# HGDMr_examples

#### Kevin Shook

#### 2025-12-08

# 1 Introduction

The Hysteretic and Gatekeeping Depressions Model (HGDM) was developed to model the time-varying effects of depressional storage on the connected/contributing fractions of basins in the Canadian Prairies. The hydrology of this region is dominated by the presence of millions of depressions (“potholes” or “sloughs”) which trap water from precipitation (rainfall, snowmelt) on the depressions and upland runoff into the depressions. When depressions fill to their spill point, additional inputs of water can flow overland to other depressions and may eventually reach a drainage network. Because the landscape is often underlain by deep deposits of glacial till, the infiltration rates from the depressions are often very small. Most of the removal of water from the depressions is usually due to evaporation.

The behaviours of complexes of depressions are very different when filling and emptying. As water is added and the depressions fill,their connected/contributing fraction increases. When water is removed, all depressions will be below their spill elevation, meaning that their connected/contributing fraction is zero. Thus the connected/contributing fractions of depressions are hysteretic withe respect to the volume of water stored in them.

Several models have been developed to simulate the variable contributing fractions of prairie basins including the Wetland DEM Ponding Model (WDPM) Shook et al. (2021b), and the Pothole Cascade Model (PCM) [Shook et al. (2013)] (https://doi.org/10.1002/hyp.9867). The WDPM is too slow to be used as part of a hydrological model, although it can be useful for mapping the extent of water coverage in a region. The Pothole Cascade Model has been shown to work well in modelling prairie basins with variable contributing fractions, but is complex to use in practice. The HGDM was developed to be simple to implement for operational hydrology.

Shook et al. (2021b) demonstrated that when the largest depression in a basin was relatively small (having an area less than ~5% of the total depressional area), its effect on the varying connected/contributing fraction of the basin was relatively small. The hysteretic relationship between the connected/contributing fraction is represented as shown in @fig-hystereis. At point (a) water is removed from the depressions, causing the connected/contributing fraction to be zero (b). As more water is removed, the connected/contributing fraction remains at zero, while the depressional storage decreases (c). As water is added, the contributing fraction increases approximately linearly, on a trajectory to (1,1) (d). The red line 1:1 represents the maximum possible trajectory of increase of the contributing fraction - the modelled increases will never cross this line.

Shook et al. (2021b) showed that the increasing trajectory is not necessarily completely linear and Clark and Shook (2022) showed a method for modelling this, at the cost of greater complexity. However, Shook and Pomeroy (2025) demonstrated that the linear approximation of hysteresis worked well in a model of a depression-dominated prairie basin.

HGDM relationship between water storage and connected/contributing fraction

HGDM uses a single “meta” depression to model all the small depressions, using the hysteretic relationship plotted in @fig-hystereis. An advantage of this method is that it only needs a single set of parameters, which can be determined from GIS, without the need for calibration.

Shook et al. (2021b) also showed that when the area of the largest depression is more than about 5% of the total depressional area, then it results in gatekeeping, where the depression blocks inflows from upstream until it is filled. Because the depression is modelled individually, its behaviour is not hysteretic with respect to the volume of storage.

HGDM combines the hysteretic behaviour of small depressions and the gatekeeping behaviour of a the largest depression (if it is present) in a single model. The basin is modelled as shown in @fig-schematic. The region dominated by small depressions is shaded in gray, and has the hysteretic relationship between water storage and connected/contributing fraction. The region shown in white is without any depressions.

Schematic diagram of prairie basin as modelled by HGDM

The fluxes of water among the landscape units are shown in the flowchart. The upland runoff is partitioned among the “meta” depression, the large depression and directly to the outlet, through the areas without depressions. The outflows from the “meta” depression are divided among the flows to the largest depression (if it exists), and directly to the outlet.

HGDM flowchart

# 2 HGDM

The main function in the package is `HGDM()` which executes the model. The function has many parameters, most of which can be derived from GIS.

