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organization into subsectsion
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jerabekjak committed Mar 11, 2024
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ep is the effective precipitation intensity (the potential precipitation
reduced by the interception zone and the surface retention)
($\mathrm{L.t^{-1}}$), and inf is the infiltration rate ($\mathrm{L.t^{-1}}$).
The kinematic wave approach is used in the calculation of the overland flow.
The momentum is therefore expressed by the power-law:

\begin{equation}
q = ah^b
\label{eq:powerlaw}
\end{equation}
where a and b are power-law parameters. Equation \ref{eq:powerlaw} can be
expressed in the form of the Manning–Strickler formula


\begin{equation}
q = n^{-1} s^Y h^b,
\label{eq:powerlaw}
\end{equation}
where n is roughness, Y empirical parameter and s is the surface slope ($\mathrm{L.L^{-1}}$).

The infiltration component of Equation \ref{eq:bilance} is calculated by the
Philip infiltration equation \citep{philip1957}
\begin{equation}
inf = \frac{1}{2}St^{-1/2}+K_s.
\label{eq:infiltration}
\end{equation}

where S stands for sorptivity ($\mathrm{L.t^{1/2}}$) and ($\mathrm{K_s}$)
stands for saturation hydraulic conductivity ($\mathrm{L.t^{-1}}$).

The SMODERP2D model is subjected to uniform rainfall. The potential
precipitation is reduced due to surface retention. The surface retention is the
storage that needs to be filled before surface runoff can occur.

The flow routing of the surface runoff is based on the D8 one-direction flow
algorithm \cite{o1984extraction}. The inflow to cell i is defined as the sum of the sheet
outflows from the adjacent cells, as:

\begin{equation}
q^{in}_{i,t-1} = \sum_j^m q^{out}_{j,t-1},
\label{eq:d8}
\end{equation}
where j is the index of the adjacent up-slope cells identified by the D8 flow
algorithm.

The time derivative in Equation \ref{eq:bilance} is calculated using an
explicit method. The computation is therefore sensitive to the size of the time
step. The size of the time step is controlled by the Courant criterion, which
needs to be kept below the theoretical maximum value of 1, while the maximum
value in practise is lower than 1
\cite{zhang1989modeling, esteves2000overland}.


The sheet flow water level of the next time t + 1 step in Equation
\ref{eq:bilance} which incorporates the sum \ref{eq:d8} is calculated with the
explicit time discretisation scheme for cell i as:
\begin{equation}
h_{i,t} =h_{i,t} + \mathrm{d}t (es_{i,t-1} + \sum_j^m q^{out}_{j,t-1}-
inf_{i,t-1} - q^{out}_{i,t-1}),
\label{eq:bilance}
\end{equation}

%The principle of the model is the cell-by-cell mass balance calculated in each
%time step. The change in the water level of the shear flow in any cell in time
Expand Down Expand Up @@ -116,69 +59,136 @@
%(6)

\subsubsection{Interception component (JJ)}
\subsubsection{Infiltration component (JJ) }
The SMODERP2D model is subjected to uniform rainfall. The potential
precipitation is reduced due to surface retention. The surface
retention is the storage that needs to be filled before surface
runoff can occur.

TODO

\subsubsection{Infiltration component (JJ) }
The infiltration component of Equation \ref{eq:bilance} is calculated by the
Philip infiltration equation \citep{philip1957}
\begin{equation}
inf = \frac{1}{2}St^{-1/2}+K_s.
\label{eq:infiltration}
\end{equation}
where S stands for sorptivity ($\mathrm{L.t^{1/2}}$) and ($\mathrm{K_s}$)
stands for saturation hydraulic conductivity ($\mathrm{L.t^{-1}}$).

{\bf GA}

%Phillip infiltration
%
%The infiltration component of Equation (1) is calculated by the Philip infiltration equation [44]
%The infiltration component of Equation (1) is calculated by the
%Philip infiltration equation [44]
%
%[Equation],
%
%
%(4)
%
%where S stands for sorptivity (L.t1/2) and Ks stands for saturation hydraulic conductivity (L.t−1).
%where S stands for sorptivity (L.t1/2) and Ks stands for
%saturation hydraulic conductivity (L.t−1).
%
%GA???



