---
title: "Simulated vs observed upstream flows in calibration of semi-distributed GR4J model"
author: "David Dorchies"
bibliography: V00_airgr_ref.bib
output: rmarkdown::html_vignette
vignette: >
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteIndexEntry{Simulated vs observed upstream flows in calibration of semi-distributed GR4J model}
  %\VignetteEncoding{UTF-8}
---

```{r, include=FALSE, fig.keep='none', results='hide'}
library(airGR)
options(digits = 3)
```

# 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**](https://cran.r-project.org/package=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 [@delavenne_regularization_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

```{r, warning=FALSE, include=FALSE}
library(airGR)
options(digits = 3)
```

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

```{r, warning=FALSE}
## 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.

```{r}
QObsDown <- (BasinObs$Qmm + c(0, 0, BasinObs$Qmm[1:(length(BasinObs$Qmm)-2)])) / 2
options(digits = 5)
summary(cbind(QObsUp = BasinObs$Qmm, QObsDown))
options(digits = 3)
```

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

```{r}
Velocity <- 100 * 1e3 / (2 * 86400)
paste("Velocity: ", format(Velocity), "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](V01_get_started.html) vignette.

```{r}
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$DatesR, format = "%Y-%m-%d") == "1999-12-31"))
RunOptionsUp <- CreateRunOptions(FUN_MOD = RunModel_GR4J,
                                 InputsModel = InputsModelUp,
                                 IndPeriod_WarmUp = NULL, IndPeriod_Run = Ind_Run,
                                 IniStates = NULL, IniResLevels = NULL)
# Error criterion is KGE computed on the root-squared discharges
InputsCritUp <- CreateInputsCrit(FUN_CRIT = ErrorCrit_KGE, InputsModel = InputsModelUp,
                                 RunOptions = RunOptionsUp,
                                 VarObs = "Q", Obs = BasinObs$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)
```

And see the result of the simulation:

```{r}
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:

```{r}
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]
)
```

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

```{r}
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]
)
```


## 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:

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

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

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

Performance of the model validation is then:

```{r}
KGE_down1 <- ErrorCrit_KGE(InputsCritDown, OutputsModelDown1)
```


## 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:

```{r}
OutputsCalibDown2 <- Calibration_Michel(InputsModel = InputsModelDown2,
                                        RunOptions = RunOptionsDown,
                                        InputsCrit = InputsCritDown,
                                        CalibOptions = CalibOptionsDown,
                                        FUN_MOD = RunModel_GR4J)
ParamDown2 <- OutputsCalibDown2$ParamFinalR
```

## Calibration with upstream simulated flow and parameter regularisation

The regularisation follow the method proposed by @delavenne_regularization_2019.

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

```{r}
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.

```{r}
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

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

The KGE is then calculated for performance comparisons:

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


# Discussion

## Identification of Velocity parameter

Both calibrations overestimate this parameter:

```{r}
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)
```

## 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:

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


## Parameters and performance of each subcatchment for all calibrations

```{r}
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)
```

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
