library(airGR) options(digits = 3)

In the **airGR** package, the classical way to calibrate a model is to use the Michel's algorithm (see the `Calibration_Michel()`

function).

The Michel's algorithm combines a global and a local approach. A screening is first performed either based on a rough predefined grid (considering various initial values for each parameter) or from a list of initial parameter sets. The best set identified in this screening is then used as a starting point for the steepest descent local search algorithm.

In some specific situations, for example if the calibration period is too short and by consequence non representative of the catchment behaviour, a local calibration algorithm can give poor results.

In this vignette, we show a method using a known parameter set that can be used as an alternative for the grid-screening calibration procedure, and we well compare to two methods using the `Calibration_Michel()`

function. The generalist parameters sets introduced here are taken from @andreassian_seeking_2014.

We load an example data set from the package and the GR4J parameter sets.

## loading catchment data data(L0123001) ## loading generalist parameter sets data(Param_Sets_GR4J)

The given GR4J **X4u** variable does not correspond to the actual GR4J **X4** parameter. As explained in @andreassian_seeking_2014 [section 2.1], the given GR4J **X4u** value has to be adjusted (rescaled) using catchment area (S) [km2] as follows: `X4 = X4u / 5.995 * S^0.3`

(please= note that **the formula is erroneous in the publication**).

It means we need first to transform the **X4** parameter.

Param_Sets_GR4J$X4 <- Param_Sets_GR4J$X4u / 5.995 * BasinInfo$BasinArea^0.3 Param_Sets_GR4J$X4u <- NULL Param_Sets_GR4J <- as.matrix(Param_Sets_GR4J)

Please, find below the summary of the `r nrow(Param_Sets_GR4J)`

sets of the `r ncol(Param_Sets_GR4J)`

parameters.

```
summary(Param_Sets_GR4J)
```

We assume that the R global environment contains data and functions from the Get Started vignette.

The calibration period has been defined from **1990-01-01** to **1990-02-28**, and the validation period from **1990-03-01** to **1999-12-31**.

As a consequence, in this example the calibration period is very short, less than 6 months.

## preparation of the InputsModel object InputsModel <- CreateInputsModel(FUN_MOD = RunModel_GR4J, DatesR = BasinObs$DatesR, Precip = BasinObs$P, PotEvap = BasinObs$E) ## ---- calibration step ## short calibration period selection (< 6 months) Ind_Cal <- seq(which(format(BasinObs$DatesR, format = "%d/%m/%Y %H:%M")=="01/01/1990 00:00"), which(format(BasinObs$DatesR, format = "%d/%m/%Y %H:%M")=="28/02/1990 00:00")) ## preparation of the RunOptions object for the calibration period RunOptions_Cal <- CreateRunOptions(FUN_MOD = RunModel_GR4J, InputsModel = InputsModel, IndPeriod_Run = Ind_Cal) ## efficiency criterion: Nash-Sutcliffe Efficiency InputsCrit_Cal <- CreateInputsCrit(FUN_CRIT = ErrorCrit_NSE, InputsModel = InputsModel, RunOptions = RunOptions_Cal, Obs = BasinObs$Qmm[Ind_Cal]) ## ---- validation step ## validation period selection Ind_Val <- seq(which(format(BasinObs$DatesR, format = "%d/%m/%Y %H:%M")=="01/03/1990 00:00"), which(format(BasinObs$DatesR, format = "%d/%m/%Y %H:%M")=="31/12/1999 00:00")) ## preparation of the RunOptions object for the validation period RunOptions_Val <- CreateRunOptions(FUN_MOD = RunModel_GR4J, InputsModel = InputsModel, IndPeriod_Run = Ind_Val) ## efficiency criterion (Nash-Sutcliffe Efficiency) on the validation period InputsCrit_Val <- CreateInputsCrit(FUN_CRIT = ErrorCrit_NSE, InputsModel = InputsModel, RunOptions = RunOptions_Val, Obs = BasinObs$Qmm[Ind_Val])

It is recommended to use the generalist parameter sets when the calibration period is less than 6 months.

As shown in @andreassian_seeking_2014 [figure 4], a recommended way to use the `Param_Sets_GR4J`

`data.frame`

is to run the GR4J model with each parameter set and to select the best one according to an objective function (here we use the Nash-Sutcliffe Efficiency criterion).

OutputsCrit_Loop <- apply(Param_Sets_GR4J, 1, function(iParam) { OutputsModel_Cal <- RunModel_GR4J(InputsModel = InputsModel, RunOptions = RunOptions_Cal, Param = iParam) OutputsCrit <- ErrorCrit_NSE(InputsCrit = InputsCrit_Cal, OutputsModel = OutputsModel_Cal) return(OutputsCrit$CritValue) })

Find below the `r nrow(Param_Sets_GR4J)`

criteria corresponding to the different parameter sets.