## 2.1 Model fluxes

HGDM requires a data frame containing all the model forcings. The interval can be sub-daily or daily. If the interval is sub-daily then the first column must be named “datetime” and must be a POSIXct date-time. If daily values are used, then the first column must be named “date” and needs to be a date object.

The forcings used must be determined by a hydrological model, at the basin scale. The best models for modelling prairie hydrology are CRHM and MESH. Whatever model is used, it must include all the important prairie hydrological processes including erosion, transportation, deposition and sublimation of snow by the wind, snow melt driven by solar radiation and infiltration to frozen soils. If your model does not include all of these processes, then it is invalid in the Canadian Prairies and cannot give good results when forcing HGDM.

The forcings required by the functions are:

- rainfall

The depth of rain (mm) falling on the water (sloughs and lake) in the basin. Note that this does not include the snowfall.

- snowmelt

The depth of snow (mm) melting on top of the water in each interval. This is not the same as the snowfall, which could have occurred months earlier and includes the blowing snow processes as well as the energy balance snow melt processes.

- runoff

This is not the same thing as the streamflow divided by the basin area. It refers to the depth of water (mm) running off the uplands within the basin. In other words, it is the snow melt minus the infiltration to the frozen soils.

- evap

This is the depth of evaporation (mm) from the water in the basin.

Because modelling prairie hydrology is difficult, the PHyDAP project was developed. PHyDAP consists of outputs from CRHM models for the Canadian Prairies. The data are available at https://www.frdr-dfdr.ca/repo/dataset/7ce4bd7a-4bcc-4f8c-8129-32a691f46c8e. The creation of the data sets is described in detail by Shook et al.(2024).

The PHyDAP data consist of exactly the fluxes required by the functions. However, they must be combined into a single data frame with the required names for each of the variables.

The sample data set `daily_7120951600` contains CRHM fluxes for basin 7120951600 modelled using forcings from the ERA5 reanalysis data set. These values will be used for the modelling demonstrated here.

    data(HGDMr)
    summary(daily_7120951600)

    ##       date               rainfall         snowmelt           runoff
    ##  Min.   :1950-01-01   Min.   : 0.000   Min.   : 0.0000   Min.   : 0.0000
    ##  1st Qu.:1967-10-01   1st Qu.: 0.000   1st Qu.: 0.0000   1st Qu.: 0.0000
    ##  Median :1985-07-01   Median : 0.000   Median : 0.0000   Median : 0.0000
    ##  Mean   :1985-07-01   Mean   : 1.101   Mean   : 0.3684   Mean   : 0.1497
    ##  3rd Qu.:2003-04-01   3rd Qu.: 0.350   3rd Qu.: 0.0000   3rd Qu.: 0.0000
    ##  Max.   :2020-12-30   Max.   :75.080   Max.   :51.9995   Max.   :51.7754
    ##       evap
    ##  Min.   :0.000
    ##  1st Qu.:0.000
    ##  Median :0.000
    ##  Mean   :2.055
    ##  3rd Qu.:4.315
    ##  Max.   :7.895

## 2.2 Model parameters

`HGDM` has many parameters, although several of them are optional.

The parameters labelled as being `small_depression` refer to the “meta” depression which represents the small depressions in the basin. The `large_depression` parameters refers to the single large depression, i.e. where the area is > 5% of the total depressional area. If your basin does not have a large slough/lake then the parameters related to it can be omitted.

### 2.2.1 Dimension parameters

The basin area and volume parameters can be determined by GIS using DEMs.

HGDM dimension parameters
Name | Meaning | Units
--- | --- | ---
upland_area | Total area of uplands, which drain to the outlet, small depressions or the large depression | Area units
small_depression_area | Area of small depressions | Area units
large_depression_area | Optional area of large depression. If `0` or `NULL` large depression is not modelled | Area units
area_units | Units of all areas | Must be one of “km2” (default), “ha” or “m2”
max_small_depression_storage | Maximum depth of storage in small depressions (meta depression) | Storage units
max_large_depression_storage | Maximum depth of storage in large depression | Storage units
storage_units | Units of all storage depths | Must be one of “mm” (default), “m”, or “m3”. If a depth is specified then it will be converted to a volume by multiplying by the appropriate area.

