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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Adding Multiple Wells
This notebook shows how a WellModel can be used to fit multiple wells with one response function. The influence of the individual wells is scaled by the distance to the observation point. 
Developed by R.C. Caljé, (Artesia Water 2020), D.A. Brakenhoff, (Artesia Water 2019), and R.A. Collenteur, (Artesia Water 2018)
Step1:  Load data from a Menyanthes file
Menyanthes is timeseries analysis software used by many people in the Netherlands. In this example a Menyanthes-file with one observation-series is imported, and simulated. There are several stresses in the Menyanthes-file, among which are three groundwater extractions with a significant influence on groundwater head.
Import the Menyanthes-file with observations and stresses.
Step2:  Get the distances of the extractions to the observation well. Extraction 1 is about two times as far from the observation well as extraction 2 and 3. We will use this information later in our WellModel.
Step3:  Then plot the observations, together with the diferent stresses in the Menyanthes file.
Step4:  Create a model with a separate StressModel for each extraction
First we create a model with a separate StressModel for each groundwater extraction. First we create a model with the heads timeseries and add recharge as a stress.
Step5:  Get the precipitation and evaporation timeseries and round the index to remove the hours from the timestamps.
Step6:  Create a recharge stressmodel and add to the model.
Step7:  Get the extraction timeseries.
Step8:  Add each of the extractions as a separate StressModel.
Step9:  Solve the model. 
Note the use of ps.LmfitSolve. This is because of an issue concerning optimization with small parameter values in scipy.least_squares. This is something that may influence models containing a WellModel (which we will be creating later) and since we want to keep the models in this Notebook as similar as possible, we're also using ps.LmfitSolve here.
Step10:  Visualize the results
Plot the decomposition to see the individual influence of each of the wells.
Step11:  We can calculate the gain of each extraction (quantified as the effect on the groundwater level of a continuous extraction of ~1 Mm$^3$/yr).
Step12:  Create a model with a WellModel
We can reduce the number of parameters in the model by including the three extractions in a WellModel. This WellModel takes into account the distances from the three extractions to the observation well, and assumes constant geohydrological properties. All of the extractions now share the same response function, scaled by the distance between the extraction well and the observation well.
First we create a new model and add recharge.
Step13:  We have all the information we need to create a WellModel
Step14:  Solve the model. 
We are once again using ps.LmfitSolve. The user is notified about the preference for this solver in a WARNING when creating the WellModel (see above). 
As we can see, the fit with the measurements (EVP) is similar to the result with the previous model, with each well included separately.
Step15:  Visualize the results
Plot the decomposition to see the individual influence of each of the wells
Step16:  Plot the stacked influence of each of the individual extraction wells in the results plot
Step17:  Get parameters for each well (including the distance) and calculate the gain. The WellModel reorders the stresses from closest to the observation well, to furthest from the observation well. We have take this into account during the post-processing.
The gain of extraction 1 is lower than the gain of extraction 2 and 3. This will always be the case in a WellModel when the distance from the observation well to extraction 1 is larger than the distance to extraction 2 and 3.
Step18:  Compare individual StressModels and WellModel
Compare the gains that were calculated by the individual StressModels and the WellModel.
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Python Code:
import numpy as np
import pandas as pd
import pastas as ps
import matplotlib.pyplot as plt
ps.show_versions()
Explanation: Adding Multiple Wells
This notebook shows how a WellModel can be used to fit multiple wells with one response function. The influence of the individual wells is scaled by the distance to the observation point. 
Developed by R.C. Caljé, (Artesia Water 2020), D.A. Brakenhoff, (Artesia Water 2019), and R.A. Collenteur, (Artesia Water 2018)
End of explanation
fname = '../data/MenyanthesTest.men'
meny = ps.read.MenyData(fname)
Explanation: Load data from a Menyanthes file
Menyanthes is timeseries analysis software used by many people in the Netherlands. In this example a Menyanthes-file with one observation-series is imported, and simulated. There are several stresses in the Menyanthes-file, among which are three groundwater extractions with a significant influence on groundwater head.
Import the Menyanthes-file with observations and stresses.
End of explanation
# Get distances from metadata
xo = meny.H["Obsevation well"]['xcoord']
yo = meny.H["Obsevation well"]['ycoord']
distances = []
extraction_names = ['Extraction 2', 'Extraction 3']
for extr in extraction_names:
    xw = meny.IN[extr]["xcoord"]
    yw = meny.IN[extr]["ycoord"]
    distances.append(np.sqrt((xo-xw)**2 + (yo-yw)**2))
extraction_names = [name.replace(" ", "_") for name in extraction_names]  # replace spaces in names for Pastas
df = pd.DataFrame(distances, index=extraction_names, columns=['Distance to observation well'])
df
Explanation: Get the distances of the extractions to the observation well. Extraction 1 is about two times as far from the observation well as extraction 2 and 3. We will use this information later in our WellModel.
End of explanation
# plot timeseries
f1, axarr = plt.subplots(len(meny.IN)+1, sharex=True, figsize=(10,8))
oseries = meny.H['Obsevation well']["values"]
oseries.plot(ax=axarr[0], color='k')
axarr[0].set_title(meny.H['Obsevation well']["Name"])
for i, (name, data) in enumerate(meny.IN.items(), start=1):
    data["values"].plot(ax=axarr[i])
    axarr[i].set_title(name)
plt.tight_layout(pad=0)
Explanation: Then plot the observations, together with the diferent stresses in the Menyanthes file.
End of explanation
oseries = ps.TimeSeries(meny.H['Obsevation well']['values'].dropna(), name="heads", settings="oseries")
# create model
ml = ps.Model(oseries)
Explanation: Create a model with a separate StressModel for each extraction
First we create a model with a separate StressModel for each groundwater extraction. First we create a model with the heads timeseries and add recharge as a stress.
End of explanation
prec = meny.IN['Precipitation']['values']
prec.index = prec.index.round("D")
prec.name = "prec"
evap = meny.IN['Evaporation']['values']
evap.index = evap.index.round("D")
evap.name = "evap"
Explanation: Get the precipitation and evaporation timeseries and round the index to remove the hours from the timestamps.
End of explanation
rm = ps.RechargeModel(prec, evap, ps.Exponential, 'Recharge')
ml.add_stressmodel(rm)
Explanation: Create a recharge stressmodel and add to the model.
End of explanation
stresses = []
for name in extraction_names:
    # get extraction timeseries
    s = meny.IN[name.replace("_", " ")]['values']
    