\subsubsection{Surface retention component (JJ)}
\subsection{Sheet flow hydraulics (JJ)}
The kinematic wave approach is used in the calculation of the overland flow.
The momentum is therefore expressed by the power-law:

\begin{equation}
q = ah^b
\label{eq:powerlaw}
\end{equation}
where a and b are power-law parameters. Equation \ref{eq:powerlaw} can be
expressed in the form of the Manning–Strickler formula


\begin{equation}
q = n^{-1} s^Y h^b,
\label{eq:powerlaw}
\end{equation}
where n is roughness, Y empirical parameter and s is the surface
slope ($\mathrm{L.L^{-1}}$).
\subsubsection{Principle of the solution}

%The kinematic wave approach is used in the calculation of the overland flow. The momentum is therefore expressed by the power-law:
%The kinematic wave approach is used in the calculation of the
%overland flow. The momentum is therefore expressed by the power-law:
%
%[Equation],
%
%
%(2)
%
%where a and b are power-law parameters. Equation (2) can be expressed in the form of the Manning–Strickler formula
%where a and b are power-law parameters. Equation (2) can be
%expressed in the form of the Manning–Strickler formula
%
%[Equation]
%
%
%(3)
%
%where b and Y are empirical parameters and s is the surface slope (L.L−1). n- Manning roughness coefficient for sheet flow
%where b and Y are empirical parameters and s is the surface slope
%(L.L−1). n- Manning roughness coefficient for sheet flow
%
%
%
%XXX - tables here or link to user guide

\subsubsection{D8/ Multiple flow approach}

The flow routing of the surface runoff is based on the D8 one-direction flow
algorithm \cite{o1984extraction}. The inflow to cell i is defined as the sum of the sheet
outflows from the adjacent cells, as:

\begin{equation}
q^{in}_{i,t-1} = \sum_j^m q^{out}_{j,t-1},
\label{eq:d8}
\end{equation}
where j is the index of the adjacent up-slope cells identified by the D8 flow
algorithm.

%Two (optional) types of flow direction can be used in the model solution
%
%D8 algorithm
%
%The flow routing of the surface runoff is based on the D8 one-direction flow algorithm [45]. The inflow to cell i is defined as the sum of the sheet outflows from the adjacent cells, as:
%The flow routing of the surface runoff is based on the D8
%one-direction flow algorithm [45]. The inflow to cell i is defined
%as the sum of the sheet outflows from the adjacent cells, as:
%
%[Equation],
%
%
%(5)
%
%where j is the index of the adjacent up-slope cells identified by the D8 flow algorithm.
%where j is the index of the adjacent up-slope cells identified by
%the D8 flow algorithm.
%
%Multiple flouw algorithm
%


\subsection{Rill flow formation and hydraulics (JJ)}

%For each soil type a critical value of the tangential stress and velocity was
%estimated. From this critical value critical height in each cell is calculated.
%In principle this is a comparison of the current level and its critical value
%at each time interval. If the critical value is exceeded, the calculation
%enters the stage at which the rill starts to form. Dimensions of the rills are
%calculated from volume of the water exceeding the critical value. Sheet surface
%runoff is then calculated using the critical value level instead of current
%height in the time step. Once the level has dropped below the critical height
%value, the calculation returns only in the calculation of surface runoff. The
%resulting rasters of rill flow and speed in the rill are stored in
%user-selected directory along with vector shapefile of created rills.
%Calculation of the flow in the rill is based on Manning equation.

\subsubsection{Rill formation (JJ)}

The rill flow is also included in the calculation. In SMODERP2D, rill flow in a
cell occurs if $h>h_{crit}$, where $h_{crit}$ is the critical water level. The
water flow corresponding to the water level above the critical water level has
Expand Down Expand Up @@ -214,9 +224,51 @@
h_{rill} = MAX(h-h_{crit},0),
\label{eq:hrill}
\end{equation}
P stands for the size of the raster cell. The rill is simplified as a small
channel at the soil surface with a rectangular cross section. The rectangle has
width $b_{rill}$ and rill height $y_{rill} = 0.7b_{rill}$. The rill flow is as
P stands for the size of the raster cell.
%The rill flow is also included in the calculation. In SMODERP2D, rill flow in a
%cell occurs if [Equation], where [Equation] is the critical water level. The
%water flow corresponding to the water level above the critical water level has
%enough energy to start to carry the soil particles and to create a rill.
%
%The critical water level [Equation] is calculated as:
%
%[Equation]
%
%
%(7)
%
%where [Equation] is the water corresponding to the critical velocity, and
%[Equation] is the water level corresponding to the critical shear stress. As
%shown in Formula (7), [Equation] uses several values obtained with a different
%approach. This approach is adopted in order to remain on the safe side of the
%emergence of a rill, since [Equation] is more sensitive to the sheet flow
%velocity and [Equation] is more sensitive to the slope of the soil surface.
%
%When the condition [Equation] is fulfilled, a rill starts to develop in a given
%cell and [Equation]. In SMODERP2D, the rill is a dynamic component and can
%increase as the rill flow increases. The rill volume is controlled by the
%volume of water corresponding to the rill water level [Equation] This volume is
%calculated as:
%
%[Equation]
%
%
%(8)
%
%where:
%
%[Equation]
%
%
%(9)
%
%P stands for the size of the raster cell.