The initial state of storage in the basin can be estimated from modelling, and/or field work or remote sensing. The initial connected fraction is generally set to zero, unless you have reason to believe that the storage was increasing before the first time interval.

HGDM initial state parameters
Name | Meaning | Units
--- | --- | ---
initial_small_depression_storage | Initial depth of storage in small depressions | Storage units
initial_large_depression_storage | Initial depth of storage in large depressions | Storage units
small_depressions_initial_connected_fraction | Initial connected fraction (0-1) | dimensionless

### 2.2.2 Flow partitioning parameters

The HGDM flow partitioning parameters split the flows among the destinations as shown in the flowchart. The fractions can be extracted using GIS and DEMs. Remember that

`upland_fraction_to_small + upland_fraction_to_large + upland_fraction_to_outlet = 1`

and

`small_fraction_to_large + small_fraction_to_outlet = 1`

`upland_fraction_to_large` is the region which drains directly into the large depression `without passing though any small depressions`. It is the region directly adjacent to the large depression without any sloughs in it, shown in @HGDMflowchart. This area is typically quite small. Shook et al. (2013) showed that the larger the depression, the smaller this area is relative to the depression’s area.

`small_fraction_to_large` is the fraction of outflows from the small depressions (i.e. the meta depression) which flow into the large depression. This depends on the location of the large depression in the basin. If the large depression is at the top of the basin, then the value of `small_fraction_to_large` will be very small (near zero), and the large depression may never fill. If the large depression is near the outlet, then the value of `small_fraction_to_large` will be very large (near 1), and the large depression will gatekeep all flows above it until it fills.

If there isn’t a large depression, then `upland_fraction_to_large = 0` and `small_fraction_to_large = 0`

HGDM flow partitioning parameters
Name | Meaning | Units
--- | --- | ---
upland_fraction_to_small | Fraction of uplands draining to small depressions | dimensionless
upland_fraction_to_large | Fraction of uplands draining to large depression. This is the basin of the large depression | dimensionless
upland_fraction_to_outlet | Fraction of uplands draining directly to outlet. Analogous to the basin effective fraction | dimensionless
small_fraction_to_large | Fraction of small depression area draining into large depression. Governed by location of large depression in the basin. | dimensionless

### 2.2.3 Miscellaneous parameters

The forcings are as described above. The parameter `small_p` is an exponent used to relate the pond storage and water area, as discussed by Shook and Clark (2022).

The rating curve for the large depression can either be a single `p` value as used for the meta depression, or can be a table representing a rating curve. If a table is specified, it must be a data frame, with the variables `area` (in m2) and `volume` (in m3).

The parameter `sub_intervals` is used to divide each time interval into a specified number of sub intervals. The reason for doing so is that the state variables (water storage, water area, connected/contributing fraction) change during the application of water. If you are using a very long time interval, then it can cause error. In the case of daily or sub-daily intervals, a value of `1` should be sufficient. Note that using long (multiple day) intervals will also cause errors because it will result in combining periods with and without rainfall.

HGDM miscellaneous parameters
Name | Meaning | Units
--- | --- | ---
forcings | A data frame of time series of forcings | mm
small_p | Parameter for small depression water volume-area relationship | dimensionless
large_rating | Rating curve parameters for large depression | dimensionless
sub_intervals | Number of sub-intervals for solution of each time interval | dimensionless

# 3 Examples

In these examples, the parameter values are not based on an actual location, much less the same location as the forcing data. However, they are typical of the type of values that might be found in the Canadian Prairies.

