V05_sd_model.Rmd

---
title: "Simulating a reservoir with semi-distributed GR4J model"
author: "David Dorchies"
bibliography: V00_airgr_ref.bib
output: rmarkdown::html_vignette
vignette: >
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteIndexEntry{Simulating a reservoir with semi-distributed GR4J model}
  %\VignetteEncoding{UTF-8}
---

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

# 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.

`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 two strategies for calibrating the downstream subcatchment:

- using upstream observed flows
- using upstream simulated flows

We finally compare these calibrations with a theoretical set of parameters.

## 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$Qmm)-2)])) / 2
summary(cbind(QObsUp = BasinObs$Qmm, QObsDown), digits = 3)
```

# 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$DatesR, format = "%Y-%m-%d") == "1999-12-31"))
RunOptionsUp <- CreateRunOptions(FUN_MOD = RunModel_GR4J,
                                 InputsModel = InputsModelUp,
                                 IndPeriod_WarmUp = NULL, IndPeriod_Run = Ind_Run,
                                 IniStates = NULL, IniResLevels = NULL)
InputsCritUp <- CreateInputsCrit(FUN_CRIT = ErrorCrit_NSE, InputsModel = InputsModelUp,
                                 RunOptions = RunOptionsUp,
                                 VarObs = "Q", Obs = BasinObs$Qmm[Ind_Run])
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 with upstream flow observations

Observed flow data contain `NA` values and a complete time series is mandatory for running the Lag model. We propose to complete the observed upstream flow with linear interpolation:

```{r}
QObsUp <- imputeTS::na_interpolation(BasinObs$Qmm)
```

we need to create the `InputsModel` object completed with upstream information:

```{r}
InputsModelDown1 <- CreateInputsModel(
  FUN_MOD = RunModel_GR4J, DatesR = BasinObs$DatesR,
  Precip = BasinObs$P, PotEvap = BasinObs$E,
  Qupstream = matrix(QObsUp, ncol = 1), # upstream observed flow
  LengthHydro = 1e2 * 1e3, # distance between upstream catchment outlet & the downstream one [m]
  BasinAreas = c(180, 180) # upstream and downstream areas [km²]
)
```

And then 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_NSE, InputsModel = InputsModelDown1,
                                   RunOptions = RunOptionsDown,
                                   VarObs = "Q", Obs = QObsDown[Ind_Run])
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)
```

To run the complete model, we should substitute the observed upstream flow by the simulated one:

```{r}
InputsModelDown2 <- InputsModelDown1
InputsModelDown2$Qupstream[Ind_Run] <- OutputsModelUp$Qsim
```

`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}
CritDown1 <- ErrorCrit_NSE(InputsCritDown, OutputsModelDown1)
```


# Calibration of the downstream subcatchment 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
```


# Discussion

## Identification of Velocity parameter

The theoretical Velocity parameter should be equal to:

```{r}
Velocity <- InputsModelDown1$LengthHydro / (2 * 86400)
paste(format(Velocity), "m/s")
```

Both calibrations overestimate this parameter:

```{r}
mVelocity <- matrix(c(Velocity,
                 OutputsCalibDown1$ParamFinalR[1],
                 OutputsCalibDown2$ParamFinalR[1]),
               ncol = 1,
               dimnames = list(c("theoretical",
                                 "calibrated with observed upstream flow",
                                 "calibrated with simulated  upstream flow"),
                               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 and we know the lag time. So this set of parameter should give a better performance criteria:

```{r}
ParamDownTheo <- c(Velocity, OutputsCalibUp$ParamFinalR)
OutputsModelDownTheo <- RunModel(InputsModel = InputsModelDown2,
                                 RunOptions = RunOptionsDown,
                                 Param = ParamDownTheo,
                                 FUN_MOD = RunModel_GR4J)
CritDownTheo <- ErrorCrit_NSE(InputsCritDown, OutputsModelDownTheo)
```



## Parameters and performance of each subcatchment for all calibrations

```{r}
comp <- matrix(c(0, OutputsCalibUp$ParamFinalR,
                 rep(OutputsCalibDown1$ParamFinalR, 2),
                 OutputsCalibDown2$ParamFinalR,
                 ParamDownTheo),
               ncol = 5, byrow = TRUE)
comp <- cbind(comp, c(OutputsCalibUp$CritFinal,
                      OutputsCalibDown1$CritFinal,
                      CritDown1$CritValue,
                      OutputsCalibDown2$CritFinal,
                      CritDownTheo$CritValue))
colnames(comp) <- c("Velocity", paste0("X", 1:4), "NSE")
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",
                    "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.