Primer

This section of the primer provides a simple to understand "plain English" summary of the IHACRES model.

IHACRES provides estimations of streamflow at catchment scale by applying the conceptual model of a "leaky bucket".

The model can be conceptualized as taking the following structure:

As a general overview, catchments hold an amount of water in its soils. This water evaporates under the heat of the sun, and is used by plants having the effect of drying the catchment. This drying effect is counteracted by rainfall which replenishes the water in the catchment. In this context the proportion of rainfall that contributes to streamflow is referred to as "effective rainfall". Some rainfall may "recharge" groundwater, some proportion of which contributes to streamflow via baseflow.

The first "bucket" in IHACRES takes daily rainfall and temperature data to determine the amount of effective rainfall, recharge, and the level of "dryness" in the catchment (referred to as the Catchment Moisture Deficit, or CMD). The higher the CMD value, the drier the catchment is. A value of 0 means the catchment is "completely wet". Temperature is correlated with evapotranspiration, which is evaporation combined with how much water plants use ("transpirate"). Typically, the hotter it is, the higher evapotranspiration is. How wet, or dry, the catchment is controls the amount of effective rainfall and recharge, which then influences how much runoff occurs after a rainfall event, and the volume of streamflow in the days afterwards.

The second set of "buckets" represents the Unit Hydrograph module, which uses the effective rainfall and recharge values to provide estimates of quickflow and slowflow (i.e., overland and baseflow, respectively). The sum of these then is the total streamflow.

When used to represent a sub-catchment, the streamflow is made up of the local quick and slow flow plus any upstream flow, minus any extractions/loss that may occur.

A more detailed overview

This section of the primer gives a brief technical description of IHACRES_nim. Further detail may be found in the API documentation.

IHACRES_nim provides functions which may be composed to represent different formulations of the IHACRES model.

All formulations available in IHACRES_nim require the following three parameters (with usual bounds):

A six parameter implementation can be achieved with additional parameters:

The bilinear implementation (detailed later below) adds an additional flow threshold parameter ($d_2$) and replaces the above $\tau$ and $v_s$ parameters.

Note: Appropriate values of $a$ and $b$ can be context specific. Nevertheless, to give some guidance, these values can be set between 0.1 and 10.0 for $a$ and between 0.001 and 0.1 for $b$.

An eighth parameter is also added to account for groundwater storage:

Delving into the implementation, the flow between parameters (light blue) and functions (rounded boxes) for an application of IHACRES_GW can be seen in the diagram below can be seen in the diagram below.

Here,

• rounded boxes represent functions
• blue boxes represent input factors
• green functions indicate those relevant to the non-linear loss module
• the beige function is the relevant unit hydrograph function

Use of any component function may be replaced with an equivalent. For example, calc_ft_interim_cmd may be replaced by any other calc_*_interim_cmd function, so long as the correct parameters are passed in. See documentation for details.

An implementation example for the Python language, can be found here.

The CMD module

The CMD module starts by producing an interim CMD value (a value that does not yet take into account $ET$), as well as effective rainfall ($U$) and recharge ($r$) estimates.

There are different formulations available to do this. Those included in IHACRES_nim are the:

• linear
• trignometric
• bilinear

The linear and trignometric formulations require the CMD value for the previous time step ($M_{k-1}$), the $d$ parameter, and the rainfall for the current time step ($P_k$). The linear formulation is shown below.

$$ M_f = \begin{cases} M_{k-1} \cdot \exp(-P_k/d) & \text{if $M_{k-1} < d$} \\ \exp((-P_k + M_{k-1} - d) / d) \cdot d & \text{if $M_{k-1} < (d + P_k)$} \\ M_{k-1} - P_k & \text{otherwise} \end{cases} $$

Further implementation details can be found in the API documentation for the CMD module.

Estimates of potential evapotranspiration ($ET$) are then derived from one of temperature or evaporation data ($T$ or $E$) for time step $k$.

If calculating from temperature ($T$):

$$ ET_k = \begin{cases} 0 & \text{if $T_k \le 0$} \\ e \cdot T_k \cdot \min(1, \exp(1 - 2(M_{f}/g))) & \text{otherwise, where } g := fd \end{cases} $$

If calculating from evaporation ($E$):

$$ ET_k = \begin{cases} e \cdot E_k & \text{if $M_f \le g$, where $g := fd$} \\ e \cdot E_k \cdot \min(1, \exp(2(1 - M_f/g))) & \text{otherwise} \end{cases} $$

Implementation details are found in the climate API documentation.

The updated CMD value for the current time step ($M_k$) is then calculated from the CMD value for the previous time step ($M_{k-1}$), $P_k$, $U_k$ and $r_k$:

$$ M_k = M_{k-1} + ET_k + U_k + r_k - P_k $$

The Unit Hydrograph

The linear approach implemented in IHACRES_nim assumes two stores in parallel.

$v_q = 1 - v_s$

$A_{u,k} = U_k \cdot A$, where $A$ is the catchment area in km²

Quickflow is calculated with:

$$ Q_q = (\beta \cdot A_{u,k}) + (\alpha \cdot Q_{q,k-1}) $$

where,

$$ \alpha = \exp(-1 /\tau_q) \\ \beta = v_q(1 - \alpha) $$

Similarly, slowflow is:

$$Q_s = (\beta \cdot A_{u,k}) + (\alpha \cdot Q_{s,k-1})$$

where

$$ \alpha = \exp(-1 /\tau_s) \\ \beta = v_s(1 - \alpha) $$

Streamflow is the sum of these

$$Q_T = Q_q + Q_s$$

An additional routing module may be used to account for extractions, groundwater interactions, and other factors.

Table of input variables