We will calculate the evapotranspiration using two methods: Thornthwaite and Penman.

Download data from the IMS

Go to the Israel Meteorological Service website, and download the following data:

  1. hourly data
    • on the first page, choose all options and press continue.
    • on the next page, choose the following date range: 01/01/2020 to 01/01/2021, then press continue.
    • Choose station Bet Dagan (בית דגן 2523), then select (בחר), then continue.
    • Choose option "by station" (לפי תחנות), then produce report.
    • Download report as csv, call it "bet-dagan-3h.csv".
  2. daily data
    • on the first page, choose all options and press continue.
    • on the next page, choose the following date range: 01/01/2020 to 01/01/2021, then press continue.
    • Choose station Bet Dagan Meuyeshet (בית דגן מאוישת 2520), then select (בחר), then continue.
    • Choose option "by station" (לפי תחנות), then produce report.
    • Download report as csv, call it "bet-dagan-day-gigit.csv".
  3. radiation data
    • on the first page, choose all options, then on the bottom right option "radiation" (קרינה), choose kJ/m2, and then press continue.
    • on the next page, choose the following date range: 01/01/2020 to 01/01/2021, then press continue.
    • Choose station Bet Dagan Krina (בית דגן קרינה 2524), then select (בחר), then continue.
    • Choose option "by station" (לפי תחנות), then produce report.
    • Download report as csv, call it "bet-dagan-krina.csv".
import matplotlib.pyplot as plt
import matplotlib
import numpy as np
import pandas as pd
from pandas.plotting import register_matplotlib_converters
register_matplotlib_converters()  # datetime converter for a matplotlib
import seaborn as sns
sns.set(style="ticks", font_scale=1.5)

import hourly data

df = pd.read_csv('bet-dagan-3h.csv', encoding = 'unicode_escape', na_values=["-"])
# find out what hebrew gibberish means: http://www.pixiesoft.com/flip/
name_conversion_dictionary = {"ùí úçðä": "station name",
                              "îñôø úçðä": "station number",
                              "úàøéê": "Date",
                              "ùòä-LST": "LST time",
                              "èîôøèåøä(C°)": "T",
                              "èîôøèåøä ìçä(C°)": "wet-bulb temperature (°C)",
                              "èîôøèåøú ð÷åãú äèì(C°)": "dew_point_T",
                              "ìçåú éçñéú(%)": "relative humidity (%)",
                              "îäéøåú äøåç(m/s)": "wind_speed",
                              "ëéååï äøåç(îòìåú)": "wind direction (degrees)",
                              "ìçõ áâåáä äúçðä(hPa)": "Pressure",
                              "ìçõ áâåáä ôðé äéí(hPa)": "pressure at sea level (hPa)",
                              """äúàãåú éåîéú îâéâéú ñåâ à'(î"î)""": "pan evaporation (mm)",
                              "ñåâ ÷øéðä()": "radiation type",
                             }
# units
# T = temperature (°C)
# dew_point_T = dew point temperature (°C)
# wind_speed = wind speed (m/s)
# Pressure = pressure at station height (hPa = 0.1 kPa)

