+
Description
+
The model is based on the HBV-96 model. However, the hydrological
+routing represent in HBV by a triangular function controlled by the
+MAXBAS parameter has been removed. Instead, the kinematic wave function
+is used to route the water downstream. All runoff that is generated
+in a cell in one of the HBV reservoirs is added to the kinematic wave
+reservoir at the end of a timestep. There is no connection between
+the different HBV cells within the model. Wherever possible all functions
+that describe the distribution of parameters within a subbasin have
+been removed as this is not needed in a distributed application/
+
A catchment is divided into a number of grid cells. For each of the
+cells individually, daily runoff is computed through application of
+the HBV-96 of the HBV model. The use of the grid cells offers the
+possibility to turn the HBV modelling concept, which is originally
+lumped, into a distributed model.
+
+
The figure above shows a schematic view of hydrological response
+simulation with the HBV-modelling concept. The land-phase of the hydrological
+cycle is represented by three different components: a snow routine,
+a soil routine and a runoff response routine. Each component is discussed
+separately below.
+
+
The snow routine
+
Precipitation enters the model via the snow routine. If the air temperature,
+T_{a}, is below a user-defined threshold TT (\approx0^{o}C)
+precipitation occurs as snowfall, whereas it occurs as rainfall if
+T_{a}\geq TT. A another parameter TTI defines how precipitation
+can occur partly as rain of snowfall (see the figure below).
+If precipitation occurs as snowfall, it is added to the dry snow component
+within the snow pack. Otherwise it ends up in the free water reservoir,
+which represents the liquid water content of the snow pack. Between
+the two components of the snow pack, interactions take place, either
+through snow melt (if temperatures are above a threshold TT) or
+through snow refreezing (if temperatures are below threshold TT).
+The respective rates of snow melt and refreezing are:
+
+
Q_{m} = cfmax(T_{a}-TT)\;\;;T_{a}>TT
+
+Q_{r} = cfmax*cfr(TT-T_{a})\;;T_{a}<TT
+
where Q_{m} is the rate of snow melt, Q_{r} is the rate of snow
+refreezing, and $cfmax$ and $cfr$ are user defined model parameters
+(the melting factor mm/(^{o}C*day) and the refreezing factor
+respectively)
+
+
Note
+
The FoCFMAX parameter from the original HBV version is not used. instead
+the CFMAX is presumed to be for the landuse per pixel. Normally for
+forested pixels the CFMAX is 0.6 {*} CFMAX
+
+
The air temperature, T_{a}, is related to measured daily average
+temperatures. In the original HBV-concept, elevation differences within
+the catchment are represented through a distribution function (i.e.
+a hypsographic curve) which makes the snow module semi-distributed.
+In the modified version that is applied here, the temperature, T_{a},
+is represented in a fully distributed manner, which means for each
+grid cell the temperature is related to the grid elevation.
+
The fraction of liquid water in the snow pack (free water) is at most
+equal to a user defined fraction, WHC, of the water equivalent
+of the dry snow content. If the liquid water concentration exceeds
+WHC, either through snow melt or incoming rainfall, the surpluss
+water becomes available for infiltration into the soil:
+
+
Q_{in}=max\{(SW-WHC*SD);\;0.0\}
+
where Q_{in} is the volume of water added to the soil module, SW
+is the free water content of the snow pack and SD is the dry snow
+content of the snow pack.
+
+
+
+
Potential Evaporation
+
The original HBV version includes both a multiplication factor for
+potential evaporation and a exponential reduction factor for potential
+evapotranspiration during rain events. The CEVPF factor is used
+to connect potential evapotranspiration per landuse. In the original
+version the CEVPFO is used and it is used for forest landuse only.
+
+
+
Interception
+
The parameters ICF0 and ICFI introduce interception storage
+for forested and non-forested zones respectively in the original model.
+Within our application this is replaced by a single $ICF$ parameter
+assuming the parameter is set for each grid cell according to the
+land-use. In the original application it is not clear if interception
+evaporation is subtracted from the potential evaporation. In this
+implementation we dos subtract the interception evaporation to ensure
+total evaporation does not exceed potential evaporation. From this
+storage evaporation equal to the potential rate ET_{p} will occur
+as long as water is available, even if it is stored as snow. All water
+enters this store first, there is no concept of free throughfall (e.g.
+through gaps in the canopy). In the model a running water budget is
+kept of the interception store:
+
+
+- The available storage (ICF-Actual storage) is filled with the water
+coming from the snow routine (Q_{in})
+- Any surplus water now becomes the new Q_{in}
+- Interception evaporation is determined as the minimum of the current
+interception storage and the potential evaporation
+
+
+
+
+
The soil routine
+
The incoming water from the snow and interception routines, Q_{in},
+is available for infiltration in the soil routine. The soil layer
+has a limited capacity, F_{c}, to hold soil water, which means
+if F_{c} is exceeded the abundant water cannot infiltrate and,
+consequently, becomes directly available for runoff.
