Modeling of Subsurface Throughflow in Urban Pervious Areas

Using the time-area model for rainfall-runoff modeling with variable rainfall intensity, the catchment is divided into several subcatchments to simulate the effect of water retention on the catchment surface. The time-dependent hydrograph is calculated by Eq. (2) (Butler and Davies 2004)where $Q$ = outflow from the catchment [${\mathrm{m}}^3{\mathrm{s}}^{-1}$]; $i_{(n)}$ [$\mathrm{m}{\mathrm{s}}^{-1}$] = rainfall intensity in the $n$th [-] subcatchment cell with a duration of $\mathrm{\Delta }t$ [s]; $N_A$ = total number of subcatchment cells [-]; ${\mathrm{\Delta }\mathrm{A}}_{(j)}$ = surface area of the subcatchment cell contributing to runoff [${\mathrm{m}}^2$]; $\phi$ [-] = runoff coefficient; and $j$ = subcatchment cell in which rainfall, $i_n$, precipitates. The value of $j$ is calculated

The runoff coefficient is 0.18, as derived in Eq. (1), for soil water content conditions above 0.34 m³ ${\mathrm{H}}_{\mathrm{2}}\mathrm{O}/{\mathrm{m}}^3$ soil. Under these soil water conditions, the infiltration to the lower soil layer is constant. Furthermore, no water is transferred to the time-area model if the soil volumetric water content from measured data is below $0.34{\mathrm{m}}^3{\mathrm{H}}_2\mathrm{O}/{\mathrm{m}}^3$ soil. Additionally, when this soil water content level is reached, rainfall must still exceed an initial loss of 7.1 mm before runoff.

The value of $N_A$ is calculated based on the concentration time, $t_c$ [s], and chosen time increment duration, $\mathrm{\Delta }t$ (DHI 2010)

For traditional surface runoff models, the concentration time expresses the time water from the most distant area of the catchment takes to reach the catchment out. In this case, the concentration time expresses the transport time in the soil from the most distant uphill areas in the soil matrix. Assuming a rectangular catchment, the surface area increments are calculatedwhere $A$ = total catchment surface area [${\mathrm{m}}^2$].

The physically-based kinematic wave model is based on the continuity equation and Manning’s equation (Butler and Davies 2004). Assuming that flow is one-dimensional, the following continuity equation is formed of water storage in a catchment:where $y$ = water depth in the catchment [m]; $t$ = time [s]; and $i$ = rainfall intensity [$\mathrm{m}{\mathrm{s}}^{-1}$]. Outflow from the catchment is calculated by Manning’s equation (Munoz-Carpena et al. 1999)where $M$ = Manning’s number [${\mathrm{m}}^{1/3}{\mathrm{s}}^{-1}$]; $B$ = width of the flow channel or runoff contributing catchment [m]; and $I$ = catchment slope [-].

Eq. (6) can be approximated in an explicit finite-difference scheme, assuming that the timestep $\mathrm{\Delta }t$ is small, so that the change in water depth $y$ is approximately linear between timesteps

Now, substituting Eqs. (7) and (8) into Eq. (6) gives the kinematic wave approximation

Generally, it is assumed that no water ponds in the upper soil layer but infiltrates to deeper aquifer if the soil water content is below $0.34{\mathrm{m}}^3{\mathrm{H}}_2\mathrm{O}/{\mathrm{m}}^3$ soil. If the soil water content level is exceeded, an initial loss of 7.1 mm must be still be exceeded before any ponding occurs in the upper soil layer.

The linear reservoir model is also based on the continuity equation in Eq. (6). However, instead of relying on Manning’s equation, the flow is linearly related to the water depth, $y$, with the proportionality constant, $k_1$ [${\mathrm{m}}^2{\mathrm{s}}^{-1}$]. This results in the following approximation consisting of one reservoir:

Like the kinematic wave model, no water ponds in the upper soil layer unless the soil water content is above $0.34{\mathrm{m}}^3{\mathrm{H}}_2\mathrm{O}/{\mathrm{m}}^3$ soil, and the initial loss of 7.1 must be exceeded.

