A Basic Model

In this example application it is shown how a simple time series model can be developed to simulate groundwater levels. The recharge (calculated as precipitation minus evaporation) is used as the explanatory time series.

import pandas as pd

import pastas as ps

ps.show_versions()

Pastas version: 1.11.0
Python version: 3.11.12
NumPy version: 2.2.6
Pandas version: 2.3.2
SciPy version: 1.16.1
Matplotlib version: 3.10.6
Numba version: 0.61.2

1. Importing the dependent time series data

In this codeblock a time series of groundwater levels is imported using the read_csv function of pandas. As pastas expects a pandas Series object, the data is squeezed. To check if you have the correct data type (a pandas Series object), you can use type(oseries) as shown below.

The following characteristics are important when importing and preparing the observed time series:

• The observed time series are stored as a pandas Series object.

• The time step can be irregular.

The head is stored in the variable gwdata in the code cell below. The heads can be plotted using standard pandas commands or the built-in ps.plots.series method, which is shown below.

# Import groundwater time seriesm and squeeze to Series object
gwdata = pd.read_csv(
    "data/head_nb1.csv", parse_dates=["date"], index_col="date"
).squeeze()
print("The data type of the oseries is: %s" % type(gwdata))

# Plot the observed groundwater levels
ps.plots.series(head=gwdata, figsize=(8, 3), labels=["head (m)"], grid=True);

The data type of the oseries is: <class 'pandas.core.series.Series'>

2. Import the independent time series

Two explanatory series are used: the precipitation and the potential evaporation. These need to be pandas Series objects, as for the observed heads.

Important characteristics of these time series are:

• All series are stored as pandas Series objects.

• The series may have irregular time intervals, but then it will be converted to regular time intervals when creating the time series model later on.

• It is preferred to use the same length units as for the observed heads.

# Import observed precipitation series
precip = pd.read_csv(
    "data/rain_nb1.csv", parse_dates=["date"], index_col="date"
).squeeze()
print("The data type of the precip series is: %s" % type(precip))

# Import observed evaporation series
evap = pd.read_csv(
    "data/evap_nb1.csv", parse_dates=["date"], index_col="date"
).squeeze()
print("The data type of the evap series is: %s" % type(evap))

# Calculate the recharge to the groundwater
recharge = precip - evap
recharge.name = "recharge"  # set name if pandas series
print("The data type of the recharge series is: %s" % type(recharge))

# Plot the time series of the head, precipitation. and evaporation
ps.plots.series(
    head=gwdata,
    stresses=[precip, evap],
    figsize=(8, 6),
    labels=["head (m)", "precip (m/d)", "evap (m/d)"],
    grid=True,
);

The data type of the precip series is: <class 'pandas.core.series.Series'>
The data type of the evap series is: <class 'pandas.core.series.Series'>
The data type of the recharge series is: <class 'pandas.core.series.Series'>

3. Create the time series model

In this code block the actual time series model is created. First, an instance of the Model class is created (named ml here). Second, the different components of the time series model are created and added to the model. The imported time series are automatically checked for missing values and other inconsistencies. The keyword argument fillnan can be used to determine how missing values are handled. If any nan-values are found this will be reported by pastas.

# Create a model object by passing it the observed series
ml = ps.Model(gwdata, name="GWL")

# Add the recharge data as explanatory variable
sm = ps.StressModel(recharge, ps.Gamma(), name="recharge", settings="evap")
ml.add_stressmodel(sm)

The Time Series 'recharge' has nan-values. Pastas will use the fill_nan settings to fill up the nan-values.

