Developed by Mark Bakker, TU Delft
Required files to run this notebook (all available from the data subdirectory):
data
Head files: head_nb1.csv, B58C0698001_1.csv, B50H0026001_1.csv, B22C0090001_1.csv, headwell.csv
head_nb1.csv
B58C0698001_1.csv
B50H0026001_1.csv
B22C0090001_1.csv
headwell.csv
Pricipitation files: rain_nb1.csv, neerslaggeg_HEIBLOEM-L_967.txt, neerslaggeg_ESBEEK_831.txt, neerslaggeg_VILSTEREN_342.txt, rainwell.csv
rain_nb1.csv
neerslaggeg_HEIBLOEM-L_967.txt
neerslaggeg_ESBEEK_831.txt
neerslaggeg_VILSTEREN_342.txt
rainwell.csv
Evaporation files: evap_nb1.csv, etmgeg_380.txt, etmgeg_260.txt, evapwell.csv
evap_nb1.csv
etmgeg_380.txt
etmgeg_260.txt
evapwell.csv
Well files: well1.csv, well2.csv
well1.csv
well2.csv
Figure: b58c0698_dino.png
b58c0698_dino.png
Pastas is a computer program for hydrological time series analysis and is available from the Pastas Github . Pastas makes heavy use of pandas timeseries. An introduction to pandas timeseries can be found, for example, here. The Pastas documentation is available here.
pandas
timeseries
[1]:
import pandas as pd import pastas as ps import matplotlib.pyplot as plt ps.set_log_level("ERROR") ps.show_versions()
Python version: 3.7.8 | packaged by conda-forge | (default, Jul 31 2020, 02:37:09) [Clang 10.0.1 ] Numpy version: 1.18.5 Scipy version: 1.4.0 Pandas version: 1.1.2 Pastas version: 0.16.0b Matplotlib version: 3.1.3
The first step in time series analysis is to load a time series of head observations. The time series needs to be stored as a pandas.Series object where the index is the date (and time, if desired). pandas provides many options to load time series data, depending on the format of the file that contains the time series. In this example, measured heads are stored in the csv file head_nb1.csv. The heads are read from a csv file with the read_csv function of pandas and are then squeezed to create a pandas Series object. To check if you have the correct data type, use the type command as shown below.
pandas.Series
read_csv
pandas Series
type
[2]:
ho = pd.read_csv('../data/head_nb1.csv', parse_dates=['date'], index_col='date', squeeze=True) print('The data type of the oseries is:', type(ho))
The data type of the oseries is: <class 'pandas.core.series.Series'>
The variable ho is now a pandas Series object. To see the first five lines, type ho.head().
ho
ho.head()
[3]:
date 1985-11-14 27.61 1985-11-28 27.73 1985-12-14 27.91 1985-12-28 28.13 1986-01-13 28.32 Name: head, dtype: float64
The series can be plotted as follows
[4]:
ho.plot(style='.', figsize=(12, 4)) plt.ylabel('Head [m]'); plt.xlabel('Time [years]');
The head variation shown above is believed to be caused by two stresses: rainfall and evaporation. Measured rainfall is stored in the file rain_nb1.csv and measured potential evaporation is stored in the file evap_nb1.csv. The rainfall and potential evaporation are loaded and plotted.
[5]:
rain = pd.read_csv('../data/rain_nb1.csv', parse_dates=['date'], index_col='date', squeeze=True) print('The data type of the rain series is:', type(rain)) evap = pd.read_csv('../data/evap_nb1.csv', parse_dates=['date'], index_col='date', squeeze=True) print('The data type of the evap series is', type(evap)) plt.figure(figsize=(12, 4)) rain.plot(label='rain') evap.plot(label='evap') plt.xlabel('Time [years]') plt.ylabel('Rainfall/Evaporation (m/d)') plt.legend(loc='best');
The data type of the rain series is: <class 'pandas.core.series.Series'> The data type of the evap series is <class 'pandas.core.series.Series'>
As a first simple model, the recharge is approximated as the measured rainfall minus the measured potential evaporation.
[6]:
recharge = rain - evap plt.figure(figsize=(12, 4)) recharge.plot() plt.xlabel('Time [years]') plt.ylabel('Recharge (m/d)');
Once the time series are read from the data files, a time series model can be constructed by going through the following three steps:
Creat a Model object by passing it the observed head series. Store your model in a variable so that you can use it later on.