df = df.rename(columns=name_conversion_dictionary)
df['timestamp'] = df['Date'] + ' ' + df['LST time']
df['timestamp'] = pd.to_datetime(df['timestamp'], dayfirst=True)
df = df.set_index('timestamp')
df
station name station number Date LST time T wet-bulb temperature (°C) dew_point_T relative humidity (%) wind_speed wind direction (degrees) ... pressure at sea level (hPa) ëîåú òððéí ëåììú(÷åã) ëîåú òððéí ðîåëéí(÷åã) âåáä áñéñ òððéí ðîåëéí(÷åã) ñåâ äòððéí äðîåëéí(÷åã) ñåâ äòððéí äáéðåðééí(÷åã) ñåâ äòððéí äâáåäéí(÷åã) îæâ àååéø ðåëçé(÷åã) îæâ àååéø ùçìó(÷åã) øàåú àô÷éú(÷åã)
timestamp
2020-01-01 02:00:00 áéú ãâï ... 2523 01-01-2020 02:00 7.9 7.2 6.4 90 1.7 117.0 ... 1018.8 NaN NaN NaN NaN NaN NaN NaN NaN NaN
2020-01-01 05:00:00 áéú ãâï ... 2523 01-01-2020 05:00 7.5 7.0 6.4 93 1.2 116.0 ... 1018.1 NaN NaN NaN NaN NaN NaN NaN NaN NaN
2020-01-01 08:00:00 áéú ãâï ... 2523 01-01-2020 08:00 8.6 8.3 8.0 96 1.1 107.0 ... 1018.2 NaN NaN NaN NaN NaN NaN NaN NaN NaN
2020-01-01 11:00:00 áéú ãâï ... 2523 01-01-2020 11:00 15.9 13.1 10.6 71 2.4 196.0 ... 1017.4 NaN NaN NaN NaN NaN NaN NaN NaN NaN
2020-01-01 14:00:00 áéú ãâï ... 2523 01-01-2020 14:00 18.1 14.0 10.4 61 2.8 264.0 ... 1015.3 NaN NaN NaN NaN NaN NaN NaN NaN NaN
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
2021-01-31 11:00:00 áéú ãâï ... 2523 31-01-2021 11:00 19.0 13.7 8.9 52 5.6 235.0 ... 1017.3 NaN NaN NaN NaN NaN NaN NaN NaN NaN
2021-01-31 14:00:00 áéú ãâï ... 2523 31-01-2021 14:00 19.2 14.7 11.0 59 4.6 252.0 ... 1016.7 NaN NaN NaN NaN NaN NaN NaN NaN NaN
2021-01-31 17:00:00 áéú ãâï ... 2523 31-01-2021 17:00 18.2 14.8 12.2 68 0.8 203.0 ... 1017.0 NaN NaN NaN NaN NaN NaN NaN NaN NaN
2021-01-31 20:00:00 áéú ãâï ... 2523 31-01-2021 20:00 13.1 12.3 11.7 91 1.2 79.0 ... 1018.2 NaN NaN NaN NaN NaN NaN NaN NaN NaN
2021-01-31 23:00:00 áéú ãâï ... 2523 31-01-2021 23:00 10.8 10.6 10.3 97 1.7 111.0 ... 1018.9 NaN NaN NaN NaN NaN NaN NaN NaN NaN

3172 rows × 21 columns

import daily data with pan evaporation

df2 = pd.read_csv('bet-dagan-day-gigit.csv', encoding = 'unicode_escape', na_values=["-"])
df2 = df2.rename(columns=name_conversion_dictionary)
df2['Date'] = pd.to_datetime(df2['Date'], dayfirst=True)
df2 = df2.set_index('Date')
df2
station name station number èîôøèåøú î÷ñéîåí(C°) èîôøèåøú îéðéîåí(C°) èîôøèåøú îéðéîåí ìéã ä÷ø÷ò(C°) îùê æäéøú ùîù(ã÷åú) pan evaporation (mm) ÷åã äúàãåú éåîéú()
Date
2020-01-01 áéú ãâï îàåéùú ... 2520 NaN NaN NaN NaN 0.8 0.0
2020-01-02 áéú ãâï îàåéùú ... 2520 NaN NaN NaN NaN NaN NaN
2020-01-03 áéú ãâï îàåéùú ... 2520 NaN NaN NaN NaN NaN NaN
2020-01-04 áéú ãâï îàåéùú ... 2520 NaN NaN NaN NaN NaN NaN
2020-01-05 áéú ãâï îàåéùú ... 2520 NaN NaN NaN NaN 2.4 0.0
... ... ... ... ... ... ... ... ...
2021-01-27 áéú ãâï îàåéùú ... 2520 NaN NaN NaN NaN 2.5 0.0
2021-01-28 áéú ãâï îàåéùú ... 2520 NaN NaN NaN NaN 1.2 0.0
2021-01-29 áéú ãâï îàåéùú ... 2520 NaN NaN NaN NaN NaN NaN
2021-01-30 áéú ãâï îàåéùú ... 2520 NaN NaN NaN NaN NaN NaN
2021-01-31 áéú ãâï îàåéùú ... 2520 NaN NaN NaN NaN 2.6 0.0

397 rows × 8 columns

import daily data with radiation

df3 = pd.read_csv('bet-dagan-krina.csv', encoding = 'unicode_escape', na_values=["-"])
df3 = df3.rename(columns=name_conversion_dictionary)
df3['Date'] = pd.to_datetime(df3['Date'], dayfirst=True)
df3 = df3.set_index('Date')
df3 = df3.replace({"éùéøä": "direct",
                   "îôåæøú": "diffuse",
                   "âìåáàìéú": "global"})
df3['daily_radiation_MJ_per_m2_per_day'] = df3.iloc[:, 3:].sum(axis=1)/1000
df_radiation = df3.loc[df3["radiation type"] == "global", "daily_radiation_MJ_per_m2_per_day"].to_frame()
df_radiation
daily_radiation_MJ_per_m2_per_day
Date
2020-01-01 10.0296
2020-01-02 4.3128
2020-01-03 11.6748
2020-01-04 1.6452
2020-01-05 6.8544
... ...
2021-01-27 12.2652
2021-01-28 7.1640
2021-01-29 7.2936
2021-01-30 9.3276
2021-01-31 13.5468

