# Simulated vs observed upstream flows in calibration of semi-distributed GR4J model

#### David Dorchies

# Introduction

## Scope

The airGR package implements semi-distributed model capabilities using a lag model between subcatchments. It allows to chain together several lumped models as well as integrating anthropogenic influence such as reservoirs or withdrawals.

Here we explain how to implement the semi-distribution with airGR. For everyday use, however, it is easier to use the airGRiwrm package.

`RunModel_Lag` documentation gives an example of simulating the influence of a reservoir in a lumped model. Try `example(RunModel_Lag)` to get it.

In this vignette, we show how to calibrate 2 sub-catchments in series with a semi-distributed model consisting of 2 GR4J models. For doing this we compare 3 strategies for calibrating the downstream subcatchment:

- using upstream observed flows
- using upstream simulated flows
- using upstream simulated flows and parameter regularisation (de Lavenne et al. 2019)

We finally compare these calibrations with a theoretical set of parameters. This comparison is based on the Kling-Gupta Efficiency computed on the root-squared discharges as performance criterion.

## Model description

We use an example data set from the package that unfortunately contains data for only one catchment.

    ## loading catchment data
    data(L0123001)

Let’s imagine that this catchment of 360 km² is divided into 2 subcatchments:

- An upstream subcatchment of 180 km²
- 100 km downstream another subcatchment of 180 km²

We consider that meteorological data are homogeneous on the whole catchment, so we use the same pluviometry `BasinObs$P` and the same evapotranspiration `BasinObs$E` for the 2 subcatchments.

For the observed flow at the downstream outlet, we generate it with the assumption that the upstream flow arrives at downstream with a constant delay of 2 days.

    QObsDown <- (BasinObs$Qmm + c(0, 0, BasinObs$Qmm[1:(length(BasinObs$Qmm)-2)])) / 2
    options(digits = 5)
    summary(cbind(QObsUp = BasinObs$Qmm, QObsDown))

    ##      QObsUp           QObsDown
    ##  Min.   : 0.0168   Min.   : 0.0204
    ##  1st Qu.: 0.3936   1st Qu.: 0.4104
    ##  Median : 0.9768   Median : 0.9984
    ##  Mean   : 1.4732   Mean   : 1.4740
    ##  3rd Qu.: 1.8840   3rd Qu.: 1.9060
    ##  Max.   :23.8800   Max.   :19.8000
    ##  NAs    :802       NAs    :820

    options(digits = 3)

With a delay of 2 days between the 2 gauging stations, the theoretical Velocity parameter should be equal to:

    Velocity <- 100 * 1e3 / (2 * 86400)
    paste("Velocity: ", format(Velocity), "m/s")

    ## [1] "Velocity:  0.579 m/s"

# Calibration of the upstream subcatchment

The operations are exactly the same as the ones for a GR4J lumped model. So we do exactly the same operations as in the Get Started vignette.

    InputsModelUp <- CreateInputsModel(FUN_MOD = RunModel_GR4J, DatesR = BasinObs$DatesR,
                                       Precip = BasinObs$P, PotEvap = BasinObs$E)
    Ind_Run <- seq(which(format(BasinObs$DatesR, format = "%Y-%m-%d") == "1990-01-01"),
                   which(format(BasinObs$DatesR, format = "%Y-%m-%d") == "1999-12-31"))
    RunOptionsUp <- CreateRunOptions(FUN_MOD = RunModel_GR4J,
                                     InputsModel = InputsModelUp,
                                     IndPeriod_WarmUp = NULL, IndPeriod_Run = Ind_Run,
                                     IniStates = NULL, IniResLevels = NULL)

    ## Warning in CreateRunOptions(FUN_MOD = RunModel_GR4J, InputsModel = InputsModelUp, : model warm up period not defined: default configuration used
    ##   the year preceding the run period is used

    # Error criterion is KGE computed on the root-squared discharges
    InputsCritUp <- CreateInputsCrit(FUN_CRIT = ErrorCrit_KGE, InputsModel = InputsModelUp,
                                     RunOptions = RunOptionsUp,
                                     VarObs = "Q", Obs = BasinObs$Qmm[Ind_Run],
                                     transfo = "sqrt")
    CalibOptionsUp <- CreateCalibOptions(FUN_MOD = RunModel_GR4J, FUN_CALIB = Calibration_Michel)
    OutputsCalibUp <- Calibration_Michel(InputsModel = InputsModelUp, RunOptions = RunOptionsUp,
                                         InputsCrit = InputsCritUp, CalibOptions = CalibOptionsUp,
                                         FUN_MOD = RunModel_GR4J)

