The Numerical Formulation of Simple Hysteretic Models to Simulate the LargeScale Hydrological Impacts of Prairie Depressions
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 ensembledepression model) explicitly simulate storage dynamics for a distribution of discrete depressions. In these spatially distributed models, the largescale storagecontributing fraction relationships are an emergent property of the modeled system. The spatially integrated models (the metadepression model) parameterize the largescale storagedischarge 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 ensembledepression and metadepression models have similar largescale 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 largescale 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 & MiguezMacho, 2011; Melton et al., 2013; Wik et al., 2016).
In this note, we focus on the general problem of simulating storagedischarge dynamics in complexes of depressional wetlands (Evenson, Golden, et al., 2018; Fan & MiguezMacho, 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 catchmentscale 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 MiguezMacho (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 MiguezMacho (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 areavolume relationships for the HEW were defined using a powerlaw as A = βV^{α}, where is the wetland area, is the wetland volume, and β and α are coefficients in the areavolume 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 groundwaterwetland interactions (subsurface inflows and wetland seepage), and improving simulations of evapotranspiration (Evenson et al., 2016; Evenson, Jones, et al., 2018). Recent largedomain applications of the depressionintegrated model for the Upper Mississippi River basin demonstrate that including depressions in the model substantially improves largedomain model simulations (Rajib et al., 2020).
Other model development efforts focus on how the thresholddriven 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 fillandspill 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; Trompvan Meerveld & McDonnell, 2006a, 2006b).
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., 2013, 2021). 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 contributingfraction curves which differ from the original filling curve shown in Figure 2 (Shook et al., 2013).

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 areavolume 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.

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 ensembledepression model) explicitly simulate storage dynamics for a distribution of discrete depressions. In these spatially distributed models the largescale storagecontributing fraction relationships are an emergent property of the modeled system. The spatially integrated models (the metadepression model) parameterize the largescale storagedischarge 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 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 areavolume distributions and the relationships between depression area and catchment area (Shook et al., 2013, 2021). Section 3 presents the ordinary differential equations and flux parameterizations for the simple hysteretic storagedischarge 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, , and volume, . We use the term “pond” to refer to the water within the depression, which is also defined by the pond area, , and pond volume, . Depressions are also characterized by their maximum depth, , 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, , 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.
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, V_{w}, is equal to the volume of the depression, V_{d}, 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 largescale 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
Equation 8 enables converting perunitarea fluxes from a host land model into volume fluxes based on timevarying changes in pond area.
The presentation in Equations 18 is a departure from Hayashi and Van der Kamp (2000), and papers that follow (e.g., Minke et al., 2010; Shook et al., 2013, 2021), that use h_{0} = 1. The use of physically meaningful quantities such as h_{d} and r_{d} has been derived before (e.g., Brooks & Hayashi, 2002) and simplifies parameterizations of depression bathymetry when information is available on the depression area A_{d} and the depression volume V_{d}. The simplified bathymetry described in Equations 18 is illustrated in Figure 4.
2.3 The Spatial Distribution of Depressions Across the Prairie Landscape
Largedomain simulations of prairie hydrology entail simulating the storage dynamics for distributions of depressions. This requires spatial information on the depression area, , the depression volume, , the area upland of each depression, , and the parameter p that defines the power law slope profile in Equation 1.
In this study, estimates of A_{d}, V_{d}, and A_{u} were derived from a limiteddomain 2m Digital Elevation Model (DEM) based on LiDAR data from the Smith Creek Research basin. The 2m 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. (2013, 2021) was run to define A_{d} and V_{d}. The upstream areas A_{u} were also derived from the resampled 10m 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 A_{d}, V_{d}, and A_{u}. This derived data set of prairie depressions is publicly available at https://doi.org/10.5281/zenodo.7008095.
