stress_factor_shape.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "consecutive-compromise",
   "metadata": {},
   "source": [
    "# **Mathematical shape of the stress factor function**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "sufficient-kenya",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Part I - Methodology \n",
    "\n",
    "## <u> Motivation </u> \n",
    "\n",
    "### Theoretical background\n",
    "\n",
    "Different mathematical shapes of the stress factor can be found in the litterature, from simplistic linear ones (Martens et al., 2017) to more exotic and complex shapes (Sellers et al., 2007). The present notebook does not aim at exploring all the different mathmatical shapes possible for the stress factor arising from the literature. We will limit our investigation to the three most commonly encountered shapes : \n",
    "1. Linear : \n",
    "\\begin{equation}\n",
    "F(\\theta) = \n",
    "\\begin{cases} \n",
    "0 & \\theta < \\theta_4\\\\\n",
    "\\frac{\\theta-\\theta_4}{\\theta_3 - \\theta_4} & \\theta_4 < \\theta < \\theta_3\\\\\n",
    "1 & \\theta  > \\theta_3\n",
    "\\end{cases}\n",
    "\\end{equation}\n",
    "\n",
    "2. Exponential concave :\n",
    "\\begin{equation}\n",
    "F(\\theta) = \n",
    "\\begin{cases} \n",
    "0 & \\theta < \\theta_4\\\\\n",
    "a.exp(b.\\theta) & \\theta_4 < \\theta < \\theta_3, b > 0\\\\\n",
    "1 & \\theta  > \\theta_3\n",
    "\\end{cases}\n",
    "\\end{equation}\n",
    "\n",
    "3. Exponential convexe :\n",
    "\\begin{equation}\n",
    "F(\\theta) = \n",
    "\\begin{cases} \n",
    "0 & \\theta < \\theta_4\\\\\n",
    "a.exp(b.\\theta) & \\theta_4 < \\theta < \\theta_3, b < 0\\\\\n",
    "1 & \\theta  > \\theta_3\n",
    "\\end{cases}\n",
    "\\end{equation}\n",
    "\n",
    "### Modelling experiements\n",
    "\n",
    "Different experiments are carried out to compare the different models and assess their behavior: \n",
    "1. All models are calibrated for a single year and their ability to reproduce an observed time serie is assessed\n",
    "2. Their prediction capability is evaluated by randomly taking one or several years of data from the Howard Springs dataset, calibrating the model for this specific year and predicting the evapo-transpiration time serie for the other years. \n",
    "3. The same procedure is repeated across different sites in Australia"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "headed-theorem",
   "metadata": {},
   "source": [
    "# Part II - Functions set up"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "distant-antarctica",
   "metadata": {},
   "source": [
    "## Importing relevant packages"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "adverse-croatia",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING (aesara.link.c.cmodule): install mkl with `conda install mkl-service`: No module named 'mkl'\n",
      "WARNING (aesara.tensor.blas): Using NumPy C-API based implementation for BLAS functions.\n"
     ]
    }
   ],
   "source": [
    "# data manipulation and plotting\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib._layoutgrid import plot_children\n",
    "from collections import OrderedDict\n",
    "from IPython.display import display\n",
    "import os # to look into the other folders of the project\n",
    "import importlib.util # to open the .py files written somewhere else\n",
    "#sns.set_theme(style=\"whitegrid\")\n",
    "\n",
    "# Sympy and sympbolic mathematics\n",
    "from sympy import (asin, cos, diff, Eq, exp, init_printing, log, pi, sin, \n",
    "                   solve, sqrt, Symbol, symbols, tan, Abs)\n",
    "from sympy.physics.units import convert_to\n",
    "init_printing() \n",
    "from sympy.printing import StrPrinter\n",
    "from sympy import Piecewise\n",
    "StrPrinter._print_Quantity = lambda self, expr: str(expr.abbrev)    # displays short units (m instead of meter)\n",
    "from sympy.printing.aesaracode import aesara_function\n",
    "from sympy.physics.units import *    # Import all units and dimensions from sympy\n",
    "from sympy.physics.units.systems.si import dimsys_SI, SI\n",
    "\n",
    "# for ESSM, environmental science for symbolic math, see https://github.com/environmentalscience/essm\n",
    "from essm.variables._core import BaseVariable, Variable\n",
    "from essm.equations import Equation\n",
    "from essm.variables.units import derive_unit, SI, Quantity\n",
    "from essm.variables.utils import (extract_variables, generate_metadata_table, markdown, \n",
    "                                  replace_defaults, replace_variables, subs_eq)\n",
    "from essm.variables.units import (SI_BASE_DIMENSIONS, SI_EXTENDED_DIMENSIONS, SI_EXTENDED_UNITS,\n",
    "                                  derive_unit, derive_baseunit, derive_base_dimension)\n",
    "\n",
    "# For netCDF\n",
    "import netCDF4\n",
    "import numpy as np\n",
    "import xarray as xr\n",
    "import warnings\n",
    "from netCDF4 import Dataset\n",
    "\n",
    "# For regressions\n",
    "from sklearn.linear_model import LinearRegression\n",
    "\n",
    "# Deactivate unncessary warning messages related to a bug in Numpy\n",
    "warnings.simplefilter(action='ignore', category=FutureWarning)\n",
    "\n",
    "# for calibration\n",
    "from scipy import optimize\n",
    "\n",
    "from random import random"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "early-excuse",
   "metadata": {},
   "source": [
    "## Path of the different files (pre-defined python functions, sympy equations, sympy variables)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "right-degree",
   "metadata": {
    "tags": [
     "parameters"
    ]
   },
   "outputs": [],
   "source": [
    "path_variable = '../../theory/pyFile_storage/theory_variable.py'\n",
    "path_equation = '../../theory/pyFile_storage/theory_equation.py' \n",
    "path_analysis_functions = '../../theory/pyFile_storage/analysis_functions.py'\n",
    "path_data = '../../../data/eddycovdata/'\n",
    "dates_fPAR = '../../../data/fpar_howard_spring/dates_v5'\n",
    "\n",
    "timeSerie_oneSite_oneYear = 'timeSerie_oneSite_oneYear.png'\n",
    "inverseModelling = \"inverseModelling.png\"\n",
    "Influence_atmo_E_dry = \"Influence_atmo_E_dry.png\"\n",
    "Influence_atmo_E_wet = \"Influence_atmo_E_wet.png\"\n",
    "Influence_atmo_rel_dry = \"Influence_atmo_rel_dry.png\"\n",
    "Influence_atmo_rel_wet = \"Influence_atmo_rel_wet.png\"\n",
    "sensitivity_parameters = \"sensitivity_parameters.png\"\n",
    "statistical_assessment = \"statistical_assessment.png\"\n",
    "different_sites = \"different_sites.png\""
   ]
  },
  {
   "cell_type": "markdown",
   "id": "centered-panama",
   "metadata": {},
   "source": [
    "## Importing the sympy variables and equations defined in the theory.ipynb notebook"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "guided-weekend",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "theta_sat\n",
      "theta_res\n",
      "alpha\n",
      "n\n",
      "m\n",
      "S_mvg\n",
      "theta\n",
      "h\n",
      "S\n",
      "theta_4\n",