## 3.1 Example #1 - modelling a basin without a large depressiom

### 3.1.1 Setting parameter values

It’s usually a good idea to begin with specifying the total basin area. That way, the other areas can be calculated as fractions of that area which is convenient and reduces the possibility of gross errors. Note that the value of `small_depression_frac` which is the areal fraction of all the small depressions in the basin when completely filled with water (`0.24`), is relatively large for the region, and is appropriate for a flat basin completely dominated by depressions.

Area parameters

    area_units <- "km2"
    basin_area <- 100
    small_depression_frac <- 0.24
    small_depression_area <- small_depression_frac * basin_area
    large_depression_area <- 0
    upland_area <- basin_area - (small_depression_area + large_depression_area)

Storage parameters

Remember that storage as a depth is averaged over the area of the unit, i.e. all the small depressions (the meta depression) and the large depression.

    storage_units <- "mm"
    max_small_depression_storage <- 500
    max_large_depression_storage <- 0

Initial state parameters

    initial_small_depression_storage <- max_small_depression_storage / 2
    initial_large_depression_storage <- max_large_depression_storage / 2
    small_depressions_initial_connected_fraction <- 0

Flow partitioning parameters

In this example 98% of the runoff from the uplands goes to the small depressions, with only 2% going directly to the outlet.

    upland_fraction_to_small <- 0.98
    upland_fraction_to_large <- 0
    upland_fraction_to_outlet <- 0.02
    small_fraction_to_large <- 0

Miscellaneous parameters

The value of p for the small depressions (`small_p`) is pretty typical for many Canadian Prairie basins. See Shook and Pomeroy (2025) for how to derive it. The value of p for the large depression is not used here, but will be used later.

    small_p <- 1.2
    sub_intervals <- 1
    large_rating <- 1.4

### 3.1.2 Executing `HGDM`

The function returns a data frame with the time intervals as the forcings.

    simulation <- HGDM(
      upland_area,
      small_depression_area,
      large_depression_area,
      area_units,
      max_small_depression_storage,
      max_large_depression_storage,
      initial_small_depression_storage,
      initial_large_depression_storage,
      storage_units,
      small_depressions_initial_connected_fraction,
      upland_fraction_to_small,
      upland_fraction_to_large,
      upland_fraction_to_outlet,
      small_fraction_to_large,
      forcings = daily_7120951600,
      small_p,
      large_rating,
      sub_intervals)

### 3.1.3 Plotting the simulation results

Plot the basin discharge volume

    p <- ggplot(simulation, aes(date, total_outflow_volume)) +
      geom_point()
    p

Plot the total connected/contributing fraction over time

    p <- ggplot(simulation, aes(date, total_contrib_frac)) +
      geom_point() +
      ylim(0, 1)
    p

Plot the small (meta) depression connected/contributing fraction vs the depressional storage, as a fraction of the total.

    simulation$small_depression_water_volume_fraction <-
      simulation$small_depression_water_depth / (max_small_depression_storage / 1000)

    p2 <- ggplot(simulation,
                aes(small_depression_water_volume_fraction,
                    small_depression_contrib_frac)) +
      geom_point() +
      xlab("Meta depression volumetric fraction") +
      ylab("Meta depression connected/contributing fraction") +
      coord_fixed(ratio = 1) +
      xlim(0, 1) +
      ylim(0, 1) +
      geom_abline(slope = 1, intercept = 0, colour = "red")
    p2

## 3.2 Example #2 - modelling a basin with a large depressiom

### 3.2.1 Setting parameter values

In this example the large depression is set near the top of the basin, where it collects 25% of the small depression’s outputs.