Figure 5 illustrates the variability in A_{d}, V_{d}, and A_{u}. Figure 5 (left plot) illustrates the areavolume relationships obtained from WDPM for the Smith Creek Research Basin. Results show that V_{d} is consistently lower than A_{d}, 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 puddletopuddle filling, spilling, splitting, and merging processes through celltocell and puddletopuddle flow routing.
Figure 5 (right plot) illustrates relationships between A_{d} and A_{b} for the Smith Creek Research Basin, where A_{b} = A_{u} + A_{d}. It is apparent from Figure 5 that the slope of the relationship between and 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 A_{d}, V_{d}, and A_{b} from limiteddomain 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 A_{d} and V_{d}. Another key issue is the control of terrain on depression storage, that is, how the terrain controls the frequency distributions of A_{d} and V_{d}. 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 A_{d} and V_{d} in regions where LiDARderived 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 largedomain 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, A_{d}, depression volume, V_{d}, and catchment area, A_{b}. Our stochastic model samples from the bivariate lognormal distributions that define the relationships between A_{d} and V_{d} and between A_{d} and A_{b} (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 A_{d} and V_{d} (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 A_{d}, V_{d}, and A_{b} in regions where highresolution DEM analyses are not available.
3 Simple Hysteretic Models of Prairie Depressions
3.1 Model Forcing
The largescale 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 ) and a depression landscape type (of area ). This coupling approach uses the following information from the host land model: surface runoff from the upland land area, , surface runoff over the depression land area, , rain + melt over the pond surface, , and the losses of water from the depression due to evapotranspiration, . 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 modelagnostic solution to simulate hysteretic storagedischarge 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 & MiguezMacho, 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 surfacesubsurface flow models (e.g., Restrepo et al., 1998; Wilsnack et al., 2001) or more conceptual models of groundwaterwetland interactions (e.g., Evenson et al., 2016; Evenson, Jones, et al., 2018; Fan & MiguezMacho, 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 fillandspill processes. The model representations of fillandspill dynamics we introduce here will still be useful for groundwatersupported wetlands, in which case these fillandspill models must be coupled with a more detailed representation of surfacesubsurface 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.
3.3 Modeling Alternatives
3.3.1 Spatially Distributed Dynamics: The EnsembleDepression Model
We first consider a spatially distributed model that explicitly simulates the storage dynamics for a distribution of discrete depressions (the ensembledepression model). The ensembledepression model is similar to the Pothole Cascade Model (PCM) described by Shook et al. (2013), where each individual depression is simulated using the volumeareadepth relationships of Hayashi & Van der Kamp (2000). The difference from PCM is that the ensembledepression model assumes that each depression, once full, is connected to the largescale river network (i.e., the ensembledepression 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 storagedischarge 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 ensembledepression model is restricted to this case. In these spatially distributed models, the largescale storagecontributing fraction relationships are an emergent property of the modeled system (Shook et al., 2015).
In Equation 15, the depression outflow is simply equal to the net forcing if the depression is full.
The ensembledepression 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 areavolume distributions and the relationships between depression area and catchment area (Shook et al., 2013, 2021). 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 MetaDepression Model
Our methods to simulate the storagedischarge dynamics for a meta depression follows the development of largescale flux parameterizations in many conventional hydrological and land models. The challenge of defining largescale flux parameterizations is often referred to as the problem of “closure” (e.g., Beven, 2006) where largescale flux parameterizations are needed to close catchmentscale water budgets. Here, we are interested in developing spatially integrated models where we directly parameterize the largescale hysteretic relationships between storage and contributing fraction.
A previous attempt to parameterize the largescale 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 catchmentaveraged 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.
Values of V_{d} 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 V_{d}. Because the drainage (or restoration) of depressions is often nonuniform, as described in Section 2.3, the value of b may also be affected.
Equation 18 sets 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 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
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 Q_{do}. We will address the specific implementation details for these model variants in the next sections.
3.4.1 The EnsembleDepression Model
It is desirable that the numerical solution of Equation 9 provides a good representation of the threshold fillandspill 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 V_{w} at V_{d}, 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.