    ## Grid-Screening in progress (0% 20% 40% 60% 80% 100%)
    ##   Screening completed (81 runs)
    ##       Param =  169.017,   -0.020,   42.098,    2.384
    ##       Crit. KGE[sqrt(Q)] = 0.8676
    ## Steepest-descent local search in progress
    ##   Calibration completed (22 iterations, 244 runs)
    ##       Param =  151.411,    0.443,   59.145,    2.423
    ##       Crit. KGE[sqrt(Q)] = 0.8906

And see the result of the simulation:

    OutputsModelUp <- RunModel_GR4J(InputsModel = InputsModelUp, RunOptions = RunOptionsUp,
                                    Param = OutputsCalibUp$ParamFinalR)

# Calibration of the downstream subcatchment

## Creation of the InputsModel objects

We need to create `InputsModel` objects completed with upstream information with upstream observed flow for the calibration of first case and upstream simulated flows for the other cases:

    InputsModelDown1 <- CreateInputsModel(
      FUN_MOD = RunModel_GR4J, DatesR = BasinObs$DatesR,
      Precip = BasinObs$P, PotEvap = BasinObs$E,
      Qupstream = matrix(BasinObs$Qmm, ncol = 1), # upstream observed flow
      LengthHydro = 100, # distance between upstream catchment outlet & the downstream one [km]
      BasinAreas = c(180, 180) # upstream and downstream areas [km²]
    )

    ## Warning in CreateInputsModel(FUN_MOD = RunModel_GR4J, DatesR = BasinObs$DatesR,
    ## : 'Qupstream' contains NA values: model outputs will contain NAs

For using upstream simulated flows, we should concatenate a vector with the simulated flows for the entire period of simulation (warm-up + run):

    Qsim_upstream <- rep(NA, length(BasinObs$DatesR))
    # Simulated flow during warm-up period (365 days before run period)
    Qsim_upstream[Ind_Run[seq_len(365)] - 365] <- OutputsModelUp$RunOptions$WarmUpQsim
    # Simulated flow during run period
    Qsim_upstream[Ind_Run] <- OutputsModelUp$Qsim

    InputsModelDown2 <- CreateInputsModel(
      FUN_MOD = RunModel_GR4J, DatesR = BasinObs$DatesR,
      Precip = BasinObs$P, PotEvap = BasinObs$E,
      Qupstream = matrix(Qsim_upstream, ncol = 1), # upstream observed flow
      LengthHydro = 100, # distance between upstream catchment outlet & the downstream one [km]
      BasinAreas = c(180, 180) # upstream and downstream areas [km²]
    )

    ## Warning in CreateInputsModel(FUN_MOD = RunModel_GR4J, DatesR = BasinObs$DatesR,
    ## : 'Qupstream' contains NA values: model outputs will contain NAs

## Calibration with upstream flow observations

We calibrate the combination of Lag model for upstream flow transfer and GR4J model for the runoff of the downstream subcatchment:

    RunOptionsDown <- CreateRunOptions(FUN_MOD = RunModel_GR4J,
                                       InputsModel = InputsModelDown1,
                                       IndPeriod_WarmUp = NULL, IndPeriod_Run = Ind_Run,
                                       IniStates = NULL, IniResLevels = NULL)

    ## Warning in CreateRunOptions(FUN_MOD = RunModel_GR4J, InputsModel = InputsModelDown1, : model warm up period not defined: default configuration used
    ##   the year preceding the run period is used

    InputsCritDown <- CreateInputsCrit(FUN_CRIT = ErrorCrit_KGE, InputsModel = InputsModelDown1,
                                       RunOptions = RunOptionsDown,
                                       VarObs = "Q", Obs = QObsDown[Ind_Run],
                                       transfo = "sqrt")
    CalibOptionsDown <- CreateCalibOptions(FUN_MOD = RunModel_GR4J,
                                           FUN_CALIB = Calibration_Michel,
                                           IsSD = TRUE) # specify that it's a SD model
    OutputsCalibDown1 <- Calibration_Michel(InputsModel = InputsModelDown1,
                                            RunOptions = RunOptionsDown,
                                            InputsCrit = InputsCritDown,
                                            CalibOptions = CalibOptionsDown,
                                            FUN_MOD = RunModel_GR4J)

    ## Grid-Screening in progress (0% 20% 40% 60% 80% 100%)
    ##   Screening completed (243 runs)
    ##       Param =    1.250,  169.017,   -0.020,   42.098,    2.384
    ##       Crit. KGE[sqrt(Q)] = 0.9399
    ## Steepest-descent local search in progress
    ##   Calibration completed (32 iterations, 542 runs)
    ##       Param =    0.804,  147.332,    0.291,   35.300,    4.551
    ##       Crit. KGE[sqrt(Q)] = 0.9611

`RunModel` is run in order to automatically combine GR4J and Lag models.