      "theta_3\n",
      "theta_2\n",
      "theta_1\n",
      "L\n",
      "Mw\n",
      "Pv\n",
      "Pvs\n",
      "R\n",
      "T\n",
      "c1\n",
      "T0\n",
      "Delta\n",
      "E\n",
      "G\n",
      "H\n",
      "Rn\n",
      "LE\n",
      "gamma\n",
      "alpha_PT\n",
      "c_p\n",
      "w\n",
      "kappa\n",
      "z\n",
      "u_star\n",
      "VH\n",
      "d\n",
      "z_om\n",
      "z_oh\n",
      "r_a\n",
      "g_a\n",
      "r_s\n",
      "g_s\n",
      "c1_e\n",
      "c2_e\n",
      "e\n",
      "T_min\n",
      "T_max\n",
      "RH_max\n",
      "RH_min\n",
      "e_a\n",
      "e_s\n",
      "iv_T\n",
      "T_kv\n",
      "P\n",
      "rho_a\n",
      "VPD\n",
      "eq_m_n\n",
      "eq_MVG_neg_case\n",
      "eq_MVG\n",
      "eq_sat_degree\n",
      "eq_MVG_h\n",
      "eq_h_FC\n",
      "eq_theta_4_3\n",
      "eq_theta_2_1\n",
      "eq_water_stress_simple\n",
      "eq_Pvs_T\n",
      "eq_Delta\n",
      "eq_PT\n",
      "eq_PM\n",
      "eq_PM_VPD\n",
      "eq_PM_g\n",
      "eq_PM_inv\n"
     ]
    }
   ],
   "source": [
    "for code in [path_variable,path_equation]:\n",
    "    name_code = code[-20:-3]\n",
    "    spec = importlib.util.spec_from_file_location(name_code, code)\n",
    "    mod = importlib.util.module_from_spec(spec)\n",
    "    spec.loader.exec_module(mod)\n",
    "    names = getattr(mod, '__all__', [n for n in dir(mod) if not n.startswith('_')])\n",
    "    glob = globals()\n",
    "    for name in names:\n",
    "        print(name)\n",
    "        glob[name] = getattr(mod, name)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "hidden-westminster",
   "metadata": {},
   "source": [
    "## Importing the performance assessment functions defined in the analysis_function.py file"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "polish-invention",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "AIC\n",
      "AME\n",
      "BIC\n",
      "CD\n",
      "CP\n",
      "IoA\n",
      "KGE\n",
      "MAE\n",
      "MARE\n",
      "ME\n",
      "MRE\n",
      "MSRE\n",
      "MdAPE\n",
      "NR4MS4E\n",
      "NRMSE\n",
      "NS\n",
      "NSC\n",
      "PDIFF\n",
      "PEP\n",
      "R4MS4E\n",
      "RAE\n",
      "RMSE\n",
      "RVE\n",
      "np\n",
      "nt\n"
     ]
    }
   ],
   "source": [
    "for code in [path_analysis_functions]:\n",
    "    name_code = code[-20:-3]\n",
    "    spec = importlib.util.spec_from_file_location(name_code, code)\n",
    "    mod = importlib.util.module_from_spec(spec)\n",
    "    spec.loader.exec_module(mod)\n",
    "    names = getattr(mod, '__all__', [n for n in dir(mod) if not n.startswith('_')])\n",
    "    glob = globals()\n",
    "    for name in names:\n",
    "        print(name)\n",
    "        glob[name] = getattr(mod, name)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "level-contrast",
   "metadata": {},
   "source": [
    "## Data import, preprocess and shape for the computations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "powerful-arizona",
   "metadata": {},
   "source": [
    "### Get the different files where data are stored\n",
    "\n",
    "Eddy-covariance data from the OzFlux network are stored in **.nc** files (NetCDF4 files) which is roughly a panda data frame with meta-data (see https://www.unidata.ucar.edu/software/netcdf/ for more details about NetCDF4 file format). fPAR data are stored in **.txt** files"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "dated-performance",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['../../../data/eddycovdata/fpar_adelaide_v5.txt', '../../../data/eddycovdata/fpar_daly_v5.txt', '../../../data/eddycovdata/fpar_dry_v5.txt', '../../../data/eddycovdata/fpar_howard_v5.txt', '../../../data/eddycovdata/fpar_sturt_v5.txt']\n",
      "['../../../data/eddycovdata/AdelaideRiver_L4.nc', '../../../data/eddycovdata/DalyUncleared_L4.nc', '../../../data/eddycovdata/DryRiver_L4.nc', '../../../data/eddycovdata/HowardSprings_L4.nc', '../../../data/eddycovdata/SturtPlains_L4.nc']\n"
     ]
    }
   ],
   "source": [
    "fPAR_files = []\n",
    "eddy_files = []\n",
    "\n",
    "for file in os.listdir(path_data):\n",
    "    if file.endswith(\".txt\"):\n",
    "        fPAR_files.append(os.path.join(path_data, file))\n",
    "    elif file.endswith(\".nc\"):\n",
    "        eddy_files.append(os.path.join(path_data, file))\n",
    "        \n",
    "fPAR_files.sort()\n",
    "print(fPAR_files)\n",
    "eddy_files.sort()\n",
    "print(eddy_files)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "steady-thought",
   "metadata": {},
   "source": [
    "### Define and test a function that process the fPAR data\n",
    "In the **.txt** files, only one value per month is given for the fPAR. The following function takes one .txt file containing data about the fPAR coefficients, and the related dates, stored in the a seperate file. The fPAR data (date and coefficients) are cleaned (good string formatting), mapped together and averaged to output one value per month (the fPAR measurement period doesn't spans the measurement period of the eddy covariance data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "electric-paraguay",
   "metadata": {},
   "outputs": [],
   "source": [
    "def fPAR_data_process(fPAR_file,dates_fPAR):\n",
    "    \n",
    "    fparv5_dates = np.genfromtxt(dates_fPAR, dtype='str', delimiter=',')\n",
    "    fparv5_dates = pd.to_datetime(fparv5_dates[:,1], format=\"%Y%m\")\n",
    "    dates_pd = pd.date_range(fparv5_dates[0], fparv5_dates[-1], freq='MS')\n",
    "\n",
    "    fparv5_howard = np.loadtxt(fPAR_file,delimiter=',', usecols=3 )\n",
    "    fparv5_howard[fparv5_howard == -999] = np.nan\n",
    "    fparv5_howard_pd = pd.Series(fparv5_howard, index = fparv5_dates)\n",
    "    fparv5_howard_pd = fparv5_howard_pd.resample('MS').max()\n",
    "\n",
    "    # convert fparv5_howard_pd to dataframe\n",
    "    fPAR_pd = pd.DataFrame(fparv5_howard_pd)\n",
    "    fPAR_pd = fPAR_pd.rename(columns={0:\"fPAR\"})\n",
    "    fPAR_pd.index = fPAR_pd.index.rename(\"time\")\n",
    "\n",
    "    # convert fPAR_pd to xarray to aggregate the data\n",
    "    fPAR_xr = fPAR_pd.to_xarray()\n",
    "    fPAR_agg = fPAR_xr.fPAR.groupby('time.month').max()\n",
    "\n",
    "    # convert back to dataframe\n",
    "    fPAR_pd = fPAR_agg.to_dataframe()\n",
    "    Month = np.arange(1,13)\n",
    "    Month_df = pd.DataFrame(Month)\n",
    "    Month_df.index = fPAR_pd.index\n",
    "    Month_df = Month_df.rename(columns={0:\"Month\"})\n",
    "\n",
    "    fPAR_mon = pd.concat([fPAR_pd,Month_df], axis = 1)\n",
    "    \n",
    "    return(fPAR_mon)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "intended-merchant",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<style scoped>\n",
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       "    .dataframe tbody tr th {\n",
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>fPAR</th>\n",
