random number from a standard normal distribution, zv,i, to represent the normalized depression volume, as follows:
    urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0113(A6)
    where, as in Equation A5, wv,i defines the ith independent random draw from a standard-normal distribution. Similarly, we obtain a correlated random number from a standard normal distribution, zu,i, to represent the normalized upstream area as follows:
    urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0114(A7)
    where wu,i defines the ith independent random draw from a standard-normal distribution. The stochastic simulations of Ad, Vd, and Au are then provided as follows:
    urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0115(A8)
    urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0116(A9)
    urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0117(A10)

    Using the data from the Smith Creek research basin, the parameters are urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0118 = 5.63, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0119 = 3.62, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0120 = 7.85, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0121 = 1.91, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0122 = 2.16, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0123 = 1.39, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0124 = 0.95, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0125 = 0.70, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0126 = 50, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0127 = 1, and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0128 = 0.0001. Note that in these stochastic models, we simulate the upstream area, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0129 and calculate the basin area as a postprocessing step, that is, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0130.

    The use of this simple bivariate stochastic model to generate a statistical ensemble of depressions is illustrated in Figure A1. It is apparent from Figure A1 that the bivariate model captures the general relationships between Ad and Vd and between Ad and Ab. This stochastic approach enables large-domain model simulations where detailed DEM analyses are not available.

    Details are in the caption following the image

    The use of a simple bivariate stochastic model to generate the spatial information on topographic depressions and their catchments. The left panel shows stochastic simulations of Ad and Vd and the right panel shows stochastic simulations of Ad and Ab, where Ab = Au + Ad.

    Appendix B: Synthetic Hydrological Time Series

    The numerical experiments in this paper rely on synthetic hydrological time series in order to mimic the information that is available from a host land model. The stochastic model developed here is very simple and only intended to be used for demonstration purposes. We rely on synthetic hydrological time series for the numerical experiments in this paper in order to present a general demonstration of model behavior and enable experimenting with many different realizations of the hydrological time series.

    In the stochastic forcing model, we generate time series of surface runoff over the upslope land areas, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0131, rain + melt over the pond surface, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0132, and the losses of water from the depression due to evapotranspiration, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0133. As defined in Equation 11, the ensemble- and meta-depression models also require information on surface runoff from the dry portion of the depression, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0134, and we assume here that urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0135.

    The hydrological time series are generated as follows. The temporal distribution of surface runoff is generated using a two-parameter lognormal distribution as follows:
    urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0136(B1)
    where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0137 is the length of the time interval that is used to discretize the probability distribution, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0138 is the elapsed time since the start of the simulation, and μt and σt are the parameters of the lognormal distribution, defined as μt = 1.5 and σt = 0.5. The runoff time series is then generated as follows:
    urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0139(B2)
    where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0140 is the total seasonal runoff, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0141 is the runoff coefficient (here rc = 0.05), and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0142 is the precipitation rate. Precipitation is generated by sampling from an exponential distribution with scale parameter λp as follows:
    urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0143(B3)
    where p0 is the probability of precipitation occurrence, u0,i and up,i are the ith independent realization from a uniform distribution that are used to generate estimates of precipitation occurrence and the precipitation rate, respectively. In Equation B3 we make use of the analytical quantile function for the exponential distribution (the inverse cumulative distribution function), such that urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0144 on time intervals that precipitation occurs. The scale parameter λp = Δtnp/pseas, where np is the number of time intervals with precipitation and pseas is the seasonal precipitation total. In Equation B3, precipitation is generated for each time interval independently, so Equation B3 is most suitable for use when Δt = 1 day. The rain + melt over the pond area is then given simply as follows:
    urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0145(B4)
    Finally, pond evaporation is defined as follows:
    urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0146(B5)
    where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0147 is the seasonality of pond evaporation, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0148 is the maximum evaporation rate. In Equation B5 urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0149, where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0150 is the time since the winter solstice.

    The synthetic hydrological time series are generated by sampling seasonal statistics from uniform distributions (see Figure B1). Sampling the seasonal statistics is a convenient way to maintain seasonality and persistence in the generated time series of runoff and evapotranspiration.

    Details are in the caption following the image

    Synthetic hydrological time series generated by sampling seasonal statistics from uniform distributions, where urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0151, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0152, urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0153, and urn:x-wiley:00431397:media:wrcr26352:wrcr26352-math-0154.

    Data Availability Statement

    The derived data set of prairie depressions for the Smith Creek Research Basin is publicly available at https://doi.org/10.5281/zenodo.7008095.