396 rows × 1 columns

Part 1: Thornthwaite estimation

$$ \begin{equation} E = 16\left[ \frac{10\,T^\text{monthly mean}}{I} \right]^a, \end{equation} $$

where $$ \begin{equation} I = \sum_{i=1}^{12} \left[ \frac{T_i^\text{monthly mean}}{5} \right]^{1.514}, \end{equation} $$ and $$ \begin{align} a &= 6.75\times 10^{-7}I^3 \\ &- 7.71\times 10^{-5}I^2 \nonumber\\ &+ 1.792\times 10^{-2}I \nonumber\\ &+ 0.49239 \nonumber \end{align} $$

  • $E$ is the monthly potential ET (mm)
  • $T_\text{monthly mean}$ is the mean monthly temperature in °C
  • $I$ is a heat index
  • $a$ is a location-dependent coefficient

From df, make a new dataframe, df_th, that stores monthly temperatures means. Use resample function.

# monthly data
df_th = (df['T'].resample('MS')  # MS assigns mean to first day in the month
                .mean()
                .to_frame()
        )
# we now add 14 days to the index, so that all monthly data is in the middle of the month
# not really necessary, makes plot look better
df_th.index = df_th.index + pd.DateOffset(days=14)
df_th
T
timestamp
2020-01-15 12.484274
2020-02-15 14.046983
2020-03-15 16.439113
2020-04-15 18.512500
2020-05-15 23.166532
2020-06-15 24.600000
2020-07-15 27.353226
2020-08-15 28.090323
2020-09-15 28.462500
2020-10-15 25.120161
2020-11-15 19.308475
2020-12-15 15.916129
2021-01-15 14.123790

Calculate $I$, then $a$, and finally $E_p$. Add $E_p$ as a new column in df_th.

# Preparing "I" for the Thornthwaite equation
I = np.sum( (df_th['T']/5)**(1.514) )

# Preparing "a" for the Thornthwaite equation
a = (+6.75e-7 * I**3 
     -7.71e-5 * I**2
     +1.792e-2 * I
     + 0.49239)

# The final Thornthwaite model for monthly potential ET (mm)
df_th['Ep'] = 16*((10*df_th['T']/I)**a)
df_th
T Ep
timestamp
2020-01-15 12.484274 20.163427
2020-02-15 14.046983 27.179636
2020-03-15 16.439113 40.472053
2020-04-15 18.512500 54.671821
2020-05-15 23.166532 96.461219
2020-06-15 24.600000 112.296873
2020-07-15 27.353226 146.898516
2020-08-15 28.090323 157.128632
2020-09-15 28.462500 162.453109
2020-10-15 25.120161 118.406386
2020-11-15 19.308475 60.820862
2020-12-15 15.916129 37.291178
2021-01-15 14.123790 27.557481

Plot the Thornthwaite ET that you calculated.

fig, ax = plt.subplots(1, figsize=(10,7))
ax.plot(df_th['Ep'])
ax.set(xlabel="date",
       ylabel=r"$E_p$ (mm)",
       title="Thornthwaite potential evapotranspiration");

Part 2: Penman

The Penman model is almost entirely a theory based formula for predicting evaporative flux. It can run on a much finer timescale, and requires a much wider variety of data than most models. In addition to temperature, the Penman functions on measurements of radiation, wind speed, elevation above sea level, vapour-pressure deficit, and heat flux density to the ground.