       "      <th>Month</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>month</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.78</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.84</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.79</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.84</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.71</td>\n",
       "      <td>5</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.75</td>\n",
       "      <td>6</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.60</td>\n",
       "      <td>7</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.54</td>\n",
       "      <td>8</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>0.52</td>\n",
       "      <td>9</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>0.67</td>\n",
       "      <td>10</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>0.73</td>\n",
       "      <td>11</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>0.78</td>\n",
       "      <td>12</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       fPAR  Month\n",
       "month             \n",
       "1      0.78      1\n",
       "2      0.84      2\n",
       "3      0.79      3\n",
       "4      0.84      4\n",
       "5      0.71      5\n",
       "6      0.75      6\n",
       "7      0.60      7\n",
       "8      0.54      8\n",
       "9      0.52      9\n",
       "10     0.67     10\n",
       "11     0.73     11\n",
       "12     0.78     12"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fPAR_data_process(fPAR_files[3],dates_fPAR)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "egyptian-prefix",
   "metadata": {},
   "source": [
    "### fPARSet function\n",
    "Map the fPAR time serie to the given eddy-covariance data. Takes two dataframes as input (one containing the fPAR data, the other containing the eddy-covariance data) and returns a data frame where the fPAR monthly values have been scaled to the time scale of the eddy covariance data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "capable-melbourne",
   "metadata": {},
   "outputs": [],
   "source": [
    "def fPARSet(df_add, fPAR_pd):\n",
    "    \n",
    "    # construct the time serie of the fPAR coefficients\n",
    "    dummy_len = df_add[\"Fe\"].size\n",
    "    fPAR_val = np.zeros((dummy_len,))\n",
    "    \n",
    "    dummy_pd = df_add\n",
    "    dummy_pd.reset_index(inplace=True)\n",
    "    dummy_pd.index=dummy_pd.time\n",
    "    \n",
    "    month_pd = dummy_pd['time'].dt.month\n",
    "    \n",
    "    for i in range(dummy_len):\n",
    "        current_month = month_pd.iloc[i]\n",
    "        line_fPAR = fPAR_pd[fPAR_pd['Month'] == current_month]\n",
    "        fPAR_val[i] = line_fPAR['fPAR']\n",
    "    \n",
    "    # transform fPAR_val into dataframe to concatenate to df:\n",
    "    fPAR = pd.DataFrame(fPAR_val, index = df_add.index)\n",
    "    df_add = pd.concat([df_add,fPAR], axis = 1)\n",
    "    df_add = df_add.rename(columns = {0:\"fPAR\"})\n",
    "    \n",
    "    return(df_add)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "successful-beach",
   "metadata": {},
   "source": [
    "### DataChose function\n",
    "\n",
    "Function taking the raw netcdf4 data file from the eddy covariance measurement and shape it such that it can be used for the computations. Only relevant variables are kept (latent heat flux, net radiation, ground heat flux, soil water content, wind speed, air temperature, VPD, bed shear stress). The desired data period is selected and is reshaped at the desired time scale (daily by default). Uses the fPARSet function defined above\n",
    "\n",
    "List of variable abbreviation : \n",
    "* `Rn` : Net radiation flux\n",
    "* `G` : Ground heat flux \n",
    "* `Sws` : soil moisture\n",
    "* `Ta` : Air temperature\n",
    "* `RH` : Relative humidity\n",
    "* `W` : Wind speed\n",
    "* `E` : measured evaporation\n",
    "* `VPD` : Vapour pressure deficit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "median-boards",
   "metadata": {},
   "outputs": [],
   "source": [
    "def DataChose(ds_ref, period_sel, fPAR_given, Freq = \"D\", sel_period_flag = True):\n",
    "    \"\"\"Take subset of dataset if Flag == True, entire dataset else\n",
    "    \n",
    "    ds_ref: xarray object to be considered as the ref for selecting attributes\n",
    "    agg_flag: aggregate the data at daily time scale if true\n",
    "    Flag: select specific period if true (by default)\n",
    "    period: time period to be selected\n",
    "    ----------\n",
    "    Method : \n",
    "    - transform the xarray in panda dataframe for faster iteration\n",
    "    - keep only the necessary columns : Fe, Fn, Fg, Ws, Sws, Ta, ustar, RH\n",
    "    - transform / create new variables : Temperature in °C, T_min/T_max, RH_min/RH_max\n",
    "    - create the Data vector (numpy arrays)\n",
    "    - create back a xarray \n",
    "    - return an xarray\n",
    "    ----------\n",
    "    \n",
    "    Returns an xarray and a Data vector\n",
    "    \"\"\"\n",
    "    \n",
    "    if sel_period_flag:\n",
    "        df = ds_ref.sel(time = period_sel) \n",
    "        # nameXarray_output = period + \"_\" + nameXarray_output\n",
    "    else : \n",
    "        df = ds_ref\n",
    "        \n",
    "    # keep only the columns of interest\n",
    "    df = df[[\"Fe\",\"Fn\",\"Fg\",\"Ws\",\"Sws\",\"Ta\",\"ustar\",\"RH\", \"VPD\",\"ps\"]]\n",
    "    \n",
    "    # convert to dataframe\n",
    "    df = df.to_dataframe()\n",
    "    \n",
    "    # aggregate following the rule stated in freq\n",
    "    pd_Tmin = df.Ta.groupby([pd.Grouper(level = \"latitude\"), pd.Grouper(level = \"longitude\"), pd.Grouper(level = \"time\", freq = Freq)]).min()\n",
    "               \n",
    "    pd_Tmax = df.Ta.groupby([pd.Grouper(level = \"latitude\"), pd.Grouper(level = \"longitude\"), pd.Grouper(level = \"time\", freq = Freq)]).max()\n",
    "    \n",
    "    pd_RHmin = df.RH.groupby([pd.Grouper(level = \"latitude\"), pd.Grouper(level = \"longitude\"), pd.Grouper(level = \"time\", freq = Freq)]).min()\n",
    "    \n",
    "    pd_RHmax = df.RH.groupby([pd.Grouper(level = \"latitude\"), pd.Grouper(level = \"longitude\"), pd.Grouper(level = \"time\", freq = Freq)]).max()\n",
    "    \n",
    "    df = df.groupby([pd.Grouper(level = \"latitude\"), pd.Grouper(level = \"longitude\"), pd.Grouper(level = \"time\", freq = Freq)]).mean()\n",
    "    df = pd.DataFrame(df)\n",
    "    \n",
    "    pd_Tmin = pd.DataFrame(pd_Tmin, index = df.index)\n",
    "    pd_Tmin = pd_Tmin.rename(columns = {\"Ta\":\"Ta_min\"})\n",
    "    \n",
    "    pd_Tmax = pd.DataFrame(pd_Tmax, index = df.index)\n",
    "    pd_Tmax = pd_Tmax.rename(columns = {\"Ta\":\"Ta_max\"})\n",
    "    \n",
    "    pd_RHmin = pd.DataFrame(pd_RHmin, index = df.index)\n",
    "    pd_RHmin = pd_RHmin.rename(columns = {\"RH\":\"RH_min\"})\n",
    "    \n",
    "    pd_RHmax = pd.DataFrame(pd_RHmax, index = df.index)\n",
    "    pd_RHmax = pd_RHmax.rename(columns = {\"RH\":\"RH_max\"})\n",
    "    \n",
    "    df = pd.concat([df,pd_Tmin,pd_Tmax,pd_RHmin,pd_RHmax],axis = 1)\n",
    "    \n",
    "    # convert data to the good units : \n",