The criterion values are quite low (from `r OutputsCrit_Loop[which.min(OutputsCrit_Loop)]`

to `r OutputsCrit_Loop[which.max(OutputsCrit_Loop)]`

), which can be expected as this does not represents an actual calibration.

OutputsCrit_Loop

The parameter set corresponding to the best criterion is the following:

Param_Best <- unlist(Param_Sets_GR4J[which.max(OutputsCrit_Loop), ]) Param_Best ## validation OutputsModel_Val <- RunModel_GR4J(InputsModel = InputsModel, RunOptions = RunOptions_Val, Param = Param_Best) OutputsCrit_Val <- ErrorCrit_NSE(InputsCrit = InputsCrit_Val, OutputsModel = OutputsModel_Val)

Now we can compute the Nash-Sutcliffe Efficiency criterion on the validation period. A quite good value (`r OutputsCrit_Val$CritValue`

) is found.

As seen above, the Michel's calibration algorithm is based on a local search procedure.

It is **not recommanded** to use the `Calibration_Michel()`

function when the **calibration period is less than 6 month**.
We will show below its application on the same short period for two configurations of the grid-screening step to demonstrate that it is less efficient than the generalist parameters sets calibration.

By default, the predefined grid used by the `Calibration_Michel()`

function contains parameters quantiles computed after recursive calibrations on 1200 basins (from Australia, France and USA).

CalibOptions <- CreateCalibOptions(FUN_MOD = RunModel_GR4J, FUN_CALIB = Calibration_Michel)

## calibration OutputsCalib <- Calibration(InputsModel = InputsModel, RunOptions = RunOptions_Cal, InputsCrit = InputsCrit_Cal, CalibOptions = CalibOptions, FUN_MOD = RunModel_GR4J, FUN_CALIB = Calibration_Michel) OutputsModel_Cal <- RunModel(InputsModel = InputsModel, RunOptions = RunOptions_Cal, Param = OutputsCalib$ParamFinalR, FUN = RunModel_GR4J) OutputsCrit_Cal <- ErrorCrit_NSE(InputsCrit = InputsCrit_Cal, OutputsModel = OutputsModel_Cal) ## validation OutputsModel_Val <- RunModel_GR4J(InputsModel = InputsModel, RunOptions = RunOptions_Val, Param = OutputsCalib$ParamFinalR) OutputsCrit_Val <- ErrorCrit_NSE(InputsCrit = InputsCrit_Val, OutputsModel = OutputsModel_Val)

The parameter set corresponding to the best criterion is the following:

names(OutputsCalib$ParamFinalR) <- paste0("X", 1:4) OutputsCalib$ParamFinalR

The Nash-Sutcliffe Efficiency criterion computed on the calibration period is better (`r OutputsCrit_Cal$CritValue`

) than with the generalist parameter sets, but the one computed on the validation period is lower (`r OutputsCrit_Val$CritValue`

).
This shows that the generalist parameter sets give more robust model in this case.

It is also possible to give to the `CreateCalibOptions()`

function a matrix of parameter sets used for the grid-screening calibration procedure.
So, it possible is to use by this way the GR4J generalist parameter sets.

CalibOptions <- CreateCalibOptions(FUN_MOD = RunModel_GR4J, FUN_CALIB = Calibration_Michel, StartParamList = Param_Sets_GR4J)

## calibration OutputsCalib <- Calibration(InputsModel = InputsModel, RunOptions = RunOptions_Cal, InputsCrit = InputsCrit_Cal, CalibOptions = CalibOptions, FUN_MOD = RunModel_GR4J, FUN_CALIB = Calibration_Michel) OutputsModel_Cal <- RunModel(InputsModel = InputsModel, RunOptions = RunOptions_Cal, Param = OutputsCalib$ParamFinalR, FUN = RunModel_GR4J) OutputsCrit_Cal <- ErrorCrit_NSE(InputsCrit = InputsCrit_Cal, OutputsModel = OutputsModel_Cal) ## validation OutputsModel_Val <- RunModel_GR4J(InputsModel = InputsModel, RunOptions = RunOptions_Val, Param = OutputsCalib$ParamFinalR) OutputsCrit_Val <- ErrorCrit_NSE(InputsCrit = InputsCrit_Val, OutputsModel = OutputsModel_Val)

Here is the parameter set corresponding to the best criteria found.

names(OutputsCalib$ParamFinalR) <- paste0("X", 1:4) OutputsCalib$ParamFinalR

The results are the same here. The Nash-Sutcliffe Efficiency criterion computed on the calibration period is better (`r OutputsCrit_Cal$CritValue`

), but the one computed on the validation period is just a little bit lower (`r OutputsCrit_Val$CritValue`

) than the classical calibration.

Generally, the advantage of using GR4J parameter sets rather than the GR4J generalist parameter quantiles is that they make more sense than a simple exploration of combinations of quantiles of parameter distributions (each parameter set represents a consistent ensemble). In addition, for the first step, the number of iterations is smaller (27 runs instead of 81), which can save time if one wants to run a very large number of calibrations.

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