$$ \begin{equation} E = \frac{1}{\lambda}\left[ \frac{\Delta}{\Delta+\gamma}Q_{ne}+ \frac{\gamma}{\Delta+\gamma}E_A \right], \end{equation} $$

where $Q_n$ is the available energy flux density

$$ \begin{equation} Q_n = R_n - G, \end{equation} $$

and $E_A$ is the drying power of the air

$$ \begin{equation} E_A = 6.43\cdot f(u)\cdot\text{VPD}. \end{equation} $$$$ \begin{equation} \gamma = \frac{c_p\, P}{\lambda\cdot MW_\text{ratio}} \end{equation} $$

$$ \begin{equation} P = 101.3-0.01055 H \end{equation} $$

$$ \begin{equation} \lambda = 2.501 - 2.361\times 10^{-3}\,T \end{equation} $$

  • $MW_\text{ratio}=0.622$: ratio molecular weight of water vapor/dry air
  • $P$: atmospheric pressure (kPa). Can be either measured or inferred from station height above sea level (m).
  • $\lambda$: latent heat of water vaporization (MJ kg$^{-1}$)

$$R_n = (1-\alpha)R_s\!\! \downarrow -R_b \!\! \uparrow,$$

where $\alpha$ (dimensionless) is the albedo. The net outgoing thermal radiation $R_b$ is given by

$$R_b = \left( a\frac{R_s}{R_{so}+b} \right)R_{bo},$$

where $R_{so}$ is the solar radiation on a cloudless day, and it depends on latitude and day of the year. $R_{bo}$ is given by

$$R_{bo} = \epsilon\, \sigma\, T^4_{Kelvin},$$

where $\sigma=4.903\times 10^{-9}$ MJ m$^{-2}$ d$^{-1}$ K$^{-4}$, and $\epsilon$ is net net emissivity:

$$\epsilon=-0.02+0.261 \exp\left(-7.77\times10^{-4}T_{Celcius}^2\right).$$

The parameters $a$ and $b$ are determined for the climate of the area:

  • $a=1.0$, $b=0.0$ for humid areas,
  • $a=1.2$, $b=-0.2$ for arid areas,
  • $a=1.1$, $b=-0.1$ for semihumid areas.
$$ G = 4.2\displaystyle\frac{T_{i+1}-T_{i-1}}{\Delta t}, $$

$$\text{VPD} = e_s - e_d.$$

For temperatures ranging from 0 to 50 °C, the saturation vapor pressure can be calculated with

$$ \begin{equation} e_s = \exp \left[ \frac{16.78\, T -116.9}{T+237.3} \right], \end{equation} $$

and the actual vapor pressure is given by

$$ \begin{equation} e_d = e_s \frac{RH}{100}, \end{equation} $$$$ \begin{equation} \Delta = \frac{\text{d} e_s}{\text{d}T} = e_s(T)\cdot \frac{4098.79}{(T+237.3)^2}. \end{equation} $$$$ \begin{equation} f(u) = 0.26(1.0 + 0.54\, u_2) \end{equation} $$

The various components of the equations above are:

$$ \begin{equation} \Delta = 0.200 \cdot (0.00738\,T + 0.8072)^7 - 0.000116 \end{equation} $$

$$ \begin{equation} \gamma = \frac{c_p\, P}{0.622 \lambda} \end{equation} $$

$$ \begin{equation} P = 101.3-0.01055 H \end{equation} $$

$$ \begin{equation} \lambda = 2.501 - 2.361\times 10^{-3}\,T \end{equation} $$

$$ \begin{equation} f_e(u) = 1.0 + 0.53\, u_2 \end{equation} $$

$$ \begin{equation} G = 4.2\frac{T_{i+1}-T_{i-1}}{\Delta t} \end{equation} $$

$$ \begin{equation} e_s = \exp \left[ \frac{16.78\, T -116.9}{T+237.3} \right] \end{equation} $$

$$ \begin{equation} e_d = e_s \frac{RH}{100} \end{equation} $$ where $\Delta t$ is the time in days between midpoints of time periods $i+1$ and $i−1$, and $T$ is the air temperature (°C).

  • $\Delta$: slope of the saturation water vapor pressure curve (kPa °C$^{-1}$)
  • $\gamma$: psychrometric constant (kPA °C$^{-1}$)
  • $c_p=0.001013$: specific heat of water at constant pressure (MJ kg$^{-1}$ °C$^{-1}$)
  • $P$: atmospheric pressure (kPa)
  • $H$: elevation above sea level (m)
  • $\lambda$: latent heat of vaporization (MJ kg$^{-1}$)
  • $R_n$: net radiation (MJ m$^{-2} d^{-1}$)
  • $G$: heat flux density to the ground (MJ m$^{-2} d^{-1}$)
  • $u_{2}$: wind speed measured 2 m above ground (m s$^{-1}$)
  • $e_{s} - e_{d}$: vapor pressure deficit (kPa)
  • $e_{s}$: saturation vapor pressure (kPa)
  • $e_{d}$: actual vapor pressure (kPa)

Calculate daily means for the following columns: temperature T, wind speed wind_speed, atmospheric pressure Pressure, and relative humidity relative humidity (%). Remember that pressure data was given in hectopascal, 1 hPa = 0.1 kPa. Store all the calculated values in a new dataframe, called df_pen.