Model
Add the stresses that are expected to cause the observed head variation to the model. In this example, this is only the recharge series. For each stess, a StressModel object needs to be created. Each StressModel object needs three input arguments: the time series of the stress, the response function that is used to simulate the effect of the stress, and a name. In addition, it is recommended to specified the kind of series, which is used to perform a number of checks on the series and fix problems when needed. This checking and fixing of problems (for example, what to substitute for a missing value) depends on the kind of series. In this case, the time series of the stress is stored in the variable recharge, the Gamma function is used to simulate the response, the series will be called 'recharge', and the kind is prec which stands for precipitation. One of the other keyword arguments of the StressModel class is up, which means that a positive stress results in an increase (up) of the head. The default value is True, which we use in this case as a positive recharge will result in the heads going up. Each StressModel object needs to be stored in a variable, after which it can be added to the model.
StressModel
kind
recharge
'recharge'
prec
up
True
When everything is added, the model can be solved. The default option is to minimize the sum of the squares of the errors between the observed and modeled heads.
[7]:
ml = ps.Model(ho) sm1 = ps.StressModel(recharge, ps.Gamma, name='recharge', settings='prec') ml.add_stressmodel(sm1) ml.solve(tmin='1985', tmax='2010')
Fit report head Fit Statistics ================================================== nfev 24 EVP 92.02 nobs 518 R2 0.92 noise True RMSE 0.13 tmin 1985-11-14 00:00:00 AIC 5.81 tmax 2010-01-01 00:00:00 BIC 27.06 freq D Obj 1.70 warmup 3650 days 00:00:00 ___ solver LeastSquares Interpolated No Parameters (5 were optimized) ================================================== optimal stderr initial vary recharge_A 748.829183 ±4.80% 215.674528 True recharge_n 1.049160 ±1.53% 1.000000 True recharge_a 134.445190 ±6.74% 10.000000 True constant_d 27.547749 ±0.08% 27.900078 True noise_alpha 59.098170 ±12.39% 15.000000 True Parameter correlations |rho| > 0.5 ================================================== recharge_A recharge_a 0.85 constant_d -0.77 recharge_n recharge_a -0.72 recharge_a constant_d -0.66
The solve function has a number of default options that can be specified with keyword arguments. One of these options is that by default a fit report is printed to the screen. The fit report includes a summary of the fitting procedure, the optimal values obtained by the fitting routine, and some basic statistics. The model contains five parameters: the parameters \(A\), \(n\), and \(a\) of the Gamma function used as the response function for the recharge, the parameter \(d\), which is a constant base level, and the parameter \(\alpha\) of the noise model, which will be explained a little later on in this notebook. The results of the model are plotted below.
solve
[8]:
ml.plot(figsize=(12, 4));
[9]:
ml = ps.Model(ho) sm1 = ps.StressModel(recharge, ps.Gamma, name='recharge', settings='prec') ml.add_stressmodel(sm1) ml.solve(tmin='1985', tmax='2010', solver=ps.LeastSquares)
[10]:
ml = ps.Model(ho) sm1 = ps.StressModel(recharge, ps.Gamma, name='recharge', settings='prec') ml.add_stressmodel(sm1) ml.set_parameter('recharge_n', vary=False) ml.solve(tmin='1985', tmax='2010', solver=ps.LeastSquares)
Fit report head Fit Statistics =================================================== nfev 28 EVP 91.52 nobs 518 R2 0.92 noise True RMSE 0.13 tmin 1985-11-14 00:00:00 AIC 5.69 tmax 2010-01-01 00:00:00 BIC 26.94 freq D Obj 1.74 warmup 3650 days 00:00:00 ___ solver LeastSquares Interpolated No Parameters (4 were optimized) =================================================== optimal stderr initial vary recharge_A 778.164968 ±5.06% 215.674528 True recharge_n 1.000000 ±nan% 1.000000 False recharge_a 154.464068 ±5.44% 10.000000 True constant_d 27.534125 ±0.08% 27.900078 True noise_alpha 63.362004 ±12.68% 15.000000 True Parameter correlations |rho| > 0.5 =================================================== recharge_A recharge_a 0.92 constant_d -0.78 recharge_a constant_d -0.72
[11]:
ml.plot(figsize=(10, 4));