+
+
Q_{dr}=max\{(SM+Q_{in}-F_{c});\;0.0\}
+
where Q_{dr} is the abundant soil water (also referred to as direct
+runoff) and SM is the soil moisture content. Consequently, the
+net amount of water that infiltrates into the soil, I_{net}, equals:
+
Part of the infiltrating water, I_{net}, will runoff through the
+soil layer (seepage). This runoff volume, SP, is related to the
+soil moisture content, SM, through the following power relation:
+
+
SP=\left(\frac{SM}{F_{c}}\right)^{\beta}I_{net}\label{eq:SP}
+
where \beta is an empirically based parameter. Application of this equation
+implies that the amount of seepage water increases with
+increasing soil moisture content. The fraction of the infiltrating
+water which doesn’t runoff, I_{net}-SP, is added
+to the available amount of soil moisture, SM. The \beta parameter
+affects the amount of supply to the soil moisture reservoir that is
+transferred to the quick response reservoir. Values of \beta vary
+generally between 1 and 3. Larger values of \beta reduce runoff
+and indicate a higher absorption capacity of the soil (see Figure
+ref{fig:HBV-Beta}).
+
+
+
A percentage of the soil moisture will evaporate. This percentage
+is related to the measured potential evaporation and the available
+amount of soil moisture:
+
+
E_{a} = \frac{SM}{T_{m}}E_{p\;\;};SM<T_{m}\\
+
+E_{a} = E_{p}\;\;\;;SM\geq T_{m}
+
where E_{a} is the actual evaporation, E_{p} is the potential
+evaporation and T_{m} (\leq F_{c}) is a user defined threshold,
+above which the actual evaporation equals the potential evaporation.
+T_{m} is defined as LP*F_{c}\; in which LP is a soil dependent
+evaporation factor (LP\leq1).
+
In the original model (Berglov, 2009 XX), a correction to Ea is
+applied in case of interception. If Ea from the soil moisture storage
+plus Ei exceeds ETp - Ei (Ei = interception
+evaporation) then the exceeding part is multiplied by a factor (1-ered),
+where the parameter ered varies between 0 and 1. This correction is presently not present in the wflow_hbv model.
+
+
+
The runoff response routine
+
The volume of water which becomes available for runoff, S_{dr}+SP,
+is transferred to the runoff response routine. In this routine the
+runoff delay is simulated through the use of a number of linear reservoirs.
+
Two linear reservoir are defined to simulate the different runoff
+processes: the upper zone (generating quick runoff and interflow)
+and the lower zone (generating slow runoff). The available runoff
+water from the soil routine (i.e. direct runoff, S_{dr}, and seepage,
+SP) in principle ends up in the lower zone, unless the percolation
+threshold, PERC, is exceeded, in which case the redundant water
+ends up in the upper zone:
+
+
\triangle V_{LZ} = min\{PERC;(S_{dr}+SP)\}\\
+
+\triangle V_{UZ} = max\{0.0;(S_{dr}+SP-PERC)\}
+
where V_{UZ} is the content of the upper zone, V_{LZ} is the
+content of the lower zone and \triangle means increase
+of.
+
Capillary flow from the upper zone to the soil moisture reservoir
+is modeled according to:
+
+
Q_{cf}=cflux*(F_{c}-SM)/F_{c}
+
where cflux is the maximum capilary flux in mm/day.
+
The Upper zone generates quick runoff (Q_{q}) using:
+
+
Q_{q}=K*UZ^{(1+alpha)}
+
here K is the upper zone recession coefficient, and \alpha determines
+the amount of non-linearity. Within HBV-96, the value of K is determined
+from three other parameters: \alpha, KHQ, and HQ (mm/day).
+The value of HQ represents an outflow rate of the upper zone for
+which the recession rate is equal to KHQ. if we define UZ_{HQ} to
+be the content of the upper zone at outflow rate HQ we can write
+the following equation:
+
+
HQ=K*UZ_{HQ}^{(1+\alpha)}=KHQ*UZ_{HQ}
+
If we eliminate UZ_{HQ} we obtain:
+
+
HQ=K*\left(\frac{HQ}{KHQ}\right)^{(1+\alpha)}
+
Rewriting for K results in:
+
+
K=KQH^{(1-alpha)}HQ^{-alpha}
+
+
Note
+
Note that the HBV-96 manual mentions that for a recession rate larger
+than 1 the timestap in the model will be adjusted.
+
+
The lower zone is a linear reservoir, which means the rate of slow
+runoff, Q_{LZ}, which leaves this zone during one time step equals:
+
where K_{LZ} is the reservoir constant.
+
The upper zone is also a linear reservoir, but it is slightly more
+complicated than the lower zone because it is divided into two zones:
+A lower part in which interflow is generated and an upper part in
+which quick flow is generated (see Figure ref{fig:upper}).
+
+
If the total water content of the upper zone, V_{UZ}, is lower
+than a threshold UZ1, the upper zone only generates interflow.
+On the other hand, if V_{UZ} exceeds UZ1, part of the upper
+zone water will runoff as quick flow:
+
+
Q_{i} = K_{i}*min\{UZ1;V_{uz}\}
+
+Q_{q} = K_{q}*max\{(V_{UZ}-UZ1);0.0\}
+
Where Q_{i} is the amount of generated interflow in one time step,
+Q_{q} is the amount of generated quick flow in one time step and
+K_{i} and K_{q} are reservoir constants for interflow and quick
+flow respectively.
+
The total runoff rate, Q, is equal to the sum of the three different
+runoff components:
+
The runoff behaviour in the runoff response routine is controlled
+by two threshold values P_{m} and UZ1 in combination with three
+reservoir parameters, K_{LZ}, K_{i} and K_{q}. In order to
+represent the differences in delay times between the three runoff
+components, the reservoir constants have to meet the following requirement:
+
+