    "    df[\"Fe\"] = df[\"Fe\"]/2.45e6 # divide by latent heat of vaporization\n",
    "    df[\"Ta\"] = df[\"Ta\"]+273 # convert to kelvin\n",
    "    df[\"VPD\"] = df[\"VPD\"]*1000 # convert from kPa to Pa\n",
    "    \n",
    "    # construct the time serie of the fPAR coefficients\n",
    "    df = fPARSet(df,fPAR_given)\n",
    "    \n",
    "    # initialise array for the error\n",
    "    Error_obs = np.zeros((df.Fe.size,))\n",
    "    \n",
    "    for i in range(df.Fe.size):\n",
    "        size_window_left, size_window_right = min(i,7),min(df.Fe.size - i-1, 7)\n",
    "        #print(size_window_left, size_window_right)\n",
    "        sub_set = df.Fe[i-size_window_left : i+size_window_right].to_numpy()\n",
    "        mean_set = np.mean(sub_set)\n",
    "        sdt_set = np.std(sub_set)\n",
    "        error_obs = 2*sdt_set\n",
    "        Error_obs[i] = error_obs\n",
    "    \n",
    "    ErrorObs = pd.DataFrame(Error_obs, index = df.index)\n",
    "    \n",
    "    df = pd.concat([df,ErrorObs], axis = 1)\n",
    "    df = df.rename(columns = {0:\"error\"})\n",
    "\n",
    "        \n",
    "    return(df)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "utility-astronomy",
   "metadata": {},
   "source": [
    "## Compile the different functions defined in the symbolic domain\n",
    "All functions defined with sympy and ESSM are defined in the symbolic domain. In order to be efficiently evaluated, they need to be vectorized to allow computations with numpu arrays. We use the *aesara* printing compiler from the sympy package. Note that this printer replace the older one (*theano*) which is deprecated. A comparison of the performances between the two packages can be found in the aesara repository."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cathedral-pilot",
   "metadata": {},
   "source": [
    "### Water stress functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "split-message",
   "metadata": {},
   "outputs": [],
   "source": [
    "def rising_slope_compiled():\n",
    "    \"\"\"Compile the slope of the function between theta 4 and theta 3\"\"\"\n",
    "    \n",
    "    rising_slope = aesara_function([theta,theta_3, theta_4], [eq_theta_4_3.rhs], dims = {theta:1, theta_3:1, theta_4:1})\n",
    "    \n",
    "    return(rising_slope)\n",
    "\n",
    "def desc_slope_compiled():\n",
    "    \"\"\"Compile the slope of the function between theta 2 and theta 1\"\"\"\n",
    "    \n",
    "    desc_slope = aesara_function([theta,theta_1, theta_2], [eq_theta_2_1.rhs], dims = {theta:1, theta_1:1, theta_2:1})\n",
    "    \n",
    "    return(desc_slope)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "circular-midnight",
   "metadata": {},
   "source": [
    "### Soil water potential"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "vital-physiology",
   "metadata": {},
   "outputs": [],
   "source": [
    "def relative_saturation_compiled(ThetaRes, ThetaSat):\n",
    "    \"\"\"Compile the relative saturation function of a soil\"\"\"\n",
    "    Dict_value = {theta_res : ThetaRes, theta_sat : ThetaSat}\n",
    "    \n",
    "    S_mvg_func = aesara_function([theta], [eq_sat_degree.rhs.subs(Dict_value)], dims = {theta:1})\n",
    "    \n",
    "    return(S_mvg_func)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "resident-centre",
   "metadata": {},
   "outputs": [],
   "source": [
    "def Psi_compiled(alphaVal, nVal):\n",
    "    \"\"\"Compile the soil water retention function as function of theta\"\"\"\n",
    "    mVal = 1-1/nVal\n",
    "    \n",
    "    Dict_value = {alpha : alphaVal, n : nVal, m : mVal}\n",
    "    \n",
    "    Psi_function = aesara_function([S_mvg], [eq_MVG_h.rhs.subs(Dict_value)], dims = {S_mvg:1})\n",
    "    \n",
    "    return(Psi_function)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "previous-hobby",
   "metadata": {},
   "source": [
    "### Penman-Monteith"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "single-execution",
   "metadata": {},
   "outputs": [],
   "source": [
    "def Delta_compiled():\n",
    "    \"\"\"Compile the Delta function\"\"\"\n",
    "    \n",
    "    # creating the dictionnary with all default values from the above defined constants\n",
    "    var_dict = Variable.__defaults__.copy()\n",
    "    \n",
    "    # computing delta values out of temperature values (slope of the water pressure curve)\n",
    "    Delta_func = aesara_function([T],[eq_Delta.rhs.subs(var_dict)], dims = {T:1})\n",
    "    \n",
    "    return(Delta_func)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "special-celebration",
   "metadata": {},
   "outputs": [],
   "source": [
    "def VH_func_compiled(z_val):\n",
    "    \"\"\"Compute the vegetation height function\n",
    "    --------------------------------------------------------\n",
    "    z : height of the measurements (m)\n",
    "    kappa : Von Karman constant\n",
    "    \n",
    "    w : wind velocity (m/s)\n",
    "    u_star : shear stress velocity (m/s)\n",
    "    --------------------------------------------------------\n",
    "    \n",
    "    Return a function with w and u_star as degrees of freedom\n",
    "    \"\"\"\n",
    "    # get the constant values\n",
    "    Dict_value = {z:z_val,kappa:kappa.definition.default}\n",
    "    \n",
    "    # compile the function\n",
    "    VH_func = aesara_function([w,u_star],[VH.definition.expr.subs(Dict_value)], dims = {w:1, u_star:1})\n",
    "    \n",
    "    return(VH_func)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "shaped-object",
   "metadata": {},
   "outputs": [],
   "source": [
    "def d_func_compiled():\n",
    "    \"\"\"Compile the zero plane displacement height function\n",
    "    --------------------------------------------------------\n",
    "    VH : Vegetation height\n",
    "    --------------------------------------------------------\n",
    "    \n",
    "    return a function with VH as degree of freedom\n",
    "    \"\"\"\n",
    "    \n",
    "    # compile the function \n",
    "    d_func = aesara_function([VH], [d.definition.expr], dims = {VH:1})\n",
    "    \n",
    "    return(d_func)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "decreased-wilson",
   "metadata": {},
   "outputs": [],
   "source": [
    "def zom_func_compiled():\n",
    "    \"\"\"Compile the characteristic momentum height exchange\n",
    "    --------------------------------------------------------\n",
    "    VH : Vegetation height\n",
    "    --------------------------------------------------------\n",
    "    \n",
    "    return a function with VH as degree of freedom\n",
    "    \"\"\"\n",
    "    \n",
    "    # compile the function \n",
    "    zom_func = aesara_function([VH], [z_om.definition.expr], dims = {VH:1})\n",
    "    \n",
    "    return(zom_func)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "opponent-heritage",
   "metadata": {},
   "outputs": [],
   "source": [
    "def zoh_func_compiled():\n",
    "    \"\"\"Compile the characteristic heat height exchange\n",
    "    --------------------------------------------------------\n",