    OutputsModelDown1 <- RunModel(InputsModel = InputsModelDown2,
                                  RunOptions = RunOptionsDown,
                                  Param = OutputsCalibDown1$ParamFinalR,
                                  FUN_MOD = RunModel_GR4J)

Performance of the model validation is then:

    KGE_down1 <- ErrorCrit_KGE(InputsCritDown, OutputsModelDown1)

    ## Crit. KGE[sqrt(Q)] = 0.8940

    ##  SubCrit. KGE[sqrt(Q)] cor(sim, obs, "pearson") = 0.8945
    ##  SubCrit. KGE[sqrt(Q)] sd(sim)/sd(obs)          = 0.9897
    ##  SubCrit. KGE[sqrt(Q)] mean(sim)/mean(obs)      = 1.0009

## Calibration with upstream simulated flow

We calibrate the model with the `InputsModel` object previously created for substituting the observed upstream flow with the simulated one:

    OutputsCalibDown2 <- Calibration_Michel(InputsModel = InputsModelDown2,
                                            RunOptions = RunOptionsDown,
                                            InputsCrit = InputsCritDown,
                                            CalibOptions = CalibOptionsDown,
                                            FUN_MOD = RunModel_GR4J)

    ## Grid-Screening in progress (0% 20% 40% 60% 80% 100%)
    ##   Screening completed (243 runs)
    ##       Param =    1.250,  169.017,   -0.020,   83.096,    2.384
    ##       Crit. KGE[sqrt(Q)] = 0.8827
    ## Steepest-descent local search in progress
    ##   Calibration completed (37 iterations, 591 runs)
    ##       Param =    0.330,  165.670,    0.273,   26.311,    3.857
    ##       Crit. KGE[sqrt(Q)] = 0.9031

    ParamDown2 <- OutputsCalibDown2$ParamFinalR

## Calibration with upstream simulated flow and parameter regularisation

The regularisation follow the method proposed by de Lavenne et al. (2019).

As a priori parameter set, we use the calibrated parameter set of the upstream catchment and the theoretical velocity:

    ParamDownTheo <- c(Velocity, OutputsCalibUp$ParamFinalR)

The Lavenne criterion is initialised with the a priori parameter set and the value of the KGE of the upstream basin.

    IC_Lavenne <- CreateInputsCrit_Lavenne(InputsModel = InputsModelDown2,
                                           RunOptions = RunOptionsDown,
                                           Obs = QObsDown[Ind_Run],
                                           AprParamR = ParamDownTheo,
                                           AprCrit = OutputsCalibUp$CritFinal)

The Lavenne criterion is used instead of the KGE for calibration with regularisation

    OutputsCalibDown3 <- Calibration_Michel(InputsModel = InputsModelDown2,
                                            RunOptions = RunOptionsDown,
                                            InputsCrit = IC_Lavenne,
                                            CalibOptions = CalibOptionsDown,
                                            FUN_MOD = RunModel_GR4J)

    ## Grid-Screening in progress (0% 20% 40% 60% 80% 100%)
    ##   Screening completed (243 runs)
    ##       Param =    1.250,  169.017,   -0.020,   83.096,    2.384
    ##       Crit. Composite    = 0.8926
    ## Steepest-descent local search in progress
    ##   Calibration completed (26 iterations, 482 runs)
    ##       Param =    0.520,  149.905,    0.443,   58.557,    2.462
    ##       Crit. Composite    = 0.9116
    ##  Formula: sum(0.86 * KGE[sqrt(Q)], 0.14 * GAPX[ParamT])

The KGE is then calculated for performance comparisons:

    OutputsModelDown3 <- RunModel(InputsModel = InputsModelDown2,
                                  RunOptions = RunOptionsDown,
                                  Param = OutputsCalibDown3$ParamFinalR,
                                  FUN_MOD = RunModel_GR4J)
    KGE_down3 <- ErrorCrit_KGE(InputsCritDown, OutputsModelDown3)

    ## Crit. KGE[sqrt(Q)] = 0.8983

    ##  SubCrit. KGE[sqrt(Q)] cor(sim, obs, "pearson") = 0.9102
    ##  SubCrit. KGE[sqrt(Q)] sd(sim)/sd(obs)          = 0.9542
    ##  SubCrit. KGE[sqrt(Q)] mean(sim)/mean(obs)      = 1.0130

