Ensemble Predictions#

R.A. Collenteur, Eawag, 2025

This notebook shows how to use Pastas and meteorological ensemble forecasts to generate ensemble groundwater predictions (EGPs). The goal is to forecast the groundwater levels for a well in Switzerland (Gossau), one month ahead. Meteorological ensemble forecasts of the ECMWF with 51 ensemble members are used as data input. These members represent the uncertainty in the meteorological input data.

The Pastas model is calibrated on 10 years of head data prior to the start of the forecast, using meteorological data provided by MeteoSwiss. In this example, meteorological forecasts are used, but it is straightforward to extend this to other input data such as ensembles of pumping forecasts.

Note: Collenteur et al. (In submission) Ensemble groundwater predictions (EGP) in alluvial aquifers in Switzerland.

0. Import Python Packages#

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

import pastas as ps

ps.set_log_level("ERROR")
ps.show_versions()

Pastas     : 2.0.0
Python     : 3.14.6
Numpy      : 2.4.6
Pandas     : 3.0.3
Scipy      : 1.18.0
Matplotlib : 3.11.0
Numba      : 0.65.1

1. Load data#

Below the head and meteorological data is loaded for the well in Gossau, Switzerland.

head = pd.read_csv("data_forecast/heads.csv", index_col=0, parse_dates=True).squeeze()
prec = pd.read_csv("data_forecast/prec.csv", index_col=0, parse_dates=True).squeeze()
evap = pd.read_csv("data_forecast/evap.csv", index_col=0, parse_dates=True).squeeze()
temp = pd.read_csv("data_forecast/temp.csv", index_col=0, parse_dates=True).squeeze()

ps.plots.series(
    head,
    [prec, evap, temp],
    tmin="2004",
    tmax="2014",
    titles=False,
    labels=["Head\n[m]", "Prec.\n[mm/d]", "Evap.\n[mm/d]", "Temp.\n[C]"],
    table=True,
)
plt.tight_layout()

../_images/4ba38b9b76dccb9da4cbc6051399d6ab8ecdd8ecb47af6b15ef6aecb7d0e729b.png

2. Make Pastas Model#

We now make a Pastas model to simulate the heads for this monitoring well in Gossau. Only meterological data, which is also available as forecast data, is used to model the groundwater levels. A nonlinear recharge model including a snow module is applied to compute the recharge. The model is calibrated on weekly groundwater level data in the period 2004-2014.

ml = ps.Model(head)
ml.add_stressmodel(
    ps.RechargeModel(
        prec=prec,
        evap=evap,
        rfunc=ps.Gamma(),
        recharge=ps.rch.FlexModel(snow=True),
        temp=temp,
        name="rch",
    )
)

ml.set_parameter("rch_tt", vary=False)
ml.solve(
    tmin="2004", tmax="2014-02-01", report=True, fit_constant=False, freq_obs="10D"
)
ml.add_noisemodel(ps.ArNoiseModel())
ml.set_parameter("rch_srmax", vary=False)
ml.solve(
    tmin="2004", tmax="2014-02-01", initial=False, fit_constant=False, freq_obs="10D"
)

Fit report solver                   Fit Statistics
==================================================
nfev     34                     EVP          80.91
nobs     370                    R2            0.81
noise    False                  RMSE          0.21
tmin     2004-01-01 00:00:00    AICc      -1141.69
tmax     2014-02-01 00:00:00    BIC       -1110.78
freq     D                      Obj            nan
freq_obs 10D                    ___               
warmup   3650 days 00:00:00     Interp.         No

Parameters (8 optimized)
==================================================
               optimal     initial   vary
rch_A         0.459714    0.405356   True
rch_n         1.262503    1.000000   True
rch_a        19.925843   10.000000   True
rch_srmax    72.381308  250.000000   True
rch_lp        0.250000    0.250000  False
rch_ks       34.242572  100.000000   True
rch_gamma     7.900758    2.000000   True
rch_kv        1.097887    1.000000   True
rch_simax     2.000000    2.000000  False
rch_tt        0.000000    0.000000  False
rch_k         5.480549    2.000000   True
constant_d  637.514501    0.000000  False
Fit report solver                   Fit Statistics
==================================================
nfev     33                     EVP          79.79
nobs     370                    R2            0.80
noise    True                   RMSE          0.22
tmin     2004-01-01 00:00:00    AICc      -1409.07
tmax     2014-02-01 00:00:00    BIC       -1378.16
freq     D                      Obj            nan
freq_obs 10D                    ___               
warmup   3650 days 00:00:00     Interp.         No