    "    VH : Vegetation height\n",
    "    --------------------------------------------------------\n",
    "    \n",
    "    return a function with VH as degree of freedom\n",
    "    \"\"\"\n",
    "    \n",
    "    # compile the function \n",
    "    zoh_func = aesara_function([z_om], [z_oh.definition.expr], dims = {z_om:1})\n",
    "    \n",
    "    return(zoh_func)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "undefined-mailing",
   "metadata": {},
   "outputs": [],
   "source": [
    "def ra_func_compiled(z_val):\n",
    "    \"\"\"Substitutes the different terms of the r_a expression\n",
    "    --------------------------------------------------------\n",
    "    z : height of the measurement (m)\n",
    "    \n",
    "    d : zero plane displacement height (m)\n",
    "    zoh_val : characteristic height of the heat transfert\n",
    "    zom_val : characteristic height of the momentum transfert\n",
    "    --------------------------------------------------------\n",
    "    \n",
    "    returns the compiled expression of r_a evaluable according to the wind speed\n",
    "    \"\"\"\n",
    "    # evaluate the values in the expression\n",
    "    Dict_value = {z:z_val,kappa:kappa.definition.default}\n",
    "    \n",
    "    # compile the function\n",
    "    ra_func = aesara_function([z_om,z_oh,d,w], [r_a.definition.expr.subs(Dict_value)], dims = {z_om:1,z_oh:1,d:1,w:1})\n",
    "    \n",
    "    return(ra_func)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "diverse-bulgarian",
   "metadata": {},
   "outputs": [],
   "source": [
    "def ea_func_compiled():\n",
    "    \"\"\" Compute the actual water vapour deficit with RH and T (min/max) left as degree of freedom\n",
    "    --------------------------------------------------------\n",
    "    c1 : internal variable \n",
    "    c2 : internal variable \n",
    "    \n",
    "    T_min : time serie of daily min temperature\n",
    "    T_max : time serie of daily max temperature \n",
    "    RH_min : time serie of daily min relative humidity \n",
    "    RH_max : time serie of daily max relative humidity\n",
    "    --------------------------------------------------------\n",
    "    \n",
    "    retunrs the compiled expression of the e_a function with four degrees of freedom \n",
    "    \"\"\"\n",
    "    \n",
    "    # get the constants\n",
    "    Dict_value = {c1_e:c1_e.definition.default, c2_e:c2_e.definition.default}\n",
    "    \n",
    "    #compile the function\n",
    "    ea_func = aesara_function([T_min,T_max,RH_min,RH_max],[e_a.definition.expr.subs(Dict_value)], dims={T_min:1, T_max:1, RH_min:1, RH_max:1})\n",
    "    \n",
    "    return(ea_func)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "deadly-spice",
   "metadata": {},
   "outputs": [],
   "source": [
    "def es_func_compiled():\n",
    "    \"\"\"Compute the saturation water vapour deficit with T (min / max) left as degree of freedom \n",
    "    --------------------------------------------------------\n",
    "    c1 : internal variable \n",
    "    c2 : internal variable \n",
    "    \n",
    "    T_min : time serie of daily min temperature\n",
    "    T_max : time serie of daily max temperature \n",
    "    --------------------------------------------------------\n",
    "    \n",
    "    retunrs the compiled expression of the e_s function with four degrees of freedom \n",
    "    \"\"\"\n",
    "    \n",
    "    # get the constants\n",
    "    Dict_value = {c1_e:c1_e.definition.default, c2_e:c2_e.definition.default}\n",
    "    \n",
    "    #compile the function\n",
    "    ea_func = aesara_function([T_min,T_max],[e_s.definition.expr.subs(Dict_value)], dims={T_min:1, T_max:1})\n",
    "    \n",
    "    return(ea_func)    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "pressed-familiar",
   "metadata": {},
   "outputs": [],
   "source": [
    "def PM_compiled():\n",
    "    \"\"\" Compute the compiled version of the PM VPD equation\n",
    "    --------------------------------------------------------\n",
    "    c_p : specific heat of the air\n",
    "    rho_a : mean air density\n",
    "    gamma : psychrometric constant\n",
    "    L : Latent heat flux\n",
    "    \n",
    "    r_s : Surface resistance -> given (can also be modeled)\n",
    "    G : ground heat flux -> comes from the data\n",
    "    Rn : net radiation flux -> comes from the data \n",
    "    Delta : slope of the saturation curve -> computed above\n",
    "    r_a : aerodynamic resistance -> computed above\n",
    "    --------------------------------------------------------\n",
    "    \n",
    "    returns the evaporation flux (in mm/time) with Delta, G, Rn, e_a, e_s, r_a as degrees of freedom\n",
    "    \"\"\"\n",
    "    \n",
    "    # get the constant values\n",
    "    Dict_value = {c_p:c_p.definition.default, rho_a:rho_a.definition.default, L:L.definition.default, gamma:gamma.definition.default}\n",
    "    \n",
    "    # compile the function\n",
    "    PM_func = aesara_function([G,Rn,Delta,VPD,g_a, g_s], [eq_PM_g.rhs.subs(Dict_value)], dims = {G:1,Rn:1,Delta:1,VPD:1,g_a:1, g_s:1})\n",
    "    \n",
    "    return(PM_func)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "civilian-toner",
   "metadata": {},
   "outputs": [],
   "source": [
    "def PT_compiled():\n",
    "    \"\"\"Compute the Priestley-Taylor equation,\n",
    "    need 4 input : temperature, net radiations, ground heat flux and alpha PT parameter\n",
    "    Each must be given as np.array\n",
    "    Return the compiled function\n",
    "    \"\"\"\n",
    "    \n",
    "    # creating the dictionnary with all default values from the above defined constants\n",
    "    var_dict = Variable.__defaults__.copy()\n",
    "    \n",
    "    # Computing the values of evaporation using the Priestley-Taylor model\n",
    "    PT_func = aesara_function([Delta,Rn,G, alpha_PT],[eq_PT.rhs.subs(var_dict)],dims = {Delta:1,Rn:1,G:1, alpha_PT:1})\n",
    "\n",
    "    return(PT_func)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "absent-arthritis",
   "metadata": {},
   "outputs": [],
   "source": [
    "def rs_PM_inv():\n",
    "    \"Derive the time serie of the surface resistance out of the observed latent heat fluxes --> inverse modelling of the PM equation\"\n",
    "    \n",
    "    #get the constant values \n",
    "    Dict_value = {c_p:c_p.definition.default, rho_a:rho_a.definition.default, L:L.definition.default, gamma:gamma.definition.default}\n",
    "    \n",
    "    # compile the function\n",
    "    Inv_PM = aesara_function([E,G,Rn,Delta,VPD,r_a], [eq_PM_inv.rhs.subs(Dict_value)], dims = {E:1,G:1,Rn:1,Delta:1,VPD:1,r_a:1})\n",
    "    \n",
    "    return(Inv_PM)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "breathing-manual",
   "metadata": {},
   "source": [
    "### Assign the different compiled functions to variables functions (create the functions in python)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "robust-maker",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Water_stress properties: \n",
