`vignettes/Notes_on_the_implementation_of_Dynamic_TOPMODEL.Rmd`

`Notes_on_the_implementation_of_Dynamic_TOPMODEL.Rmd`

The `dynatop`

package represents an implementation of the concepts behind a dynamic TOPMODEL.
One of the underlying principles of dynamic TOPMODEL is that the landscape can be broken up into
hydrologically similar regions, or Hydrological Response Units (HRUs), where
all the area within a HRU behaves in a hydrologically similar fashion.

While a Hydrological Response Unit (HRU) has typically been thought of as an area of hill slope with, for example, similar topographic, soil and upslope area characteristics the idea may be generalised. In doing this a catchments is conceptualised as a collection of HRUs which may represent of different types of landscape (e.g. hill slope, channel, lake) which exchange fluxes at specified levels (e.g. saturated zone, surface). Currently a single type of HRU is present in the model, which may be parameterised to represent a number of different physical features.

The aim of this document is to outline

- the computational sequence and linking the HRUs
- conventions for the input data
- other miscellaneous information that may help developers

Each HRU has a unique reference number in the `id`

variable. In principle the HRUs are
solved in reverse order of `id`

; that is from the largest `id`

value to the
smallest. Currently this is strictly enforced. Future releases may relax this
to enable parallel solutions.

Fluxes between the HRUs are represented as occurring at two levels, the surface and the saturated zones. For the \(i\)th HRU the outflow in \(m^3/s\) at time \(t+\Delta t\) is \(\left.q_{sf}^{\left[i\right]}\right\rvert_{t+\Delta t}\) at the surface and \(\left.q_{sz}^{\left[i\right]}\right\rvert_{t+\Delta t}\) in the saturated zone.

The fraction of outflow going from the \(i\) th to the \(j\) th HRU is \(f_{i,j}\). The redistribution is conservative as does not vary in time so \[ \sum\limits_{j} f_{i,j} = 1 \] Since there is no exchange between the surface and saturated zones during redistribution the inflows at time \(t+\Delta t\) to the \(k\)th HRU can be computed as \[ \sum\limits_{i>k} f_{i,k}\left.q_{sf}^{\left[i\right]}\right\rvert_{t+\Delta t} \] and \[ \sum\limits_{i>k} f_{i,k}\left.q_{sz}^{\left[i\right]}\right\rvert_{t+\Delta t} \]

The values of \(f_{i,j}\) are specified in the `flow_direction`

data.frame of
the model.

It is expected that the precipitation and potential evapotranspiration inputs series are given in \(m\) accrued over the proceeding time step. So if the data has a time step \(\Delta t\) the value given at \(t+\Delta t\) is accrued in the interval between time \(t\) and time \(t+\Delta t\).

To aid the readability of the code the variables are labelled consistently with regards to the vignettes. For example:

- Single subscript: \(l_{sz}\) becomes
`l_sz`

, \(s_{uz}\) becomes`s_uz`

etc. - Directional variables: \(q_{rz \rightarrow uz}\) becomes
`q_rz_uz`

, etc. - Greek letters are spelt: \(\lambda\) becomes
`lambda`

However the notation used in the vignettes for intermediate values of fluxes or states (e.g. ) is dropped from the code where it can be inferred from the computational sequence (or comments)

Currently the code uses a purely implicit scheme for the solution of the
HRUs. This requires a solving two *zero finding* problems for each HRU
at each (sub)time step. Currently a bisection algorithm is used with a
user specified tolerance and maximum number of iterations. The tolerance is
defined as the difference between the upper and lower limits of the interval
containing the zero point. Notionally faster algorithms (such as the TOMS-748
algorithm) could be used.