# Resampling hourly data over same day and taking mean, to obtain daily averages
df_pen = (df['T'].resample('D')
                 .mean()
                 .to_frame()
         )
df_pen['dew_point'] = (df['dew_point_T'].resample('D')
                                        .mean()
                      )
df_pen['u'] = (df['wind_speed'].resample('D')
                               .mean()
              )
df_pen['P'] = (df['Pressure'].resample('D')
                             .mean()
              )/10
df_pen['RH'] = (df['relative humidity (%)'].resample('D')
                                           .mean()
               )
df_pen
T dew_point u P RH
timestamp
2020-01-01 12.3625 9.0625 1.5250 101.30875 81.500
2020-01-02 11.9750 9.8250 1.9250 101.20125 87.000
2020-01-03 13.0500 4.9750 5.1750 101.37125 58.500
2020-01-04 10.8625 6.6875 5.5625 101.15500 78.375
2020-01-05 12.9375 9.2125 4.5625 101.23625 79.125
... ... ... ... ... ...
2021-01-27 13.8125 8.2375 1.8875 100.83750 72.375
2021-01-28 14.4000 10.2250 3.5250 101.12750 76.750
2021-01-29 12.3500 7.9125 5.0250 101.22125 75.250
2021-01-30 12.9625 7.6500 4.4250 101.49500 71.375
2021-01-31 15.0625 8.3125 3.7500 101.32500 66.000

397 rows × 5 columns

With average $T$ for every day of the year, we can now calculate daily latent heat of vaporization $\lambda$, the slope of the saturation-vapor pressure-temperature curve $\Delta$, and the heat flux density to the ground $G$. Add each of these to dataframe df_pen.

Calculate also the wind function using the data for wind speed, and add this to df_pen.

def lambda_latent_heat(T):
    """daily latent heat of vaporization (MJ/kg)"""
    return 2.501 - 2.361e-3*T

def Delta(T):
    """slope of saturation-vapor curve (kPa/°C)"""
    return 0.2000*(0.00738*T + 0.8072)**7 - 0.000116

def G(T):
    """heat flux density to the ground, G (MJ/m2/d)"""
    return 4.2*np.gradient(T.values)

cp = 0.001013  # (MJ kg−1 °C−1) 
df_pen['lambda'] = lambda_latent_heat(df_pen['T'])
df_pen['Delta'] = Delta(df_pen['T'])
df_pen['G'] = G(df_pen['T'])
df_pen['gamma'] = (cp*df_pen['P'])/(0.622*df_pen['lambda'])
df_pen['f_wind'] = 1.0 + 0.53 * df_pen['u']
df_pen
T dew_point u P RH lambda Delta G gamma f_wind
timestamp
2020-01-01 12.3625 9.0625 1.5250 101.30875 81.500 2.471812 0.094385 -1.62750 0.066750 1.808250
2020-01-02 11.9750 9.8250 1.9250 101.20125 87.000 2.472727 0.092300 1.44375 0.066654 2.020250
2020-01-03 13.0500 4.9750 5.1750 101.37125 58.500 2.470189 0.098185 -2.33625 0.066835 3.742750
2020-01-04 10.8625 6.6875 5.5625 101.15500 78.375 2.475354 0.086530 -0.23625 0.066553 3.948125
2020-01-05 12.9375 9.2125 4.5625 101.23625 79.125 2.470455 0.097554 4.25250 0.066739 3.418125
... ... ... ... ... ... ... ... ... ... ...
2021-01-27 13.8125 8.2375 1.8875 100.83750 72.375 2.468389 0.102551 4.77750 0.066532 2.000375
2021-01-28 14.4000 10.2250 3.5250 101.12750 76.750 2.467002 0.106028 -3.07125 0.066760 2.868250
2021-01-29 12.3500 7.9125 5.0250 101.22125 75.250 2.471842 0.094317 -3.01875 0.066691 3.663250
2021-01-30 12.9625 7.6500 4.4250 101.49500 71.375 2.470396 0.097694 5.69625 0.066911 3.345250
2021-01-31 15.0625 8.3125 3.7500 101.32500 66.000 2.465437 0.110070 8.82000 0.066933 2.987500

397 rows × 10 columns

It's time to calculate net radiation $R_n$. The monthly mean solar radiation $R_{so}$ for latitude 30 degrees is
[17.46, 21.65, 25.96, 29.85, 32.11, 33.20, 32.66, 30.44, 26.67, 22.48, 18.30, 16.04] (MJ m$^{-2}$ d$^{-1}$)
(Israel's latitude is ~ 31 degrees).