Parameters (8 optimized)
==================================================
                optimal    initial   vary
rch_A          0.451175   0.459714   True
rch_n          1.098605   1.262503   True
rch_a         24.987849  19.925843   True
rch_srmax     72.381308  72.381308  False
rch_lp         0.250000   0.250000  False
rch_ks        32.704805  34.242572   True
rch_gamma      5.529698   7.900758   True
rch_kv         1.042876   1.097887   True
rch_simax      2.000000   2.000000  False
rch_tt         0.000000   0.000000  False
rch_k          5.713153   5.480549   True
constant_d   637.471217   0.000000  False
noise_alpha   32.274831  10.000000   True

Visualize the model results#

ml.plots.results();

../_images/288806bd85597a2f464d5c44bfb118cd646308ef4bac7c8de120927da9d9a7e9.png

3. Prepare the forecast ensembles#

Now that we have a calibrated Pastas model, we can prepare the forecast data used to generate the groundwater ensemble predictions. The forecast data should be prepared carefully as a dictionary. For each stressmodel, one item in the dictionary should be provided where the key is the stressmodel name (i.e., same as in ml.stressmodels). The value of that dictionary should be a dictionary as well with the key the name of the stress argument, provided to the stressmodel and the value the pandas.DataFrame with the ensemble time series.

Each DataFrame should have the same DateimeIndex with the dates of the forecasts of the input data. It should also have the same number of columns, where each column represents an ensemble member (i.e., the precipitation and evaporation data need the same number of members). All these properties are internally checked by the internal method ps.forecast._check_forecast_data and an error is raised if something is wrong.

fc = {
    "rch": {
        "prec": pd.read_csv(
            "data_forecast/ensemble_prec.csv", index_col=0, parse_dates=True
        ),
        "evap": pd.read_csv(
            "data_forecast/ensemble_evap.csv", index_col=0, parse_dates=True
        ),
        "temp": pd.read_csv(
            "data_forecast/ensemble_temp.csv", index_col=0, parse_dates=True
        ),
    }
}

fig, axes = plt.subplots(1, 3, figsize=(10, 3))
names = ["Cum. Precipitation", "Cum. Evaporation", "Temperature"]
for i, (key, name) in enumerate(zip(fc["rch"], names)):
    series = fc["rch"][key]
    data = series.cumsum() if "Cum" in name else series
    data.plot(legend=False, ax=axes[i], color="k", alpha=0.7, title=name)

../_images/ca449536e74daf0a9b0fde861ec41b85db7921ab846f8680317b112b1f872261.png

4. Compute GWL forecasts#

We can now generate the forecasts of the groundwater levels using the calibrated model. We can include the parameter uncertainty, by drawing multiple parameter sets (i.e., \(N\) parameter sets). The model is run with each of the \(N\) parameter sets for all \(M\) ensemble member sets (i.e., set of precipitation, evaporation and temperature), resulting in \(N\) x \(M\) forecasts.

For each individual forecast the mean forecast and the variance of the error or noise is returned, depending on whether or not post_process is True or False. If True, the forecast is:

corrected using the last GWL measurement and the fitted noisemodel, and
the variance of the error is computed using the noise and the noisemodel.

If False, the forecast is not corrected and the variance of the residuals is returned. Thus, for each combination of a forecast of the ensemble members and the parameter sets a mean forecast and the variance of theb forecast is returned. These can be used to compute a prediction interval.

# Draw parameter sets
params = ml.solver.get_parameter_sample(n=10)
params[:2]

array([[ 4.57179234e-01,  1.15326258e+00,  2.43777746e+01,
         7.23813081e+01,  2.50000004e-01,  3.30105132e+01,
         6.24739298e+00,  1.01051130e+00,  2.00000000e+00,
        -3.77854317e-25,  5.59329378e+00,  6.37471217e+02,
         2.96395710e+01],
       [ 4.31212953e-01,  1.09157135e+00,  2.47062890e+01,
         7.23813081e+01,  2.49999997e-01,  3.28412285e+01,
         5.35771475e+00,  9.74035250e-01,  2.00000000e+00,
        -3.74302543e-25,  5.71317259e+00,  6.37471217e+02,
         2.99592906e+01]])

# Generate the forecast
df_nopp = ps.forecast.forecast(ml, fc, p=params, post_process=False)
df_pp = ps.forecast.forecast(ml, fc, p=params, post_process=True)

df_pp.head()

