Pastas Noise model

Developed by Stijn Klop and Mark Bakker

This Notebook contains a number of examples and tests with synthetic data. The purpose of this notebook is to demonstrate the noise model of Pastas.

In this Notebook, heads are generated with a known response function. Next, Pastas is used to solve for the parameters of the model it is verified that Pastas finds the correct parameters back. Several different types of errors are introduced in the generated heads and it is tested whether the confidence intervals computed by Pastas are reasonable.

The first step is to import all the required python packages.



In [1]:

%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy.special import gammainc, gammaincinv
import pandas as pd
import pastas as ps



The rainfall and reference evaporation are read from file and truncated for the period 1980 - 2000. The rainfall and evaporation series are taken from KNMI station De Bilt. The reading of the data is done using Pastas.

Heads are generated with a Gamma response function which is defined below.



In [2]:

rain = rain['1980':'1999']
evap = evap['1980':'1999']




INFO: Inferred frequency from time series RH 260: freq=D
INFO: Time Series RH 260: 30 nan-value(s) was/were found and filled with: 0.0
INFO: Inferred frequency from time series EV24 260: freq=D




In [3]:

def gamma_tmax(A, n, a, cutoff=0.99):
return gammaincinv(n, cutoff) * a

def gamma_step(A, n, a, cutoff=0.99):
tmax = gamma_tmax(A, n, a, cutoff)
t = np.arange(0, tmax, 1)
s = A * gammainc(n, t / a)
return s

def gamma_block(A, n, a, cutoff=0.99):
# returns the gamma block response starting at t=0 with intervals of delt = 1
s = gamma_step(A, n, a, cutoff)
return np.append(s[0], s[1:] - s[:-1])



The Gamma response function requires 3 input arguments; A, n and a. The values for these parameters are defined along with the parameter d, the base groundwater level. The response function is created using the functions defined above.



In [4]:

Atrue = 800
ntrue = 1.1
atrue = 200
dtrue = 20
h = gamma_block(Atrue, ntrue, atrue) * 0.001
tmax = gamma_tmax(Atrue, ntrue, atrue)
plt.plot(h)
plt.xlabel('Time (days)')
plt.ylabel('Head response (m) due to 1 mm of rain in day 1')
plt.title('Gamma block response with tmax=' + str(int(tmax)));






Create synthetic observations

Rainfall is used as input series for this example. No errors are introduced. A Pastas model is created to test whether Pastas is able to . The generated head series is purposely not generated with convolution. Heads are computed for the period 1990 - 2000. Computations start in 1980 as a warm-up period. Convolution is not used so that it is clear how the head is computed. The computed head at day 1 is the head at the end of day 1 due to rainfall during day 1. No errors are introduced.



In [5]:

step = gamma_block(Atrue, ntrue, atrue)[1:]
lenstep = len(step)
h = dtrue * np.ones(len(rain) + lenstep)
for i in range(len(rain)):
h[i:i + lenstep] += rain[i] * step

plt.figure(figsize=(12,5))
plt.legend(loc=0)
plt.xlabel('Time (years)');




/Applications/anaconda3/envs/py37_pastas/lib/python3.7/site-packages/pandas/plotting/_converter.py:129: FutureWarning: Using an implicitly registered datetime converter for a matplotlib plotting method. The converter was registered by pandas on import. Future versions of pandas will require you to explicitly register matplotlib converters.

To register the converters:
>>> from pandas.plotting import register_matplotlib_converters
>>> register_matplotlib_converters()
warnings.warn(msg, FutureWarning)



Create Pastas model

The next step is to create a Pastas model. The head generated using the Gamma response function is used as input for the Pastas model.

A StressModel instance is created and added to the Pastas model. The StressModel intance takes the rainfall series as input aswell as the type of response function, in this case the Gamma response function ( ps.Gamma).

The Pastas model is solved without a noise model since there is no noise present in the data. The results of the Pastas model are plotted.



In [6]:

sm = ps.StressModel(rain, ps.Gamma, name='recharge', settings='prec')
ml.solve(noise=False)
ml.plots.results();




INFO: Inferred frequency from time series 0: freq=D
INFO: Inferred frequency from time series RH 260: freq=D

Model Results 0                Fit Statistics
=============================================
nfev     11                     EVP    100.00
nobs     3652                   R2       1.00
noise    False                  RMSE     0.00
tmin     1990-01-01 00:00:00    AIC       nan
tmax     1999-12-31 00:00:00    BIC       nan
freq     D                      ___
warmup   3650 days 00:00:00     ___
solver   LeastSquares           ___

Parameters (4 were optimized)
=============================================
optimal  stderr     initial  vary
recharge_A    800.0  ±0.00%  224.669629  True
recharge_n      1.1  ±0.00%    1.000000  True
recharge_a    200.0  ±0.00%   10.000000  True
constant_d     20.0  ±0.00%   21.783161  True

Parameter correlations |rho| > 0.5
=============================================
recharge_A recharge_a  0.64
constant_d -0.98
recharge_n recharge_a -0.87
recharge_a constant_d -0.56



The results of the Pastas model show the calibrated parameters for the Gamma response function. The parameters calibrated using pastas are equal to the Atrue, ntrue, atrue and dtrue parameters defined above. The Explained Variance Percentage for this example model is 100%.