Equations 2123 hence preserve the threshold behavior in Equation 15.
3.4.2 The MetaDepression Model
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 metadepression model are formulated as smooth continuously differentiable functions of pond volume.
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 V_{w} = V_{d}). These filling experiments are performed for a set of 48,295 real depressions as well as for 10,000 realizations of 1000depression complexes generated using the stochastic methods defined in Appendix A. The results presented in Figure 7 reveal simple largescale relationships between fractional pond volume and fractional contributing area, supporting the largescale parameterizations of fractional contributing area defined in Equations 17 and 25. Similar largescale relationships were documented previously by Shook et al. (2021).
Next, we evaluate the storage dynamics and contributing area simulations from the ensemble and metadepression 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., 2013, 2021), the ensembledepression 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 18. We use the parameter τ_{dix} = 0.01 to define the time constant for infiltration defined in Equation 13. The simulations from the ensembledepression 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 metadepression model (fourth plot in Figure 8) is run using the same parameters in the ensembledepression model (p = 1.72 and τ_{dix} = 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 ensembledepression model, the metadepression model simulates the cessation of connectivity during depletion periods. An interesting difference between the ensembledepression model and the metadepression model is that the ensembledepression 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.
Finally, we evaluate the hysteretic behavior of the ensembledepression and metadepression models. To this end, we run the ensembledepression 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 metadepression model using the same parameters defined above and the same 10,000 realizations of synthetic forcing. In both the ensembledepression and metadepression 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 ensembledepression and the metadepression 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 F_{c} will in their metadepression simulations. Note also that the metadepression 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.
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 ensembledepression model) explicitly simulate storage dynamics for a distribution of discrete depressions. In these spatially distributed models, the largescale storagecontributing fraction relationships are an emergent property of the modeled system. The spatially integrated models (the metadepression model) parameterize the largescale storagedischarge 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 freewater evaporation fluxes from a conventional hydrological model.

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 depthareavolume 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.

We formulate simple mechanistic models to simulate the hysteretic storage dynamics of prairie depressions. We define an ensembledepression model and a metadepression 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 ensembledepression and metadepression models have similar largescale behavior. An interesting difference between the ensembledepression model and the metadepression model is that the ensembledepression 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., 2013, 2021; Shook & Pomeroy, 2011) and the numerical formulations presented here provide a stronger basis for further model development.
The ensembledepression and metadepression models defined here resolve many of the conceptual limitations with previous attempts to parameterize the largescale 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 basinaverage 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 singlevalued function of catchmentaverage 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 metadepression 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 ensembledepression model. If information is only required on the largescale hydrological impacts of depressions, then this information can be obtained using the metadepression 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 largedomain 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 highresolution LiDAR DEM for the Smith Creek Research Basin that was used to estimate A_{d}, V_{d}, and A_{b}. We also thank Louise Arnal for her help producing Figure 1.
Appendix A: Stochastic Simulations of A_{d}, V_{d}, and A_{u}
The stochastic simulations of depression area, A_{d}, depression volume, V_{d}, and upslope area, A_{u}, are obtained by sampling from the joint distribution of A_{d} and V_{d} and the joint distribution of A_{d} and A_{u}. We do not explicitly simulate the dependence between V_{d} and A_{u}.
Using the data from the Smith Creek research basin, the parameters are = 5.63, = 3.62, = 7.85, = 1.91, = 2.16, = 1.39, = 0.95, = 0.70, = 50, = 1, and = 0.0001. Note that in these stochastic models, we simulate the upstream area, and calculate the basin area as a postprocessing step, that is, .
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 A_{d} and V_{d} and between A_{d} and A_{b}. This stochastic approach enables largedomain model simulations where detailed DEM analyses are not available.
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, , rain + melt over the pond surface, , and the losses of water from the depression due to evapotranspiration, . As defined in Equation 11, the ensemble and metadepression models also require information on surface runoff from the dry portion of the depression, , and we assume here that .
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.
Open Research
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.