    max_large_depression_storage <- 2000
    total_depression_area <- small_depression_area
    large_depression_frac <- 0.3
    large_depression_area <- large_depression_frac * total_depression_area
    small_depression_area <- (1 - large_depression_frac) * total_depression_area
    upland_area <- basin_area - (small_depression_area + large_depression_area)
    initial_large_depression_storage <- max_large_depression_storage / 2

    upland_fraction_to_small <- 0.96
    upland_fraction_to_large <- 0.02
    upland_fraction_to_outlet <- 0.02
    small_fraction_to_large <- 0.25

    simulation_large_pond <- HGDM(
      upland_area,
      small_depression_area,
      large_depression_area,
      area_units,
      max_small_depression_storage,
      max_large_depression_storage,
      initial_small_depression_storage,
      initial_large_depression_storage,
      storage_units,
      small_depressions_initial_connected_fraction,
      upland_fraction_to_small,
      upland_fraction_to_large,
      upland_fraction_to_outlet,
      small_fraction_to_large,
      forcings = daily_7120951600,
      small_p,
      large_rating,
      sub_intervals
      )

### 3.2.2 Plotting the simulation results

Plot the total connected/contributing fraction over time

    p3 <- ggplot(simulation_large_pond, aes(date, total_contrib_frac)) +
      geom_point()
    p3

    max_water_volume <- ((max_large_depression_storage / 1000) *
                           (large_depression_area * 1e6)) +
      ((max_small_depression_storage / 1000) * (small_depression_area * 1e6))

    simulation_large_pond$total_water_volume_fraction <-
      (simulation_large_pond$small_depression_water_volume +
         simulation_large_pond$large_depression_water_volume) / max_water_volume

    p4 <- ggplot(simulation_large_pond,
                 aes(total_water_volume_fraction, total_contrib_frac)) +
      geom_point() +
      xlab("Total volumetric fraction") +
      ylab("Total connected/contributing fraction") +
      coord_fixed(ratio = 1) +
      xlim(0, 1) +
      ylim(0, 1)
    p4

# 4 References

Shook, Kevin R., and John W. Pomeroy. “The Hysteretic and Gatekeeping Depressions Model - A New Model for Variable Connected Fractions of Prairie Basins.” Journal of Hydrology 654 (June 1, 2025): 132821. https://doi.org/10.1016/j.jhydrol.2025.132821.

Shook, Kevin R., Zhihua He, John W. Pomeroy, Chris Spence, and Colin J. Whitfield. “A Practitioner-Oriented Regional Hydrology Data Product for Use in Site-Specific Hydraulic Applications.” Scientific Data 11, no. 1 (October 14, 2024): 1125. https://doi.org/10.1038/s41597-024-03962-1.

Clark, Martyn P., and Kevin R. Shook. “The Numerical Formulation of Simple Hysteretic Models to Simulate the Large-Scale Hydrological Impacts of Prairie Depressions.” Water Resources Research 58, no. 12 (2022): e2022WR032694. https://doi.org/10.1029/2022WR032694.

Shook, Kevin, Raymond J. Spiteri, John W. Pomeroy, Tonghe Liu, and Oluwaseun Sharomi. “WDPM: The Wetland DEM Ponding Model.” Journal of Open Source Software 6, no. 64 (August 13, 2021): 2276. https://doi.org/10.21105/joss.02276.

Shook, Kevin, Simon Papalexiou, and John W. Pomeroy. “Quantifying the Effects of Prairie Depressional Storage Complexes on Drainage Basin Connectivity.” Journal of Hydrology 593 (February 1, 2021): 125846. https://doi.org/10.1016/j.jhydrol.2020.125846.

Shook, Kevin, Raymond J. Spiteri, John W. Pomeroy, Tonghe Liu, and Oluwaseun Sharomi. “WDPM: The Wetland DEM Ponding Model.” Journal of Open Source Software 6, no. 64 (August 13, 2021): 2276. https://doi.org/10.21105/joss.02276.

Shook, Kevin, John W Pomeroy, Christopher Spence, and Lyle Boychuk. “Storage Dynamics Simulations in Prairie Wetland Hydrology Models: Evaluation and Parameterization.” Hydrological Processes 27, no. 13 (June 2013): 1875–89. https://doi.org/10.1002/hyp.9867.