    # convert index to end-of-month timeseries
    s.index = s.index.to_period("M").to_timestamp("M")
    
    # resample to daily values
    s_daily = ps.utils.timestep_weighted_resample_fast(s, "D")
    
    # create pastas.TimeSeries object
    stress = ps.TimeSeries(s_daily.dropna(), name=name, settings="well")
    
    # append to stresses list
    stresses.append(stress)
Explanation: Get the extraction timeseries.
End of explanation
for stress in stresses:
    sm = ps.StressModel(stress, ps.Hantush, stress.name, up=False)
    ml.add_stressmodel(sm)
Explanation: Add each of the extractions as a separate StressModel.
End of explanation
ml.solve(solver=ps.LmfitSolve)
Explanation: Solve the model. 
Note the use of ps.LmfitSolve. This is because of an issue concerning optimization with small parameter values in scipy.least_squares. This is something that may influence models containing a WellModel (which we will be creating later) and since we want to keep the models in this Notebook as similar as possible, we're also using ps.LmfitSolve here.
End of explanation
ml.plots.decomposition();
Explanation: Visualize the results
Plot the decomposition to see the individual influence of each of the wells.
End of explanation
for i in range(len(extraction_names)):
    name = extraction_names[i]
    sm = ml.stressmodels[name]
    p = ml.get_parameters(name)
    gain = sm.rfunc.gain(p) * 1e6 / 365.25
    print(f"{name}: gain = {gain:.3f} m / Mm^3/year")
    df.at[name, 'gain StressModel'] = gain
Explanation: We can calculate the gain of each extraction (quantified as the effect on the groundwater level of a continuous extraction of ~1 Mm$^3$/yr).
End of explanation
ml_wm = ps.Model(oseries, oseries.name + "_wm")
rm = ps.RechargeModel(prec, evap, ps.Gamma, 'Recharge')
ml_wm.add_stressmodel(rm)
Explanation: Create a model with a WellModel
We can reduce the number of parameters in the model by including the three extractions in a WellModel. This WellModel takes into account the distances from the three extractions to the observation well, and assumes constant geohydrological properties. All of the extractions now share the same response function, scaled by the distance between the extraction well and the observation well.
First we create a new model and add recharge.
End of explanation
w = ps.WellModel(stresses, ps.HantushWellModel, "Wells", distances, settings="well")
ml_wm.add_stressmodel(w)
Explanation: We have all the information we need to create a WellModel:
- timeseries for each of the extractions, these are passed as a list of stresses
- distances from each extraction to the observation point, note that the order of these distances must correspond to the order of the stresses.
Note: the WellModel only works with a special version of the Hantush response function called HantushWellModel. This is because the response function must support scaling by a distance $r$. The HantushWellModel response function has been modified to support this. The Hantush response normally takes three parameters: the gain $A$, $a$ and $b$. This special version accepts 4 parameters: it interprets that fourth parameter as the distance $r$, and uses it to scale the parameters accordingly. 
Create the WellModel and add to the model.
End of explanation
ml_wm.solve(solver=ps.LmfitSolve)
Explanation: Solve the model. 
We are once again using ps.LmfitSolve. The user is notified about the preference for this solver in a WARNING when creating the WellModel (see above). 
As we can see, the fit with the measurements (EVP) is similar to the result with the previous model, with each well included separately.
End of explanation
ml_wm.plots.decomposition();
Explanation: Visualize the results
Plot the decomposition to see the individual influence of each of the wells
End of explanation
ml_wm.plots.stacked_results(figsize=(10, 8));
Explanation: Plot the stacked influence of each of the individual extraction wells in the results plot
End of explanation
wm = ml_wm.stressmodels["Wells"]
for i in range(len(extraction_names)):
    # get parameters
    p = wm.get_parameters(model=ml_wm, istress=i)
    # calculate gain
    gain = wm.rfunc.gain(p) * 1e6 / 365.25
    name = wm.stress[i].name
    print(f"{name}: gain = {gain:.3f} m / Mm^3/year")
    df.at[name, 'gain WellModel'] = gain
Explanation: Get parameters for each well (including the distance) and calculate the gain. The WellModel reorders the stresses from closest to the observation well, to furthest from the observation well. We have take this into account during the post-processing.
The gain of extraction 1 is lower than the gain of extraction 2 and 3. This will always be the case in a WellModel when the distance from the observation well to extraction 1 is larger than the distance to extraction 2 and 3.
End of explanation
df.style.format("{:.4f}")
Explanation: Compare individual StressModels and WellModel
Compare the gains that were calculated by the individual StressModels and the WellModel.
End of explanation
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