The results plots show that the Pastas simulation is identical to the observed groundwater. The residuals of the simulation are shown in the plot together with the response function and the contribution for each stress.

Below the Pastas block response and the true Gamma response function are plotted.



In [7]:

plt.plot(gamma_block(Atrue, ntrue, atrue), label='Synthetic response')
plt.plot(ml.get_block_response('recharge'), '-.', label='Pastas response')
plt.legend(loc=0)
plt.ylabel('Head response (m) due to 1 m of rain in day 1')
plt.xlabel('Time (days)');






In the next test example noise is added to the observations of the groundwater head. The noise is normally distributed noise with a mean of 0 and a standard deviation of 1 and is scaled with the standard deviation of the head.

The noise series is added to the head series created in the previous example.



In [8]:

random_seed = np.random.RandomState(15892)



Create Pastas model

A pastas model is created using the head with noise. A stress model is added to the Pastas model and the model is solved.



In [9]:

sm2 = ps.StressModel(rain, ps.Gamma, name='recharge', settings='prec')
ml2.solve(noise=True)
ml2.plots.results();




INFO: Inferred frequency from time series 0: freq=D
INFO: Inferred frequency from time series RH 260: freq=D

Model Results 0                     Fit Statistics
==================================================
nfev     12                     EVP          79.43
nobs     3652                   R2            0.79
noise    True                   RMSE          0.20
tmin     1990-01-01 00:00:00    AIC          -0.04
tmax     1999-12-31 00:00:00    BIC          30.98
freq     D                      ___
warmup   3650 days 00:00:00     ___
solver   LeastSquares           ___

Parameters (5 were optimized)
==================================================
optimal   stderr     initial  vary
recharge_A   811.496524   ±1.23%  224.669629  True
recharge_n     1.047277   ±1.67%    1.000000  True
recharge_a   218.438285   ±3.33%   10.000000  True
constant_d    19.981019   ±0.10%   21.788543  True
noise_alpha    0.186436  ±65.92%    1.000000  True

Parameter correlations |rho| > 0.5
==================================================
recharge_A recharge_a  0.66
constant_d -0.98
recharge_n recharge_a -0.86
recharge_a constant_d -0.58



The results of the simulation show that Pastas is able to filter the noise from the observed groundwater head. The simulated groundwater head and the generated synthetic head are plotted below. The parameters found with the Pastas optimization are similair to the original parameters of the Gamma response function.



In [10]:

plt.figure(figsize=(12,5))
plt.plot(ml2.simulate(), label='Pastas simulation')
plt.legend(loc=0)
plt.xlabel('Date (years)');






In this example correlated noise is added to the observed head. The correlated noise is generated using the noise series created in the previous example. The correlated noise is implemented as exponential decay using the following formula:

$$n_{c}(t) = e^{-1/\alpha} \cdot n_{c}(t-1) + n(t)$$

where $n_{c}$ is the correlated noise, $\alpha$ is the noise decay parameter and $n$ is the uncorrelated noise. The noise series that is created is added to the observed groundwater head.



In [11]:

noise_corr = np.zeros(len(noise))
noise_corr[0] = noise[0]

alphatrue = 2

for i in range(1, len(noise_corr)):
noise_corr[i] = np.exp(-1/alphatrue) * noise_corr[i - 1] + noise[i]



Create Pastas model

A Pastas model is created using the head with correlated noise as input. A stressmodel is added to the model and the Pastas model is solved. The results of the model are plotted.



In [12]:

sm3 = ps.StressModel(rain, ps.Gamma, name='recharge', settings='prec')
ml3.solve(noise=True)
ml3.plots.results();




INFO: Inferred frequency from time series 0: freq=D
INFO: Inferred frequency from time series RH 260: freq=D

Model Results 0                    Fit Statistics
=================================================
nfev     16                     EVP         70.24
nobs     3652                   R2           0.70
noise    True                   RMSE         0.26
tmin     1990-01-01 00:00:00    AIC         -0.04
tmax     1999-12-31 00:00:00    BIC         30.98
freq     D                      ___
warmup   3650 days 00:00:00     ___
solver   LeastSquares           ___

Parameters (5 were optimized)
=================================================
optimal  stderr     initial  vary
recharge_A   820.677665  ±3.24%  224.669629  True
recharge_n     1.016901  ±3.60%    1.000000  True
recharge_a   234.491230  ±7.94%   10.000000  True
constant_d    19.970174  ±0.28%   21.796814  True
noise_alpha    2.076021  ±4.38%    1.000000  True

Parameter correlations |rho| > 0.5
=================================================
recharge_A recharge_a  0.66
constant_d -0.98
recharge_n recharge_a -0.82
recharge_a constant_d -0.59



The Pastas model is able to calibrate the model parameters fairly well. The calibrated parameters are close to the true values defined above. The noise_alpha parameter calibrated by Pastas is close the the alphatrue parameter defined for the correlated noise series.

Below the head simulated with the Pastas model is plotted together with the head series and the head series with the correlated noise.



In [13]:

plt.figure(figsize=(12,5))
plt.plot(ml3.simulate(), label='Pastas simulation')
plt.legend(loc=0)
plt.xlabel('Date (years)');