8  Exercises

Import relevant packages

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
# from functools import reduce
import seaborn as sns
sns.set(style="ticks", font_scale=1.5)
from pandas.plotting import register_matplotlib_converters
register_matplotlib_converters()
import urllib.request
from scipy.stats import genextreme
from scipy.optimize import curve_fit

Go to NOAA’s National Centers for Environmental Information (NCEI)
Climate Data Online: Dataset Discovery

Find station codes in this map. On the left, click on the little wrench next to “Global Summary of the Month”, then click on “identify” on the panel that just opened, and click on a station (purple circle). You will see the station’s name, it’s ID, and the period of record. For example, for Ben-Gurion’s Airport in Israel:
BEN GURION, IS
STATION ID: ISM00040180
Period of Record: 1951-01-01 to 2020-03-01

You can download daily or monthly data for each station. Use the function below to download this data to your computer. station_name can be whatever you want, station_code is the station ID.

def download_data(station_name, station_code):
    url_daily = 'https://www.ncei.noaa.gov/data/global-historical-climatology-network-daily/access/'
    url_monthly = 'https://www.ncei.noaa.gov/data/gsom/access/'
    # download daily data - uncomment the following 2 lines to make this work
    urllib.request.urlretrieve(url_daily + station_code + '.csv',
                              station_name + '_daily.csv')
    # download monthly data
    urllib.request.urlretrieve(url_monthly + station_code + '.csv',
                               station_name + '_monthly.csv')

Download daily rainfall data for Eilat, Israel. ID: IS000009972

download_data('Eilat', 'IS000009972')

Then load the data into a dataframe.
IMPORTANT!! daily precipitation data is in tenths of mm, divide by 10 to get it in mm.

df = pd.read_csv('Eilat_daily.csv', sep=",")
# make 'DATE' the dataframe index
df['DATE'] = pd.to_datetime(df['DATE'])
df = df.set_index('DATE')
# IMPORTANT!! daily precipitation data is in tenths of mm, divide by 10 to get it in mm.
df['PRCP'] = df['PRCP'] / 10
df

	STATION	LATITUDE	LONGITUDE	ELEVATION	NAME	PRCP	PRCP_ATTRIBUTES	TMAX	TMAX_ATTRIBUTES	TMIN	TMIN_ATTRIBUTES	TAVG	TAVG_ATTRIBUTES
DATE
1949-11-30	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,E	NaN	NaN	NaN	NaN	NaN	NaN
1949-12-01	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,E	NaN	NaN	NaN	NaN	NaN	NaN
1949-12-02	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,E	NaN	NaN	NaN	NaN	NaN	NaN
1949-12-03	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,E	NaN	NaN	NaN	NaN	NaN	NaN
1949-12-04	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,E	NaN	NaN	NaN	NaN	NaN	NaN
...	...	...	...	...	...	...	...	...	...	...	...	...	...
2023-04-22	IS000009972	29.55	34.95	12.0	ELAT, IS	NaN	NaN	303.0	,,S	177.0	,,S	246.0	H,,S
2023-04-23	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,S	294.0	,,S	211.0	,,S	260.0	H,,S
2023-04-24	IS000009972	29.55	34.95	12.0	ELAT, IS	NaN	NaN	328.0	,,S	199.0	,,S	265.0	H,,S
2023-04-25	IS000009972	29.55	34.95	12.0	ELAT, IS	NaN	NaN	319.0	,,S	190.0	,,S	253.0	H,,S
2023-04-26	IS000009972	29.55	34.95	12.0	ELAT, IS	NaN	NaN	334.0	,,S	179.0	,,S	267.0	H,,S

26803 rows × 13 columns

Plot precipitation data (‘PRCP’ column) and see if everything is all right.

fig, ax = plt.subplots(figsize=(10,7))
ax.plot(df['PRCP'])
ax.set_xlabel("date")
ax.set_ylabel("daily rainfall (mm)")
ax.set_title("Eilat, 1949–2021")