4. Solve the model

The next step is to compute the optimal model parameters. The default solver uses a non-linear least squares method for the optimization. The python package scipy is used (info on scipy's least_squares solver can be found here). Some standard optimization statistics are reported along with the optimized parameter values and correlations.

ml.solve()

Fit report GWL                    Fit Statistics
================================================
nfev    15                     EVP         91.78
nobs    644                    R2           0.92
noise   False                  RMSE         0.12
tmin    1985-11-14 00:00:00    AICc     -2688.47
tmax    2015-06-28 00:00:00    BIC      -2670.66
freq    D                      Obj          4.89
warmup  3650 days 00:00:00     ___              
solver  LeastSquares           Interp.        No

Parameters (4 optimized)
================================================
               optimal     initial  vary
recharge_A  674.042002  215.674528  True
recharge_n    1.185041    1.000000  True
recharge_a  106.228695   10.000000  True
constant_d   27.588701   27.900078  True

5. Plot the results

The solution can be plotted after a solution has been obtained.

ml.plot()

<Axes: xlabel='date', ylabel='Head'>

6. Advanced plotting

There are many ways to further explore the time series model. pastas has some built-in functionalities that will provide the user with a quick overview of the model. The plots subpackage contains all the options. One of these is the method plots.results which provides a plot with more information.

ml.plots.results(figsize=(10, 6))

[<Axes: xlabel='date', ylabel='Head'>,
 <Axes: >,
 <Axes: title={'right': "Stresses: ['recharge']"}, ylabel='Rise'>,
 <Axes: title={'center': 'Step response'}>,
 <Axes: title={'left': 'Model parameters ($n_c$=4)'}>]

7. Statistics

The stats subpackage includes a number of statistical functions that may applied to the model. One of them is the summary method, which gives a summary of the main statistics of the model.

ml.stats.summary()

	Value
Statistic
rmse	0.123245
sse	9.781998
mae	0.098034
nse	0.917782
evp	91.778169
rsq	0.917782
kge	0.940616
bic	-2670.657030
aic	-2688.527825
aicc	-2688.465227

8. Improvement: estimate evaporation factor

In the previous model, the recharge was estimated as precipitation minus potential evaporation. A better model is to estimate the actual evaporation as a factor (called the evaporation factor here) times the potential evaporation. First, new model is created (called ml2 here so that the original model ml does not get overwritten). Second, the RechargeModel object with a Linear recharge model is created, which combines the precipitation and evaporation series and adds a parameter for the evaporation factor f. The RechargeModel object is added to the model, the model is solved, and the results and statistics are plotted to the screen. Note that the new model gives a better fit (lower root mean squared error and higher explained variance), but that the Akiake information criterion indicates that the addition of the additional parameter does not improve the model significantly (the Akaike criterion for model ml2 is higher than for model ml).

# Create a model object by passing it the observed series
ml2 = ps.Model(gwdata)

# Add the recharge data as explanatory variable
ts1 = ps.RechargeModel(
    precip,
    evap,
    ps.Gamma(),
    name="rainevap",
    recharge=ps.rch.Linear(),
    settings=("prec", "evap"),
)
ml2.add_stressmodel(ts1)

# Solve the model
ml2.solve()

# Plot the results
ml2.plot()

# Statistics
ml2.stats.summary()

Fit report head                   Fit Statistics
================================================
nfev    14                     EVP         93.28
nobs    644                    R2           0.93
noise   False                  RMSE         0.11
tmin    1985-11-14 00:00:00    AICc     -2816.26
tmax    2015-06-28 00:00:00    BIC      -2794.01
freq    D                      Obj          4.00
warmup  3650 days 00:00:00     ___              
solver  LeastSquares           Interp.        No

Parameters (5 optimized)
================================================
               optimal     initial  vary
rainevap_A  618.958765  215.674528  True
rainevap_n    1.049253    1.000000  True
rainevap_a  146.196518   10.000000  True
rainevap_f   -1.408118   -1.000000  True
constant_d   28.019661   27.900078  True

	Value
Statistic
rmse	0.111429
sse	7.996130
mae	0.086444
nse	0.932792
evp	93.279202
rsq	0.932792
kge	0.951650
bic	-2794.010733
aic	-2816.349227
aicc	-2816.255183

Origin of the series

• The rainfall data is taken from rainfall station Heibloem in The Netherlands.

• The evaporation data is taken from weather station Maastricht in The Netherlands.

• The head data is well B58C0698, which was obtained from Dino loket