ensemble_member	0										...	50
param_member	0		1		2		3		4		...	5		6		7		8		9
forecast	mean	var	mean	var	mean	var	mean	var	mean	var	...	mean	var	mean	var	mean	var	mean	var	mean	var
2014-01-02	638.032448	0.002894	638.030963	0.002863	638.032203	0.002903	638.036304	0.002931	638.027181	0.002903	...	638.031108	0.002748	638.028596	0.002887	638.031366	0.002790	638.032493	0.002849	638.029430	0.002827
2014-01-03	638.026850	0.005598	638.024814	0.005542	638.026408	0.005621	638.032566	0.005674	638.019019	0.005621	...	638.027805	0.005355	638.024412	0.005592	638.028260	0.005420	638.030508	0.005518	638.025354	0.005485
2014-01-04	638.020433	0.008127	638.017864	0.008047	638.019751	0.008166	638.027851	0.008240	638.010114	0.008165	...	638.023049	0.007828	638.018573	0.008128	638.023650	0.007898	638.026536	0.008019	638.019878	0.007983
2014-01-05	638.014257	0.010490	638.011327	0.010391	638.013274	0.010550	638.023582	0.010640	638.001261	0.010545	...	638.070600	0.010174	638.076561	0.010504	638.077478	0.010235	638.081557	0.010363	638.067837	0.010332
2014-01-06	638.028487	0.012699	638.026881	0.012583	638.026391	0.012782	638.043525	0.012886	638.012281	0.012773	...	638.116700	0.012399	638.124923	0.012731	638.124361	0.012436	638.129203	0.012560	638.110995	0.012539

5 rows × 1020 columns

The returned variable df is a DataFrame containing the ensemble groundwater predictions. The columns of df are a MultiIndex with the first row the \(M\) ensemble members (i.e., a single meteorological forecast), the second row the \(N\) parameter sets, and the third row three columns with the mean prediction and the variance of the error that can be used to construct the prediction interval.

5. Compute the overall mean and variance#

The forecast method returns a matrix with with \(N\) x \(M\) forecasts of the mean and variance. From these, we can compute the mean forecast over all ensembles and parameter sets easily. For the variance, which we use to compute the (for example) the 95% prediction interval, we apply the law of total variance. The forecast method provides a convenience method to do this: ps.forecast.get_overall_mean_and_variance. The method takes the output dataframe df and returns two pandas.Series with the overall mean and the variance.

fig, axes = plt.subplots(1, 2, figsize=(10.0, 4.0), sharey=True, layout="tight")

suptitles = ["Without post-processing", "With post-processing"]

for i, (df, label) in enumerate(zip([df_nopp, df_pp], ["No PP", "PP"])):
    mean, var = ps.forecast.get_overall_mean_and_variance(df)
    std = np.sqrt(var)

    ax = axes[i]
    ml.oseries.series_original.loc[df.index].plot(
        ax=ax, marker=".", color="k", linestyle="None", zorder=100
    )

    mean.plot(ax=ax, alpha=0.5, zorder=100)
    ax.plot(mean, color="k", label="Mean")
    ax.fill_between(
        mean.index,
        mean - std * 1.96,
        mean + std * 1.96,
        color="k",
        alpha=0.2,
        label="1 std",
    )

    df.loc[:, (slice(None), slice(None), "mean")].plot(
        color="gray", legend=False, ax=ax
    )
    ax.set_title(suptitles[i], x=0.01, y=0.93, ha="left", va="top")

axes[-1].legend(
    ["Observations", "Mean forecast", "Ensemble Members", "95% Prediction interval"],
    loc="lower right",
    ncol=2,
    bbox_to_anchor=(1.0, 1.0),
    frameon=False,
)

axes[0].set_ylabel("Head [m]")

/tmp/ipykernel_1121/696467994.py:15: UserWarning: This axis already has a converter set and is updating to a potentially incompatible converter
  ax.plot(mean, color="k", label="Mean")
/home/docs/checkouts/readthedocs.org/user_builds/pastas/envs/latest/lib/python3.14/site-packages/pandas/plotting/_matplotlib/core.py:997: UserWarning: This axis already has a converter set and is updating to a potentially incompatible converter
  return ax.plot(*args, **kwds)
/tmp/ipykernel_1121/696467994.py:15: UserWarning: This axis already has a converter set and is updating to a potentially incompatible converter
  ax.plot(mean, color="k", label="Mean")
/home/docs/checkouts/readthedocs.org/user_builds/pastas/envs/latest/lib/python3.14/site-packages/pandas/plotting/_matplotlib/core.py:997: UserWarning: This axis already has a converter set and is updating to a potentially incompatible converter
  return ax.plot(*args, **kwds)

Text(0, 0.5, 'Head [m]')

../_images/1d7bc392627eb523a811f5b8afd109b18f1ef6b34677379c991b32502ee5bc6d.png

In the figure above we see the two ensemble predictions, without (left) and with (right) post-processing. What we can see is that the prediction intervals widen over time when using the post/processing, whereas without it, these remain more stable and only widen due to the increased uncertainty in the ensemble members. In general, we can see that the largest part of the uncertainty is due to the uncertainty in the input data, as visible bz the mean forecasts of the ensemble members (the grey lines) covering a large part of the prediction intervals.