    "rising_slope = rising_slope_compiled() # takes theta, theta_3, theta_4 in this order as input\n",
    "desc_slope = desc_slope_compiled() # takes theta, theta_1, theta_2 in this order as input\n",
    "\n",
    "# properties : \n",
    "delta_func = Delta_compiled() # takes Ta as input\n",
    "z_val = 23 # change the values of the aerodynamic constants here !!!\n",
    "VH_func = VH_func_compiled(z_val)\n",
    "d_func = d_func_compiled()\n",
    "zom_func = zom_func_compiled()\n",
    "zoh_func = zoh_func_compiled()\n",
    "ra_func = ra_func_compiled(z_val)\n",
    "eSat_func = es_func_compiled()\n",
    "ea_func = ea_func_compiled()\n",
    "\n",
    "# models\n",
    "PM_func = PM_compiled()\n",
    "PT_func = PT_compiled()\n",
    "Inv_PM_func = rs_PM_inv()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "respected-sessions",
   "metadata": {},
   "outputs": [],
   "source": [
    "def stress_factor_func(psi_vec,Psi_3,Psi_4,a):\n",
    "    \n",
    "    Length = psi_vec.size\n",
    "    \n",
    "    Stress_func = np.zeros((Length,))\n",
    "    Psi_3 = np.array([Psi_3])\n",
    "    Psi_4 = np.array([Psi_4])\n",
    "    \n",
    "    for j in range(Length):\n",
    "        try:\n",
    "            Psi_val = psi_vec[j]\n",
    "        except Exception as error:\n",
    "            print(psi_vec,error)\n",
    "        # have to convert float to arrays in order to run the compiled functions\n",
    "        Psi_val = np.array([Psi_val])\n",
    "        if (Psi_val < Psi_4):\n",
    "            Stress_func[j] = 0\n",
    "        elif (Psi_val >= Psi_4) & (Psi_val < Psi_3):\n",
    "            linear = (Psi_val-Psi_4)/(Psi_3-Psi_4)\n",
    "            if a == 0:\n",
    "                Stress_func[j] = linear\n",
    "            else:\n",
    "                Stress_func[j] = (1-np.exp(a*linear))/(1-np.exp(a))\n",
    "        else:\n",
    "            Stress_func[j] = 1\n",
    "    return(Stress_func)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dense-prairie",
   "metadata": {},
   "source": [
    "## Functions to run the different models\n",
    "Once the functions has been defined in sympy, compiled with aesara and assign to a useful python function, it is now time to link it with the data to carry out the final computations ! All the following functions take a dataframe as input, and the related parameters ($\\theta_3$ and $\\theta_4$ for the stress function) and return a numpy array and a dictionnary containing the values of the model run and some miscellaneous information."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "valuable-isaac",
   "metadata": {},
   "source": [
    "### Varying surface resistance model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "electric-kitty",
   "metadata": {},
   "outputs": [],
   "source": [
    "def PM_run_var(data, Psi_3_val, Psi_4_val,shape, gs_val = 1/70, compute_VPD = False):\n",
    "    \"\"\"run varying SR Penman Monteith model but with VPD data instead of Ea and Es\n",
    "    -----------------------------------\n",
    "    data: list containing the data /!\\ the data should be in a specific order : Data = [Ta, Fn, Fg, Sws, Fe, Ws, Ustar, Ta_min, Ta_max, RH_min, RH_max]\n",
    "    Psi_3_val: vector of the same size as the vectors in data, full of the value Psi_3\n",
    "    Psi_4_val: vector of the same size as the vectors in data, full of the value Psi_4\n",
    "    -----------------------------------\n",
    "    \"\"\"\n",
    "    \n",
    "    # unpack the variables\n",
    "    Ta_val = data[\"Ta\"].to_numpy()\n",
    "    Fn_val = data[\"Fn\"].to_numpy()\n",
    "    Fg_val = data[\"Fg\"].to_numpy()\n",
    "    Sws_val = data[\"Sws\"].to_numpy()\n",
    "    Fe_val = data[\"Fe\"].to_numpy()\n",
    "    Ws_val = data[\"Ws\"].to_numpy()\n",
    "    Ustar_val = data[\"ustar\"].to_numpy()\n",
    "    VPD_val = data[\"VPD\"].to_numpy()\n",
    "    fPAR_val = data[\"fPAR\"].to_numpy()\n",
    "    \n",
    "    if compute_VPD:\n",
    "        T_min_val = data[\"Ta_min\"]\n",
    "        T_max_val = data[\"Ta_max\"]\n",
    "        \n",
    "        RH_min_val = data[\"RH_min\"]\n",
    "        RH_max_val = data[\"RH_max\"]\n",
    "        \n",
    "        VPD_val = eSat_func(T_min_val,T_max_val) - ea_func(T_min_val,T_max_val,RH_min_val,RH_max_val)\n",
    "        \n",
    "    # --------------------------\n",
    "    # derive the two Penman models\n",
    "    # --------------------------\n",
    "    \n",
    "\n",
    "    # Stress factor:\n",
    "    FF_vec = stress_factor_func(Sws_val,Psi_3_val,Psi_4_val, shape)\n",
    "    # aerodynamic resistance:\n",
    "    VH_T = VH_func(Ws_val,Ustar_val)\n",
    "    displ_T = d_func(VH_T)\n",
    "    ZOM_T = zom_func(VH_T)\n",
    "    ZOH_T = zoh_func(ZOM_T)\n",
    "    Ra_T = ra_func(ZOM_T, ZOH_T, displ_T, Ws_val)\n",
    "    Ga_T = 1/Ra_T\n",
    "\n",
    "    # surface resistance:\n",
    "    Gs_T = gs_val*FF_vec+0.0001\n",
    "    # thermodynamic parameters\n",
    "    D_T = delta_func(Ta_val)\n",
    "\n",
    "    # compile the Ea values :\n",
    "    PM_var = fPAR_val*PM_func(Fg_val,Fn_val,D_T,VPD_val,Ga_T, Gs_T) # R_s varying with theta\n",
    "    \n",
    "    Dic_var = {\"ga\":Ga_T, \"gs\":Gs_T, \"D_T\":D_T}\n",
    "    \n",
    "    return(PM_var)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "featured-trial",
   "metadata": {},
   "source": [
    "### Constant surface conductance model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "systematic-calibration",
   "metadata": {},
   "outputs": [],
   "source": [
    "def PM_run_cst(data, Psi_3_val, Psi_4_val,shape, gs_val = 1/70, compute_VPD = False):\n",
    "    \"\"\"run constant SR Penman Monteith mode but with VPD data instead of Ea and Es\n",
    "    -----------------------------------\n",
    "    data: list containing the data /!\\ the data should be in a specific order : Data = [Ta, Fn, Fg, Sws, Fe, Ws, Ustar, Ta_min, Ta_max, RH_min, RH_max]\n",
    "    T1_val: vector of the same size as the vectors in data, full of the value T1\n",
    "    T3_val: vector of the same size as the vectors in data, full of the value T3\n",
    "    -----------------------------------\n",
    "    \"\"\"\n",
    "    \n",
    "    # unpack the variables\n",
    "    Ta_val = data[\"Ta\"].to_numpy()\n",
    "    Fn_val = data[\"Fn\"].to_numpy()\n",
    "    Fg_val = data[\"Fg\"].to_numpy()\n",
    "    Sws_val = data[\"Sws\"].to_numpy()\n",
    "    Fe_val = data[\"Fe\"].to_numpy()\n",
    "    Ws_val = data[\"Ws\"].to_numpy()\n",
    "    Ustar_val = data[\"ustar\"].to_numpy()\n",
    "    VPD_val = data[\"VPD\"].to_numpy()\n",
    "    fPAR_val = data[\"fPAR\"].to_numpy()\n",
    "    \n",
    "    if compute_VPD:\n",
    "        T_min_val = data[\"Ta_min\"]\n",
    "        T_max_val = data[\"Ta_max\"]\n",
    "        \n",
    "        RH_min_val = data[\"RH_min\"]\n",
    "        RH_max_val = data[\"RH_max\"]\n",
    "        \n",
    "        VPD_val = eSat_func(T_min_val,T_max_val) - ea_func(T_min_val,T_max_val,RH_min_val,RH_max_val)\n",
    "        \n",
    "    # --------------------------\n",