\subsubsection{Rill hydraulics (JJ)}

The rill is simplified as a small channel at the soil surface with
a rectangular cross section. The rectangle has width $b_{rill}$ and
rill height $y_{rill} = 0.7b_{rill}$. The rill flow is as
calculated with the Manning equation:

\begin{equation}
Expand Down Expand Up @@ -301,63 +353,6 @@
The rill water level is recalculated to cover the whole cell and not just the
bottom of the rill, as shown in Figures \ref{fig:rill_plneni} and
\ref{fig:rill_prazdneni}.

%For each soil type a critical value of the tangential stress and velocity was
%estimated. From this critical value critical height in each cell is calculated.
%In principle this is a comparison of the current level and its critical value
%at each time interval. If the critical value is exceeded, the calculation
%enters the stage at which the rill starts to form. Dimensions of the rills are
%calculated from volume of the water exceeding the critical value. Sheet surface
%runoff is then calculated using the critical value level instead of current
%height in the time step. Once the level has dropped below the critical height
%value, the calculation returns only in the calculation of surface runoff. The
%resulting rasters of rill flow and speed in the rill are stored in
%user-selected directory along with vector shapefile of created rills.
%Calculation of the flow in the rill is based on Manning equation.

\subsubsection{Rill formation (JJ)}

%The rill flow is also included in the calculation. In SMODERP2D, rill flow in a
%cell occurs if [Equation], where [Equation] is the critical water level. The
%water flow corresponding to the water level above the critical water level has
%enough energy to start to carry the soil particles and to create a rill.
%
%The critical water level [Equation] is calculated as:
%
%[Equation]
%
%
%(7)
%
%where [Equation] is the water corresponding to the critical velocity, and
%[Equation] is the water level corresponding to the critical shear stress. As
%shown in Formula (7), [Equation] uses several values obtained with a different
%approach. This approach is adopted in order to remain on the safe side of the
%emergence of a rill, since [Equation] is more sensitive to the sheet flow
%velocity and [Equation] is more sensitive to the slope of the soil surface.
%
%When the condition [Equation] is fulfilled, a rill starts to develop in a given
%cell and [Equation]. In SMODERP2D, the rill is a dynamic component and can
%increase as the rill flow increases. The rill volume is controlled by the
%volume of water corresponding to the rill water level [Equation] This volume is
%calculated as:
%
%[Equation]
%
%
%(8)
%
%where:
%
%[Equation]
%
%
%(9)
%
%P stands for the size of the raster cell.

\subsubsection{Rill hydraulics (JJ)}

%The rill is simplified as a small channel at the soil surface with a
%rectangular cross section. The rectangle has width[Equation] and rill height
%[Equation]. The rill flow is as calculated with the Manning equation:
Expand Down Expand Up @@ -456,11 +451,28 @@

\subsection{Implicit /Explicit approach}

\subsubsection{Implicit}
\subsubsection{Explicit}
The time derivative in Equation \ref{eq:bilance} is calculated using an
explicit method. The computation is therefore sensitive to the size of the time
step. The size of the time step is controlled by the Courant criterion, which
needs to be kept below the theoretical maximum value of 1, while the maximum
value in practise is lower than 1
\cite{zhang1989modeling, esteves2000overland}.


The sheet flow water level of the next time t + 1 step in Equation
\ref{eq:bilance} which incorporates the sum \ref{eq:d8} is calculated with the
explicit time discretisation scheme for cell i as:
\begin{equation}
h_{i,t} =h_{i,t} + \mathrm{d}t (es_{i,t-1} + \sum_j^m q^{out}_{j,t-1}-
inf_{i,t-1} - q^{out}_{i,t-1}),
\label{eq:bilance}
\end{equation}
\subsubsection{Implicit}

\subsection{Impute data requirements and description}

\input{tab_en/tabsoilveg.tex}
\subsubsection{DMR}
\subsubsection{Soil}
\subsubsection{Land Use}
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