Text(0.5, 1.0, 'Eilat, 1949–2021')

Based on what you see, you might want to exclude certain periods, e.g.:

last_date = '2018-08-01'
first_date = '1950-08-01'
df = df[((df.index < last_date) & (df.index > first_date))]
df

	STATION	LATITUDE	LONGITUDE	ELEVATION	NAME	PRCP	PRCP_ATTRIBUTES	TMAX	TMAX_ATTRIBUTES	TMIN	TMIN_ATTRIBUTES	TAVG	TAVG_ATTRIBUTES
DATE
1950-08-02	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,E	400.0	,,G	240.0	,,G	NaN	NaN
1950-08-03	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,E	410.0	,,G	260.0	,,G	NaN	NaN
1950-08-04	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,E	400.0	,,G	260.0	,,G	NaN	NaN
1950-08-05	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,E	NaN	NaN	240.0	,,G	NaN	NaN
1950-08-06	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,E	370.0	,,G	240.0	,,G	NaN	NaN
...	...	...	...	...	...	...	...	...	...	...	...	...	...
2018-07-27	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,S	414.0	,,S	NaN	NaN	359.0	H,,S
2018-07-28	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,S	386.0	,,S	NaN	NaN	329.0	H,,S
2018-07-29	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,S	NaN	NaN	268.0	,,S	334.0	H,,S
2018-07-30	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,S	375.0	,,S	277.0	,,S	327.0	H,,S
2018-07-31	IS000009972	29.55	34.95	12.0	ELAT, IS	0.0	,,S	390.0	,,S	NaN	NaN	336.0	H,,S

24836 rows × 13 columns

The rainfall data for Eilat is VERY seasonal, it’s easy to see that there is no rainfall at all during the summer. We can assume a hydrological year starting on 1 August. If you’re not sure, you can plot the monthly means (see last week’s lecture) and find what date makes sense best.

df_month = df['PRCP'].resample('M').sum().to_frame()
month_numbers = np.arange(1,13)
monthly_mean = np.array([])  # empty array
for m in month_numbers:      # cycle over months (1, 2, 3, etc)
    this_month_mean = df_month[df_month.index.month == m].mean()  # this is the monthly mean
    monthly_mean = np.append(monthly_mean, this_month_mean)    # append
    # make new df and return it
df_month = pd.DataFrame({'monthly rainfall (mm)':monthly_mean,
                          'month number':month_numbers
                         })
fig, ax = plt.subplots(figsize=(10,7))
ax.bar(df_month['month number'], df_month['monthly rainfall (mm)'])
ax.set(xlabel="month",
       ylabel="monthly rainfall (mm)",
       title="Monthly average, Eilat, 1949--2018",
       xticks=np.arange(1,13));

Let’s resample the data according to the hydrological year (1 August), and we’ll keep the maximum value:

max_annual = (df['PRCP'].resample('A-JUL')
                        .max()
                        .to_frame()
             )
max_annual

	PRCP
DATE
1951-07-31	10.8
1952-07-31	15.0
1953-07-31	34.4
1954-07-31	24.3
1955-07-31	19.0
...	...
2014-07-31	11.5
2015-07-31	2.4
2016-07-31	8.5
2017-07-31	34.5
2018-07-31	11.7

68 rows × 1 columns

Make two graphs: a) the histogram for the annual maximum (pdf) b) the cumulative probability (cdf)

fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(10,8))

h=max_annual['PRCP'].values
ax1.hist(h, bins=np.arange(0,100,10), density=True)
ax2.hist(h, bins=np.arange(0,100,10), cumulative=1, density=True)

ax1.set(ylabel="pdf")
ax2.set(xlabel="annual daily precipitation maxima (mm)",
        ylabel="cdf",
        );

How to make a cdf by yourself?