  • Add a new column Rso_monthly to df_pen, where each day has the appropriate $R_{so}$ given by the data above.
  • Add a new columns Rs with the global radiation data imported in the 3rd file.
# Rso: mean solar radiation from a cloudless sky (based on latitude)
# MJ/m2/d
Rso_monthly = np.array([17.46, 21.65, 25.96, 29.85,
                        32.11, 33.20, 32.66, 30.44,
                        26.67, 22.48, 18.30, 16.04])
  
# create empty columns
df_pen["Rso_monthly"] = ""

# every day in the month will have the same values for Rso
for i in range(12):
    df_pen.loc[df_pen.index.month==(i+1), "Rso_monthly"] = Rso_monthly[i]

df_pen["Rs"] = df_radiation["daily_radiation_MJ_per_m2_per_day"]

plt.plot(df_pen['Rso_monthly'])
plt.gcf().autofmt_xdate()
%matplotlib notebook
middle = pd.date_range(start='1/1/2020', periods=13, freq='MS') + pd.DateOffset(days=14)
new = df_pen.loc[middle, 'Rso_monthly'].astype('float')
new
df_i = (pd.DataFrame(data=new, index=new.index) #create the dataframe
           .resample("D") #resample daily
           .interpolate(method='time') #interpolate by time
       )
plt.plot(df_i, 'o')
plt.plot(df_pen['Rso_monthly'])
plt.gcf().autofmt_xdate()
plt.title("time interpolation")
Text(0.5, 1.0, 'time interpolation')

from: Ward & Trimble, "Environmental Hydrology", 2nd Edition, page 99.

  • Calculate $$R_{bo} = \epsilon\, \sigma\, T^4_{Kelvin},$$ where $$\epsilon=-0.02+0.261 \exp\left(-7.77\times10^{-4}T_{Celcius}^2\right),$$
    $$\sigma=4.903\times 10^{-9} \text{ MJ m$^{-2}$ d$^{-1}$ K$^{-4}$},$$ and $$T_{Kelvin}=T_{Celcius}+273.15$$
  • Calculate $$R_b = \left( a\frac{R_s}{R_{so}+b} \right)R_{bo},$$ where
    • for humid areas, $a=1.0$ and $b=0$,
    • for arid areas, $a=1.2$ and $b=-0.2$,
    • for semihumid areas, $a=1.1$ and $b=-0.1$
  • Finally, calculate $$R_n = (1-\alpha)R_s\!\! \downarrow -R_b \!\! \uparrow,$$ where
    • $\alpha= 0.23$ for most green crops with a full cover
    • $\alpha= 0.04$ for fresh asphalt
    • $\alpha= 0.12$ for worn-out asphalt
    • $\alpha= 0.55$ for fresh concrete

Add a new column Rn to df_pen dataframe.

Calculate the vapor pressure deficit, VPD, add a new column to df_pen.

$$e_d = e_s\cdot \frac{RH}{100}$$

$$e_s = \exp\left(\frac{16.78\,T-116.9}{T+237.3}\right)$$

Now that all variables have been defined, daily E_penman can be calculated.

$$ \begin{equation} E_{tp} = \frac{\Delta}{\Delta+\gamma}Q_{ne}+ \frac{\gamma}{\Delta+\gamma}E_A \end{equation} $$

$Q_n$ is the available energy flux density:

$$ \begin{equation} Q_n = R_n - G, \end{equation} $$

and $E_A$ is the drying power of the air:

$$ \begin{equation} E_A = f_e(u)\cdot\text{VPD} \end{equation} $$

Add a new column E_penman to df_pen.

Make a plot with the following:

  1. the Penman (daily) estimate of the potential evapotranspiration.
  2. the Thornthwaite (monthly) estimate of the potential ET.
  3. daily evaporation pan data.

Plot the mean temperatures used in the Penman calculation (daily mean) and in the Thornthwaite calculation (monthly mean).