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.

`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.

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.

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

```
## QObsUp QObsDown
## Min. : 0.02 Min. : 0.02
## 1st Qu.: 0.39 1st Qu.: 0.41
## Median : 0.98 Median : 1.00
## Mean : 1.47 Mean : 1.47
## 3rd Qu.: 1.88 3rd Qu.: 1.91
## Max. :23.88 Max. :19.80
## NA's :802 NA's :820
```

`options(digits = 3)`

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

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

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

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.

```
<- CreateInputsModel(FUN_MOD = RunModel_GR4J, DatesR = BasinObs$DatesR,
InputsModelUp Precip = BasinObs$P, PotEvap = BasinObs$E)
<- seq(which(format(BasinObs$DatesR, format = "%Y-%m-%d") == "1990-01-01"),
Ind_Run which(format(BasinObs$DatesR, format = "%Y-%m-%d") == "1999-12-31"))
<- CreateRunOptions(FUN_MOD = RunModel_GR4J,
RunOptionsUp 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
<- CreateInputsCrit(FUN_CRIT = ErrorCrit_KGE, InputsModel = InputsModelUp,
InputsCritUp RunOptions = RunOptionsUp,
VarObs = "Q", Obs = BasinObs$Qmm[Ind_Run],
transfo = "sqrt")
<- CreateCalibOptions(FUN_MOD = RunModel_GR4J, FUN_CALIB = Calibration_Michel)
CalibOptionsUp <- Calibration_Michel(InputsModel = InputsModelUp, RunOptions = RunOptionsUp,
OutputsCalibUp 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:

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

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:

```
<- CreateInputsModel(
InputsModelDown1 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):

```
<- rep(NA, length(BasinObs$DatesR))
Qsim_upstream # Simulated flow during warm-up period (365 days before run period)
seq_len(365)] - 365] <- OutputsModelUp$RunOptions$WarmUpQsim
Qsim_upstream[Ind_Run[# Simulated flow during run period
<- OutputsModelUp$Qsim
Qsim_upstream[Ind_Run]
<- CreateInputsModel(
InputsModelDown2 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
```

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

```
<- CreateRunOptions(FUN_MOD = RunModel_GR4J,
RunOptionsDown 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
```

```
<- CreateInputsCrit(FUN_CRIT = ErrorCrit_KGE, InputsModel = InputsModelDown1,
InputsCritDown RunOptions = RunOptionsDown,
VarObs = "Q", Obs = QObsDown[Ind_Run],
transfo = "sqrt")
<- CreateCalibOptions(FUN_MOD = RunModel_GR4J,
CalibOptionsDown FUN_CALIB = Calibration_Michel,
IsSD = TRUE) # specify that it's a SD model
<- Calibration_Michel(InputsModel = InputsModelDown1,
OutputsCalibDown1 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 = 11.250, 169.017, -0.020, 42.098, 2.384
## Crit. KGE[sqrt(Q)] = 0.9341
## Steepest-descent local search in progress
## Calibration completed (58 iterations, 812 runs)
## Param = 2.933, 143.296, 0.272, 31.953, 5.443
## Crit. KGE[sqrt(Q)] = 0.9635
```

`RunModel`

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

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

Performance of the model validation is then:

`<- ErrorCrit_KGE(InputsCritDown, OutputsModelDown1) KGE_down1 `

`## Crit. KGE[sqrt(Q)] = 0.8968`

```
## SubCrit. KGE[sqrt(Q)] cor(sim, obs, "pearson") = 0.8982
## SubCrit. KGE[sqrt(Q)] sd(sim)/sd(obs) = 0.9832
## SubCrit. KGE[sqrt(Q)] mean(sim)/mean(obs) = 1.0028
```

We calibrate the model with the `InputsModel`

object previously created for substituting the observed upstream flow with the simulated one:

```
<- Calibration_Michel(InputsModel = InputsModelDown2,
OutputsCalibDown2 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 = 11.250, 169.017, -0.020, 83.096, 2.384
## Crit. KGE[sqrt(Q)] = 0.8716
## Steepest-descent local search in progress
## Calibration completed (47 iterations, 695 runs)
## Param = 1.500, 157.591, 0.325, 36.234, 6.629
## Crit. KGE[sqrt(Q)] = 0.9031
```

`<- OutputsCalibDown2$ParamFinalR ParamDown2 `

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:

`<- c(Velocity, OutputsCalibUp$ParamFinalR) ParamDownTheo `

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

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

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

```
<- Calibration_Michel(InputsModel = InputsModelDown2,
OutputsCalibDown3 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 = 11.250, 169.017, -0.020, 83.096, 2.384
## Crit. Composite = 0.8165
## Steepest-descent local search in progress
## Calibration completed (44 iterations, 666 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:

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

`## 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
```

Both calibrations overestimate this parameter:

```
<- matrix(c(Velocity,
mVelocity $ParamFinalR[1],
OutputsCalibDown1$ParamFinalR[1],
OutputsCalibDown2$ParamFinalR[1]),
OutputsCalibDown3ncol = 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")))
::kable(mVelocity) knitr
```

Velocity parameter | |
---|---|

theoretical | 0.579 |

calibrated with observed upstream flow | 2.933 |

calibrated with simulated upstream flow | 1.500 |

calibrated with sim upstream flow and regularisation | 0.520 |

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

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

`## 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
```

```
<- matrix(c(0, OutputsCalibUp$ParamFinalR,
comp rep(OutputsCalibDown1$ParamFinalR, 2),
$ParamFinalR,
OutputsCalibDown2$ParamFinalR,
OutputsCalibDown3
ParamDownTheo),ncol = 5, byrow = TRUE)
<- cbind(comp, c(OutputsCalibUp$CritFinal,
comp $CritFinal,
OutputsCalibDown1$CritValue,
KGE_down1$CritFinal,
OutputsCalibDown2$CritValue,
KGE_down3$CritValue))
KGE_downTheocolnames(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")
::kable(comp) knitr
```

Velocity | X1 | X2 | X3 | X4 | KGE(vQ) | |
---|---|---|---|---|---|---|

Calibration of the upstream subcatchment | 0.000 | 151 | 0.443 | 59.1 | 2.42 | 0.891 |

Calibration 1 with observed upstream flow | 2.933 | 143 | 0.272 | 32.0 | 5.44 | 0.963 |

Validation 1 with simulated upstream flow | 2.933 | 143 | 0.272 | 32.0 | 5.44 | 0.897 |

Calibration 2 with simulated upstream flow | 1.500 | 158 | 0.325 | 36.2 | 6.63 | 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).

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.