    "    # derive the two Penman models\n",
    "    # --------------------------\n",
    "    \n",
    "\n",
    "    # Stress factor:\n",
    "    FF_vec = stress_factor_func(Sws_val,Psi_3_val,Psi_4_val,shape)\n",
    "    \n",
    "    # aerodynamic resistance:\n",
    "    VH_T = VH_func(Ws_val,Ustar_val)\n",
    "    displ_T = d_func(VH_T)\n",
    "    ZOM_T = zom_func(VH_T)\n",
    "    ZOH_T = zoh_func(ZOM_T)\n",
    "    Ra_T = ra_func(ZOM_T, ZOH_T, displ_T, Ws_val)\n",
    "    Ga_T = 1/Ra_T\n",
    "\n",
    "    # surface resistance:\n",
    "    Gs_C = gs_val*np.ones((Ta_val.size,)) # -> for constant resistance model -> Rs is constant and Ea = S(theta)*E_PM\n",
    "    # thermodynamic parameters\n",
    "    D_T = delta_func(Ta_val)\n",
    "\n",
    "    # compile the Ea values :\n",
    "    PM_cst = fPAR_val*FF_vec*PM_func(Fg_val,Fn_val,D_T,VPD_val,Ga_T, Gs_C) # R_s constant but stress factor in front of the PM evaluation\n",
    "    \n",
    "    Dic_var = {\"ga\":Ga_T, \"gs\":Gs_C, \"D_T\":D_T}\n",
    "    \n",
    "    return(PM_cst)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "together-repository",
   "metadata": {},
   "source": [
    "### Benchmark Penman-Monteith model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "worse-launch",
   "metadata": {},
   "outputs": [],
   "source": [
    "def PM_run_classic(data, gs_val = 1/70, compute_VPD = False):\n",
    "    \"\"\"run classic PM model (only Rs as calibration parameter)\n",
    "    -----------------------------------\n",
    "    data: list containing the data /!\\ the data should be in a specific order : Data = [Ta, Fn, Fg, Sws, Fe, Ws, Ustar, Ta_min, Ta_max, RH_min, RH_max]\n",
    "    Rs_val: value of the SR\n",
    "    -----------------------------------\n",
    "    \"\"\"\n",
    "    \n",
    "    # unpack the variables\n",
    "    Ta_val = data[\"Ta\"].to_numpy()\n",
    "    Fn_val = data[\"Fn\"].to_numpy()\n",
    "    Fg_val = data[\"Fg\"].to_numpy()\n",
    "    Sws_val = data[\"Sws\"].to_numpy()\n",
    "    Fe_val = data[\"Fe\"].to_numpy()\n",
    "    Ws_val = data[\"Ws\"].to_numpy()\n",
    "    Ustar_val = data[\"ustar\"].to_numpy()\n",
    "    VPD_val = data[\"VPD\"].to_numpy()\n",
    "    fPAR_val = data[\"fPAR\"].to_numpy()\n",
    "    \n",
    "    if compute_VPD:\n",
    "        T_min_val = data[\"Ta_min\"]\n",
    "        T_max_val = data[\"Ta_max\"]\n",
    "        \n",
    "        RH_min_val = data[\"RH_min\"]\n",
    "        RH_max_val = data[\"RH_max\"]\n",
    "        \n",
    "        VPD_val = eSat_func(T_min_val,T_max_val) - ea_func(T_min_val,T_max_val,RH_min_val,RH_max_val)\n",
    "            \n",
    "    # --------------------------\n",
    "    # derive the two Penman models\n",
    "    # --------------------------\n",
    "    \n",
    "    # aerodynamic resistance:\n",
    "    VH_T = VH_func(Ws_val,Ustar_val)\n",
    "    displ_T = d_func(VH_T)\n",
    "    ZOM_T = zom_func(VH_T)\n",
    "    ZOH_T = zoh_func(ZOM_T)\n",
    "    Ra_T = ra_func(ZOM_T, ZOH_T, displ_T, Ws_val)\n",
    "    Ga_T = 1/Ra_T\n",
    "\n",
    "    # surface resistance:\n",
    "    Gs_C = gs_val*np.ones((Ta_val.size,)) # -> for constant resistance model -> Rs is constant and Ea = S(theta)*E_PM\n",
    "    # thermodynamic parameters\n",
    "    D_T = delta_func(Ta_val)\n",
    "\n",
    "    # compile the Ea values :\n",
    "    PM_cst = fPAR_val*PM_func(Fg_val,Fn_val,D_T,VPD_val,Ga_T, Gs_C) # R_s constant but stress factor in front of the PM evaluation\n",
    "    \n",
    "    Dic_var = {\"ga\":Ga_T, \"gs\":Gs_C, \"D_T\":D_T}\n",
    "    \n",
    "    return(PM_cst)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "closing-rally",
   "metadata": {},
   "source": [
    "### Modified version of the PM equation\n",
    "\n",
    "2 Sided PM evapotranspiration model from *Schymanski and Or, 2017* : \n",
    "\\begin{equation}\n",
    "E = \\frac{1}{\\lambda}\\frac{\\Delta (R_n - G) + c_p \\rho_a g_a a_{sh} VPD }{\\Delta + \\gamma \\left( 1+ \\frac{g_a}{g_s} \\right) \\frac{a_{sh}}{a_s}}\n",
    "\\end{equation}\n",
    "\n",
    "with $a_{sh}$ and $a_s$ the fraction of projected area exchanging sensible heat flux with the air and fractio of one sided-leaf area covered by stomatas respectively. In the case of amphistomateous leaves, $a_s = 2$ and $a_s = 1$ for hypostomateous leaves. For the Howard Spring site, we consider $a_s = 2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "rational-improvement",
   "metadata": {},
   "outputs": [],
   "source": [
    "def PM_run_2var(data, Psi_3_val, Psi_4_val, gs_val = 1/70, compute_VPD = False):\n",
    "    \"\"\"run varying SR Penman Monteith model but with VPD data instead of Ea and Es\n",
    "    -----------------------------------\n",
    "    data: list containing the data /!\\ the data should be in a specific order : Data = [Ta, Fn, Fg, Sws, Fe, Ws, Ustar, Ta_min, Ta_max, RH_min, RH_max]\n",
    "    Psi_3_val: vector of the same size as the vectors in data, full of the value Psi_3\n",
    "    Psi_4_val: vector of the same size as the vectors in data, full of the value Psi_4\n",
    "    -----------------------------------\n",
    "    \"\"\"\n",
    "    \n",
    "    # unpack the variables\n",
    "    Ta_val = data[\"Ta\"].to_numpy()\n",
    "    Fn_val = data[\"Fn\"].to_numpy()\n",
    "    Fg_val = data[\"Fg\"].to_numpy()\n",
    "    Sws_val = data[\"Sws\"].to_numpy()\n",
    "    Fe_val = data[\"Fe\"].to_numpy()\n",
    "    Ws_val = data[\"Ws\"].to_numpy()\n",
    "    Ustar_val = data[\"ustar\"].to_numpy()\n",
    "    VPD_val = data[\"VPD\"].to_numpy()\n",
    "    fPAR_val = data[\"fPAR\"].to_numpy()\n",
    "    \n",
    "    if compute_VPD:\n",
    "        T_min_val = data[\"Ta_min\"]\n",
    "        T_max_val = data[\"Ta_max\"]\n",
    "        \n",
    "        RH_min_val = data[\"RH_min\"]\n",
    "        RH_max_val = data[\"RH_max\"]\n",
    "        \n",
    "        VPD_val = eSat_func(T_min_val,T_max_val) - ea_func(T_min_val,T_max_val,RH_min_val,RH_max_val)\n",
    "        \n",
    "    # --------------------------\n",
    "    # derive the two Penman models\n",
    "    # --------------------------\n",
    "    \n",
    "\n",
    "    # Stress factor:\n",
    "    FF_vec = stress_factor_func(Sws_val,Psi_3_val,Psi_4_val)\n",
    "    # aerodynamic resistance:\n",
    "    VH_T = VH_func(Ws_val,Ustar_val)\n",
    "    displ_T = d_func(VH_T)\n",
    "    ZOM_T = zom_func(VH_T)\n",
    "    ZOH_T = zoh_func(ZOM_T)\n",
    "    Ra_T = ra_func(ZOM_T, ZOH_T, displ_T, Ws_val)\n",
    "    Ga_T = 2/Ra_T # multiply by 2 to account for the 2 sided exchange of latent heat flux\n",
    "\n",
    "    # surface resistance:\n",
    "    Gs_T = gs_val*FF_vec+0.0001\n",
    "    # thermodynamic parameters\n",
    "    D_T = delta_func(Ta_val)\n",
    "\n",
    "    # compile the Ea values :\n",
    "    PM_var = fPAR_val*PM_func(Fg_val,Fn_val,D_T,VPD_val,Ga_T, Gs_T) # R_s varying with theta\n",
    "    \n",
    "    Dic_var = {\"ga\":Ga_T, \"gs\":Gs_T, \"D_T\":D_T}\n",
    "    \n",
