Return period - exercises
Let's calculate the intensity of once-in-a-century rainfall
Import relevant packages
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
from functools import reduce
import re
import probscale
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
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
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")
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
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
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 max (mm)",
ylabel="cdf",
);
Compute the plotting position and return time. You'll need to order the data in ascending order:
max_annual = max_annual.sort_values(by=['PRCP'], ascending=True)
$P_m=$ plotting position, or probability of occurence for each event
$n=$ total number of events
$m=$ rank of each event, where $m=1$ is the lowest value, and $m=n$ is the highest
Weibull plotting position:
$$ P_m = \frac{m}{n+1} $$Return period:
$$ \text{Return period} = T_r = \frac{1}{1-P_m} $$Plot the annual maximum against $P_m$ or against $T_r$.
fig, ax = plt.subplots(figsize=(10, 7))
# resample daily data into yearly data (maximum yearly value)
max_annual = df['PRCP'].resample('A-JUL').max().to_frame()
# sort yearly max from lowest to highest
max_annual = max_annual.sort_values(by=['PRCP'], ascending=True)
max_annual['rank'] = np.arange(1, len(max_annual) + 1)
print(max_annual)
n = len(max_annual['rank'])
m = max_annual['rank']
Pm = m / (n+1)
Tr = 1 / (1 - Pm)
# ax.plot(Tr, max_annual['PRCP'])
# ax.set(xlabel="return period (y)",
# ylabel="annual maximum (mm/24h)")
ax.plot(Pm, max_annual['PRCP'])
ax.set(xlabel="non-exeedance probability",
ylabel="annual maximum (mm/24h)");
Plot the annual maximum against the exceedance probability ($1-P_m$), in a log-log scale. Use
ax.set(xscale="log",
yscale("log")
)
See what data you'll want to use for a linear fit.
fig, ax = plt.subplots(figsize=(10, 6))
depth = max_annual['PRCP'].values
exc_prob = (1-Pm).values
ax.plot(exc_prob, depth, lw=3)
exclude = 40
depth_tofit = depth[exclude:]
exc_prob_tofit = exc_prob[exclude:]
ax.plot(exc_prob_tofit, depth_tofit, 'o')
ax.set(ylabel="annual maximum (mm/24h)",
xlabel="exceedance probability",
xscale="log",
yscale="log",
);
Let's make a linear fit. Attention! Our data is not annual_max and exceedance_prob, but their log.
We make a linear fit using:
slope, intercept = np.polyfit(xdata, ydata, 1) # the number 1 in the order of the polynomial = linear
Write a function that receives an exceedance probability and returns the corresponding rainfall depth.
fig, ax = plt.subplots(figsize=(10, 6))
depth = max_annual['PRCP'].values
exc_prob = (1-Pm).values
ax.plot(exc_prob, depth, lw=3, label="Weibull plotting position")
ax.set(ylabel="annual maximum (mm/24h)",
xlabel="exceedance probability")
ax.set_xscale("log")
ax.set_yscale("log")
exclude = 40
depth_tofit = depth[exclude:]
exc_prob_tofit = exc_prob[exclude:]
ax.plot(exc_prob_tofit, depth_tofit, 'o')
exc_prob_tofit_log = np.log(exc_prob_tofit)
depth_tofit_log = np.log(depth_tofit)
slope, intercept = np.polyfit(exc_prob_tofit_log, depth_tofit_log, 1)
def equation(p):
return np.exp(slope*np.log(p) + intercept)
prob = [1e-3,1-1e-3]
ax.plot(prob, equation(prob), lw=3, color="tab:red", alpha=0.4)
Homework
Everything we did today was for 24h rainfall events. We might be interested in extreme events in longer or shorter time scales. Using the following code, calculate the return time for 3-day rainfall events:
number_of_days = 3
df2 = (df['PRCP'].rolling(number_of_days)
.sum()
.dropna()
)
All the rest after that is the same...