# Chapter 4: Semi-distributed glacio-hydrological modeling

#### Ezequiel Toum

#### 2022-11-23

## The problem

This case is about adding to our previous example (Semi-distributed hydrological model) glaciers. Again, is a synthetic case where you will have to:

- understand how to consider glaciers in a semi-distributed HBV hydrological model.
- calibrate the glacier’s related parameters.
- study their effect (sensitivity analysis) on the basin discharge.

## Data

    library(HBV.IANIGLA)

    data("glacio_hydro_hbv")

    str(glacio_hydro_hbv)
    #> List of 5
    #>  $ baisn:'data.frame':   15 obs. of  7 variables:
    #>   ..$ elev_band      : chr [1:15] "eb_1" "eb_2" "eb_3" "eb_4" ...
    #>   ..$ soil_area(km2) : num [1:15] 10 15 17 20 25 18 17 16 14 12 ...
    #>   ..$ ice_area(km2)  : num [1:15] 0 0 0 0 0 0 0 0 0 0 ...
    #>   ..$ total_area(km2): num [1:15] 10 15 17 20 25 18 17 16 14 12 ...
    #>   ..$ rel_soil       : num [1:15] 0.0526 0.0789 0.0894 0.1052 0.1314 ...
    #>   ..$ rel_ice        : num [1:15] 0 0 0 0 0 0 0 0 0 0 ...
    #>   ..$ h(masl)        : num [1:15] 2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 ...
    #>  $ tair : num [1:5310, 1:15] 24.4 26.9 27 24.5 22.1 ...
    #>  $ prec : num [1:5310, 1:15] 0 0 0 0 0 0 0 0 0 0 ...
    #>  $ pet  : num [1:5310, 1:15] 0 0.999 0.997 0.993 0.988 ...
    #>  $ qout : num [1:5310, 1:3] 0.0021 0.0091 0.0218 0.0438 0.0617 0.0711 0.0774 0.0809 0.0858 0.0936 ...
    #>   ..- attr(*, "dimnames")=List of 2
    #>   .. ..$ : NULL
    #>   .. ..$ : chr [1:3] "total" "soil" "glacier"
    #>  - attr(*, "comment")= chr "Semi-distributed glacio-hydrological HBV model input data"

To get more details about the dataset just type `?glacio_hydro_hbv`

## Adding the glacier routines

Note that in this case we have to add the parameters and the initial conditions arguments for the glacier surface mass balance function (SnowGlacier()) and for the glacier runoff routing (Glacier_Disch()).

- Advice: before reading the code, take a look at the dataset again (`?glacio_hydro_hbv`) and try to build your own HBV glacio-hydrological model.

    ## brief arguments description
      # basin: data frame with the same structure of the data("glacio_hydro_hbv) (colnames included).
      # tair: numeric matrix with air temperature inputs.
      # precip: numeric matrix with precipitation inputs.
      # pet: numeric matrix with potential eavapotranspiration inputs.
      # param_snow: numeric vector with snow module parameters.
      # param_ice: numeric vector with glacier parameters.
      # param_soil: numeric vector with soil moisture parameters.
      # param_route: numeric vector with the routing parameters.
      # param_route_ice: numeric vector with the glacier routing parameters.
      # param_tf: numeric vector with the transfer function parameter.
      # init_snow: numeric value with initial snow water equivalent. Default value being 20 mm.
      # init_ice: numeric value with initial snow water equivalent of the glaciers. Default value
      # being 20 mm.
      # init_soil: numeric value with initial soil moisture content. Default value being 0 mm.
      # init_route: numeric vector with bucket water initial values. Default values are 0 mm.
      # init_route_ice: numeric value with glacier bucket initial value. Default values are 0 mm.
    ## output
      # simulated streamflow series.
    glacio_hydrological_hbv <- function(basin,
                                        tair,
                                        precip,
                                        pet,
                                        param_snow,
                                        param_ice,
                                        param_soil,
                                        param_route,
                                        param_route_ice,
                                        param_tf,
                                        init_snow = 20,
                                        init_ice = 20,
                                        init_soil = 0,
                                        init_route = c(0, 0, 0),
                                        init_route_ice = 0
                                        ){
      n_it <- nrow(basin)

      # create output lists
      snow_module   <- list()
      ice_module    <- list()
      soil_module   <- list()
      route_module  <- list()
      route_ice_mod <- list()
      tf_module     <- list()

      # snow and soil module in every elevation band
      for(i in 1:n_it){
        snow_module[[ i ]] <-
          SnowGlacier_HBV(model = 1, inputData = cbind(tair[ , i], precip[ , i]),
                          initCond =  c(init_snow, 2), param = param_snow)

        ice_module[[ i ]] <-
          SnowGlacier_HBV(model = 1, inputData = cbind(tair[ , i], precip[ , i]),
                          initCond =  c(init_ice, 1, basin[i, 'rel_ice']), param = param_ice)

        soil_module[[ i ]] <-
          Soil_HBV(model = 1, inputData = cbind(snow_module[[i]][ , 5] , pet[ , i]),
                   initCond = c(init_soil, basin[i, 'rel_soil']), param = param_soil )

      } # end for

      # get total soil discharge
      soil_disch <- lapply(X = 1:n_it, FUN = function(x){
        out <- soil_module[[x]][ , 1]
      })
      soil_disch <- Reduce(f = `+`, x = soil_disch)

      # get swe and total ice melt for all glacier area
      ice_disch <- lapply(X = 1:n_it, FUN = function(x){
        out <- ice_module[[x]][ , 9]
      })
      ice_disch <- Reduce(f = `+`, x = ice_disch)

      ice_swe   <- lapply(X = 1:n_it, FUN = function(x){
        out <- ice_module[[x]][ , 3] *  (basin[x, 'rel_ice'] / sum(basin[ , 'rel_ice']) )
      })
      ice_swe <- Reduce(f = `+`, x = ice_swe)

      # route module
      route_module <- Routing_HBV(model = 1, lake = F, inputData = as.matrix(soil_disch),
                                  initCond = init_route, param = param_route )

      route_ice    <- Glacier_Disch(model = 1, inputData = cbind(ice_swe, ice_disch),
                                    initCond = init_route_ice, param = param_route_ice  )

      # transfer function
      tf_soil <- round(
        UH(model = 1, Qg = route_module[ , 1], param = param_tf), 4  )

      tf_ice  <- round(
        UH(model = 1, Qg = route_ice[ , 1], param = param_tf), 4  )

      tf_out  <- tf_soil + tf_ice

      return( cbind(total = tf_out, soil = tf_soil, glacier = tf_ice) )

    }# end fun

Now, maybe is time to revisit the Semi-distributed hydrological model vignette (Calibrating the parameters section).

Your turn

I will give the correct parameters to all modules except for glacier related routines,

- `param_snow = c(1.1, 0, 0, 2.5)`
- `param_soil = c(150, 0.90, 1.5)`
- `param_route = c(0.09, 0.07, 0.05, 5, 2)`
- `param_tf = c(3.00)`

Hint: in the `param_ice()` argument, I will use the snow parameters except for the melt temperature and for the ice-melt factor.

- calibrate the glacier’s related parameters.
- is the glacier melt temperature parameter higher or lower than the one for the snow? Why?
- make a sesitivity (looking at the streamflow discharge) analysis over the calibrated parameters, which are the most sensitive?