# Discussion

## Identification of Velocity parameter

Both calibrations overestimate this parameter:

    mVelocity <- matrix(c(Velocity,
                          OutputsCalibDown1$ParamFinalR[1],
                          OutputsCalibDown2$ParamFinalR[1],
                          OutputsCalibDown3$ParamFinalR[1]),
                        ncol = 1,
                        dimnames = list(c("theoretical",
                                          "calibrated with observed upstream flow",
                                          "calibrated with simulated  upstream flow",
                                          "calibrated with sim upstream flow and regularisation"),
                                        c("Velocity parameter")))
    knitr::kable(mVelocity)

| Velocity parameter
--- | ---
theoretical | 0.579
calibrated with observed upstream flow | 0.804
calibrated with simulated upstream flow | 0.330
calibrated with sim upstream flow and regularisation | 0.520

## Value of the performance criteria with theoretical calibration

Theoretically, the parameters of the downstream GR4J model should be the same as the upstream one with the velocity as extra parameter:

    OutputsModelDownTheo <- RunModel(InputsModel = InputsModelDown2,
                                     RunOptions = RunOptionsDown,
                                     Param = ParamDownTheo,
                                     FUN_MOD = RunModel_GR4J)
    KGE_downTheo <- ErrorCrit_KGE(InputsCritDown, OutputsModelDownTheo)

    ## Crit. KGE[sqrt(Q)] = 0.8976

    ##  SubCrit. KGE[sqrt(Q)] cor(sim, obs, "pearson") = 0.9082
    ##  SubCrit. KGE[sqrt(Q)] sd(sim)/sd(obs)          = 0.9562
    ##  SubCrit. KGE[sqrt(Q)] mean(sim)/mean(obs)      = 1.0121

## Parameters and performance of each subcatchment for all calibrations

    comp <- matrix(c(0, OutputsCalibUp$ParamFinalR,
                     rep(OutputsCalibDown1$ParamFinalR, 2),
                     OutputsCalibDown2$ParamFinalR,
                     OutputsCalibDown3$ParamFinalR,
                     ParamDownTheo),
                   ncol = 5, byrow = TRUE)
    comp <- cbind(comp, c(OutputsCalibUp$CritFinal,
                          OutputsCalibDown1$CritFinal,
                          KGE_down1$CritValue,
                          OutputsCalibDown2$CritFinal,
                          KGE_down3$CritValue,
                          KGE_downTheo$CritValue))
    colnames(comp) <- c("Velocity", paste0("X", 1:4), "KGE(√Q)")
    rownames(comp) <- c("Calibration of the upstream subcatchment",
                        "Calibration 1 with observed upstream flow",
                        "Validation 1 with simulated upstream flow",
                        "Calibration 2 with simulated upstream flow",
                        "Calibration 3 with simulated upstream flow and regularisation",
                        "Validation theoretical set of parameters")
    knitr::kable(comp)

| Velocity | X1 | X2 | X3 | X4 | KGE(√Q)
--- | --- | --- | --- | --- | --- | ---
Calibration of the upstream subcatchment | 0.000 | 151 | 0.443 | 59.1 | 2.42 | 0.891
Calibration 1 with observed upstream flow | 0.804 | 147 | 0.291 | 35.3 | 4.55 | 0.961
Validation 1 with simulated upstream flow | 0.804 | 147 | 0.291 | 35.3 | 4.55 | 0.894
Calibration 2 with simulated upstream flow | 0.330 | 166 | 0.273 | 26.3 | 3.86 | 0.903
Calibration 3 with simulated upstream flow and regularisation | 0.520 | 150 | 0.443 | 58.6 | 2.46 | 0.898
Validation theoretical set of parameters | 0.579 | 151 | 0.443 | 59.1 | 2.42 | 0.898

Even if calibration with observed upstream flows gives an improved performance criteria, in validation using simulated upstream flows the result is quite similar as the performance obtained with the calibration with upstream simulated flows. The theoretical set of parameters give also an equivalent performance but still underperforming the calibration 2 one. Regularisation allows to get similar performance as the one for calibration with simulated flows but with the big advantage of having parameters closer to the theoretical ones (Especially for the velocity parameter).

# References

Lavenne, Alban de, Vazken Andréassian, Guillaume Thirel, Maria-Helena Ramos, and Charles Perrin. 2019. “A Regularization Approach to Improve the Sequential Calibration of a Semidistributed Hydrological Model.” Water Resources Research 55 (11): 8821–39. https://doi.org/10.1029/2018WR024266.