# sort the annual daily precipitation maxima, from lowest to highest
max_annual['max_sorted'] = np.sort(max_annual['PRCP'])
# let's give it a name, h
h = max_annual['max_sorted'].values
# make an array "order" of size N=len(h), from 0 to N-1
N = len(h)
order = np.arange(N)
# make a new array, "fraction"
fraction = order / N

Plot it next to the cdf that pandas’ hist makes for you. What do you see?

fig, ax = plt.subplots(1, 1)
ax.hist(h, bins=np.arange(0,100,10), cumulative=1, density=True, label="from 'hist'")
ax.plot(h, fraction, color="tab:orange", linewidth=3, label="our cdf")
ax.set_ylabel("cdf")
ax.set_xlabel("annual daily precipitation maxima (mm)")
ax.set_title("Eilat")
ax.legend()

<matplotlib.legend.Legend at 0x7faf946b35b0>

The generalized extreme value distribution has 3 parameters: shape, location, scale.

Let’s get a “best fit” estimate of these parameters for Eilat’s rainfall statistics.

params = genextreme.fit(h)
print("Best fit:")
print(f"shape = {params[0]:.2f}\nlocation = {params[1]:.2f}\nscale = {params[2]:.2f}")

Best fit:
shape = -0.44
location = 6.17
scale = 5.58

Let’s see the GEV distribution for these parameters

fig, ax = plt.subplots(1, 1)
ax.hist(h, bins=np.arange(0,100,10), density=True, label="from 'hist'")
rain = np.arange(0,80)
pdf_rain = genextreme(c=params[0], loc=params[1], scale=params[2]).pdf(rain)
ax.plot(rain, pdf_rain, color="tab:orange", lw=3, label="gev fit")
ax.set_ylabel("pdf")
ax.set_xlabel("annual daily precipitation maxima (mm)")
ax.set_title("Eilat")
ax.legend()

<matplotlib.legend.Legend at 0x7faf96631f40>

We can do the same for the cdf…

fig, ax = plt.subplots(1, 1)
ax.plot(h, fraction, color="tab:blue", linewidth=3, label="our cdf")
rain = np.arange(0,80)
cdf_rain = genextreme(c=params[0], loc=params[1], scale=params[2]).cdf(rain)
ax.plot(rain, cdf_rain, color="tab:orange", lw=3, label="gev fit")
ax.set_ylabel("cdf")
ax.set_xlabel("annual daily precipitation maxima (mm)")
ax.set_title("Eilat")
ax.legend()

<matplotlib.legend.Legend at 0x7faf8caaabb0>

We are almost there! Remember that the return time are given by:

\[ T_r(x) = \frac{1}{1-F(x)}, \]

where \(F\) is the cdf.

Survival \(= 1-F\)

The package that we are using, scipy.stats.genextreme has a method called isf, or inverse survival function, which is exactly what we want! In order to use it, you have to feed it a “quantile” q or probability. Suppose you want to know how strong is a 1 in a 100 year event, then your return period is 100 (years), and the probability is simply its inverse, 1/100.

# Compute the return levels for several return periods.
return_periods = np.array([5, 10, 20, 50, 100, 500])
return_levels = genextreme.isf(1/return_periods, *params)

print("Return levels:")
print()
print("Period    Level")
print("(years)   (mm)")

for period, level in zip(return_periods, return_levels):
    print(f'{period:4.0f}  {level:9.2f}')

Return levels:

Period    Level
(years)   (mm)
   5      18.03
  10      27.64
  20      40.38
  50      64.17
 100      89.63
 500     189.22

You might want to do the opposite: given a list of critical daily max levels, what are the return periods for them? In this case you can use the sf method, “survival function”.

levels_mm = np.array([20, 50, 100, 200, 300])
return_per = 1/genextreme.sf(levels_mm, *params)

print("Return levels:")
print()
print("Level       Period")
print("(mm)       (years)")

for level,period in zip(levels_mm, return_per):
    print(f'{level:9.2f}  {period:4.0f}')

Return levels:

Level       Period
(mm)       (years)
    20.00     6
    50.00    30
   100.00   126
   200.00   565
   300.00  1383