# Adding surface water levels#

*Developed by R.A. Collenteur & D. Brakenhoff*

In this example it is shown how to create a Pastas model that not only includes precipitation and evaporation, but also observed river levels. We will consider observed heads that are strongly influenced by river level, based on a visual interpretation of the raw data.

```
[1]:
```

```
import pandas as pd
import pastas as ps
import matplotlib.pyplot as plt
ps.show_versions()
ps.set_log_level("INFO")
```

```
Python version: 3.10.8
NumPy version: 1.23.5
Pandas version: 2.0.1
SciPy version: 1.10.1
Matplotlib version: 3.7.1
Numba version: 0.57.0
LMfit version: 1.2.1
Latexify version: Not Installed
Pastas version: 1.0.1
```

## 1. import and plot data#

Before a model is created, it is generally a good idea to try and visually interpret the raw data and think about possible relationship between the time series and hydrological variables. Below the different time series are plotted.

The top plot shows the observed heads, with different observation frequencies and some gaps in the data. Below that the observed river levels, precipitation and evaporation are shown. Especially the river level show a clear relationship with the observed heads. Note however how the range in the river levels is about twice the range in the heads. Based on these observations, we would expect the the final step response of the head to the river level to be around 0.5 [m/m].

```
[2]:
```

```
oseries = pd.read_csv("data/nb5_head.csv", parse_dates=True, index_col=0).squeeze()
rain = pd.read_csv("data/nb5_prec.csv", parse_dates=True, index_col=0).squeeze()
evap = pd.read_csv("data/nb5_evap.csv", parse_dates=True, index_col=0).squeeze()
waterlevel = pd.read_csv("data/nb5_riv.csv", parse_dates=True, index_col=0).squeeze()
ps.plots.series(oseries, [rain, evap, waterlevel], figsize=(10, 5), hist=False);
```

## 2. Create a timeseries model#

First we create a model with precipitation and evaporation as explanatory time series. The results show that precipitation and evaporation can explain part of the fluctuations in the observed heads, but not all of them.

```
[3]:
```

```
ml = ps.Model(oseries.resample("D").mean().dropna(), name="River")
sm = ps.RechargeModel(rain, evap, rfunc=ps.Exponential(), name="recharge")
ml.add_stressmodel(sm)
ml.solve(tmin="2000", tmax="2019-10-29")
ml.plots.results(figsize=(12, 8));
```

```
Fit report River Fit Statistics
==================================================
nfev 17 EVP 38.61
nobs 5963 R2 0.39
noise True RMSE 0.46
tmin 2000-01-27 00:00:00 AIC -30348.84
tmax 2019-10-29 00:00:00 BIC -30315.37
freq D Obj 18.34
warmup 3650 days 00:00:00 ___
solver LeastSquares Interp. No
Parameters (5 optimized)
==================================================
optimal stderr initial vary
recharge_A 511.044718 ±39.36% 183.785267 True
recharge_a 215.400021 ±38.66% 10.000000 True
recharge_f -1.596414 ±19.43% -1.000000 True
constant_d 8.538432 ±3.09% 8.547142 True
noise_alpha 66.994468 ±14.22% 1.000000 True
```

## 3. Adding river water levels#

Based on the analysis of the raw data, we expect that the river levels can help to explain the fluctuations in the observed heads. Here, we add a stress model (`ps.StressModel`

) to add the rivers level as an explanatory time series to the model. The model fit is greatly improved, showing that the rivers help in explaining the observed fluctuations in the observed heads. It can also be observed how the response of the head to the river levels is a lot faster than the response to precipitation
and evaporation.

```
[4]:
```

```
w = ps.StressModel(waterlevel, rfunc=ps.One(), name="waterlevel", settings="waterlevel")
ml.add_stressmodel(w)
ml.solve(tmin="2000", tmax="2019-10-29")
axes = ml.plots.results(figsize=(12, 8))
axes[-1].set_xlim(0, 10); # By default, the axes between responses are shared.
```

```
Fit report River Fit Statistics
===================================================
nfev 17 EVP 93.39
nobs 5963 R2 0.93
noise True RMSE 0.15
tmin 2000-01-27 00:00:00 AIC -39083.24
tmax 2019-10-29 00:00:00 BIC -39043.08
freq D Obj 4.24
warmup 3650 days 00:00:00 ___
solver LeastSquares Interp. No
Parameters (6 optimized)
===================================================
optimal stderr initial vary
recharge_A 206.385700 ±23.03% 183.785267 True
recharge_a 166.839116 ±22.57% 10.000000 True
recharge_f -1.265247 ±16.37% -1.000000 True
waterlevel_d 0.422350 ±0.73% 1.000000 True
constant_d 8.477319 ±1.00% 8.547142 True
noise_alpha 32.223835 ±10.56% 1.000000 True
```