It is investigated if simple regression models of catchment runoff as a function of soil water content can be used to estimate runoff. This is inspired by Kirkby and Chorley (1967) who presented a simple power function to simulate subsurface throughflow. The model defines the subsurface throughflow discharge as a function of soil water content exceeding a critical soil water content level at which water can no longer be stored in the soilwhere $k_2$ and $k_3$ = constants; $\theta$ = measured soil water content of the catchment [${\mathrm{m}}^3{\mathrm{H}}_2\mathrm{O}/{\mathrm{m}}^3$ soil]; and ${\theta }_0$ = soil water content level where the subsurface throughflow discharge is zero [${\mathrm{m}}^3{\mathrm{H}}_2\mathrm{O}/{\mathrm{m}}^3$ soil]. In this study, ${\theta }_0$ is set to $0.34{\mathrm{m}}^3{\mathrm{H}}_2\mathrm{O}/{\mathrm{m}}^3$ soil based on the relationship presented in Eq. (1) and Fig. 4. Furthermore, it is tested how a linear and exponential regression model performs compared to the power function given by Kirkby and Chorley (1967).

A two-layer feed-forward neural network model, with the structure presented in Fig. 5, is tested to evaluate the quality of purely data-driven models. Two approaches are tested in the same network structure by using a different number of input variables. The two approaches are NN1, the current data approach, and NN2, the past data approach. The current data approach relies on the neural network trained to estimate the current catchment runoff, $Q$, based on the most recent observations of rainfall and soil-water properties. The current data approach has five input variables, which are (1) rainfall, (2) average soil water content in the catchment, (3) soil water content closest to the outlet of the catchment, (4) average soil matric potential in the catchment, and (5) soil matric potential closest to the outlet of the catchment. The input variables are measured at the same time as the catchment runoff, $Q$, which is the training target of the neural network. The past data approach includes the same data as the current data approach but also relies on the observations of the average soil water content in the catchment up until runoff. This means that the past data approach includes 6–17 input variables in which Input variables 1–5 are the same as the current data approach and Input variables 6–17 are the average soil water content in the catchment 15 min, 30 min,… 180 min behind the most recent observations. The input has no prior conditioning of when runoff is most probably active, and the network must therefore identify this relationship through training.

The neural network presented in Fig. 5 contains one hidden layer consisting of L1 hidden, sigmoid-based neurons and one output layer consisting of one linear output neuron ($\mathrm{L}2=1$). The values of $v^1$and $v^2$ are the weighted sums of the weight ($w^1$ and $w^2$) and bias ($b^1$ and $b^2$) corrected input variables, $IP$, to the respective neurons. The value of $a^1$ is the output from the sigmoid function ($\sigma$) in the hidden neurons and input data to the linear output neuron (Hagan and Menhaj 1994).

When training the network, the weights and biases in the network, as seen in Fig. 5, are adjusted using the Levenberg-Marquardt backpropagation (Marquardt 1963; Hagan and Menhaj 1994); however, not the full dataset is used for network training. To avoid overconditioning, 70% of the data are used for training, 15% are used for validation of the training process, and 15% are used for independent testing. The data is subdivided randomly for each network training. Technically, the difference between the training and validation data is that training data are used to optimize the weights and biases of the neural network while validation data are used to detect overfitting. During the training of the neural network, the fit of training and validation data will often converge. However, at some point, the validation data can be used to detect overfitting. That is, if the fit based on training data seems to improve but the fit based on validation data deteriorates, there is a sign of overfitting. In this case, weights and biases should be adjusted accordingly. After the training of the neural network model has ended, the test data can be used to independently compare if these also fit the calibrated model without having been part of the training process.