    "    return(PM_var)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "owned-degree",
   "metadata": {},
   "outputs": [],
   "source": [
    "def PM_run_2cst(data, Psi_3_val, Psi_4_val, gs_val = 1/70, compute_VPD = False):\n",
    "    \"\"\"run constant SR Penman Monteith mode but with VPD data instead of Ea and Es\n",
    "    -----------------------------------\n",
    "    data: list containing the data /!\\ the data should be in a specific order : Data = [Ta, Fn, Fg, Sws, Fe, Ws, Ustar, Ta_min, Ta_max, RH_min, RH_max]\n",
    "    T1_val: vector of the same size as the vectors in data, full of the value T1\n",
    "    T3_val: vector of the same size as the vectors in data, full of the value T3\n",
    "    -----------------------------------\n",
    "    \"\"\"\n",
    "    \n",
    "    # unpack the variables\n",
    "    Ta_val = data[\"Ta\"].to_numpy()\n",
    "    Fn_val = data[\"Fn\"].to_numpy()\n",
    "    Fg_val = data[\"Fg\"].to_numpy()\n",
    "    Sws_val = data[\"Sws\"].to_numpy()\n",
    "    Fe_val = data[\"Fe\"].to_numpy()\n",
    "    Ws_val = data[\"Ws\"].to_numpy()\n",
    "    Ustar_val = data[\"ustar\"].to_numpy()\n",
    "    VPD_val = data[\"VPD\"].to_numpy()\n",
    "    fPAR_val = data[\"fPAR\"].to_numpy()\n",
    "    \n",
    "    if compute_VPD:\n",
    "        T_min_val = data[\"Ta_min\"]\n",
    "        T_max_val = data[\"Ta_max\"]\n",
    "        \n",
    "        RH_min_val = data[\"RH_min\"]\n",
    "        RH_max_val = data[\"RH_max\"]\n",
    "        \n",
    "        VPD_val = eSat_func(T_min_val,T_max_val) - ea_func(T_min_val,T_max_val,RH_min_val,RH_max_val)\n",
    "        \n",
    "    # --------------------------\n",
    "    # derive the two Penman models\n",
    "    # --------------------------\n",
    "    \n",
    "\n",
    "    # Stress factor:\n",
    "    FF_vec = stress_factor_func(Sws_val,Psi_3_val,Psi_4_val)\n",
    "    \n",
    "    # aerodynamic resistance:\n",
    "    VH_T = VH_func(Ws_val,Ustar_val)\n",
    "    displ_T = d_func(VH_T)\n",
    "    ZOM_T = zom_func(VH_T)\n",
    "    ZOH_T = zoh_func(ZOM_T)\n",
    "    Ra_T = ra_func(ZOM_T, ZOH_T, displ_T, Ws_val)\n",
    "    Ga_T = 2/Ra_T # multiply by 2 to account for the 2 sided exchange of latent heat flux\n",
    "\n",
    "    # surface resistance:\n",
    "    Gs_C = gs_val*np.ones((Ta_val.size,)) # -> for constant resistance model -> Rs is constant and Ea = S(theta)*E_PM\n",
    "    # thermodynamic parameters\n",
    "    D_T = delta_func(Ta_val)\n",
    "\n",
    "    # compile the Ea values :\n",
    "    PM_cst = fPAR_val*FF_vec*PM_func(Fg_val,Fn_val,D_T,VPD_val,Ga_T, Gs_C) # R_s constant but stress factor in front of the PM evaluation\n",
    "    \n",
    "    Dic_var = {\"ga\":Ga_T, \"gs\":Gs_C, \"D_T\":D_T}\n",
    "    \n",
    "    return(PM_cst)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "helpful-painting",
   "metadata": {},
   "source": [
    "### Priestley and Taylor model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "id": "applicable-dominant",
   "metadata": {},
   "outputs": [],
   "source": [
    "def PT_run(data,Psi_3_val, Psi_4_val, shape, alpha_val = 1.26, compute_VPD = False):\n",
    "    \"\"\"run classic PM model (only Rs as calibration parameter)\n",
    "    -----------------------------------\n",
    "    data: list containing the data /!\\ the data should be in a specific order : Data = [Ta, Fn, Fg, Sws, Fe, Ws, Ustar, Ta_min, Ta_max, RH_min, RH_max]\n",
    "    Rs_val: value of the SR\n",
    "    -----------------------------------\n",
    "    \"\"\"\n",
    "    \n",
    "    # unpack the variables\n",
    "    Ta_val = data[\"Ta\"].to_numpy()\n",
    "    Fn_val = data[\"Fn\"].to_numpy()\n",
    "    Fg_val = data[\"Fg\"].to_numpy()\n",
    "    Sws_val = data[\"Sws\"].to_numpy()\n",
    "    #Fe_val = data[\"Fe\"].to_numpy()\n",
    "    #Ws_val = data[\"Ws\"].to_numpy()\n",
    "    #Ustar_val = data[\"ustar\"].to_numpy()\n",
    "    VPD_val = data[\"VPD\"].to_numpy()\n",
    "    fPAR_val = data[\"fPAR\"].to_numpy()\n",
    "    \n",
    "    if compute_VPD:\n",
    "        T_min_val = data[\"Ta_min\"]\n",
    "        T_max_val = data[\"Ta_max\"]\n",
    "        \n",
    "        RH_min_val = data[\"RH_min\"]\n",
    "        RH_max_val = data[\"RH_max\"]\n",
    "        \n",
    "        VPD_val = eSat_func(T_min_val,T_max_val) - ea_func(T_min_val,T_max_val,RH_min_val,RH_max_val)\n",
    "        \n",
    "    # --------------------------\n",
    "    # derive the two Penman models\n",
    "    # --------------------------\n",
    "\n",
    "    # Stress factor:\n",
    "    FF_vec = stress_factor_func(Sws_val,Psi_3_val,Psi_4_val, shape)\n",
    "    \n",
    "    # surface resistance:\n",
    "    Alpha_vec = alpha_val*np.ones((Ta_val.size,))\n",
    "\n",
    "    # thermodynamic parameters\n",
    "    D_T = delta_func(Ta_val)\n",
    "\n",
    "    # compile the Ea values :\n",
    "    PT_mod = fPAR_val*FF_vec*PT_func(D_T,Fn_val,Fg_val,Alpha_vec)\n",
    "    \n",
    "    Dic_var = {\"D_T\":D_T}\n",
    "    \n",
    "    return(PT_mod)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "sticky-necessity",
   "metadata": {},
   "source": [
    "### Inverse modelling\n",
    "Compute the original $g_s$ time serie out of the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "compressed-calgary",
   "metadata": {},
   "outputs": [],
   "source": [
    "def inv_PM_run(data, compute_VPD = False):\n",
    "    \"\"\"inverse modelling of the PM equation to derive time varying surface resistance\n",
    "    -----------------------------------\n",
    "    data: list containing the data /!\\ the data should be in a specific order : Data = [Ta, Fn, Fg, Sws, Fe, Ws, Ustar, Ta_min, Ta_max, RH_min, RH_max]\n",
    "    Rs_val: value of the SR\n",
    "    -----------------------------------\n",
    "    \"\"\"\n",
    "    \n",
    "    # unpack the variables\n",
    "    Ta_val = data[\"Ta\"].to_numpy()\n",
    "    Fn_val = data[\"Fn\"].to_numpy()\n",
    "    Fg_val = data[\"Fg\"].to_numpy()\n",
    "    Sws_val = data[\"Sws\"].to_numpy()\n",
    "    Fe_val = data[\"Fe\"].to_numpy()\n",
    "    Ws_val = data[\"Ws\"].to_numpy()\n",
    "    Ustar_val = data[\"ustar\"].to_numpy()\n",
    "    VPD_val = data[\"VPD\"].to_numpy()\n",
    "    \n",
    "    if compute_VPD:\n",
    "        T_min_val = data[\"Ta_min\"]\n",
    "        T_max_val = data[\"Ta_max\"]\n",
    "        \n",
    "        RH_min_val = data[\"RH_min\"]\n",
    "        RH_max_val = data[\"RH_max\"]\n",
    "        \n",
    "        VPD_val = eSat_func(T_min_val,T_max_val) - ea_func(T_min_val,T_max_val,RH_min_val,RH_max_val)\n",
    "        \n",
    "    # --------------------------\n",
    "    # derive the two Penman models\n",
    "    # --------------------------\n",
    "    \n",
    "    # aerodynamic resistance:\n",
    "    VH_T = VH_func(Ws_val,Ustar_val)\n",
    "    displ_T = d_func(VH_T)\n",
    "    ZOM_T = zom_func(VH_T)\n",
    "    ZOH_T = zoh_func(ZOM_T)\n",
    "    Ra_T = ra_func(ZOM_T, ZOH_T, displ_T, Ws_val)\n",
    "\n",
    "    # thermodynamic parameters\n",
    "    D_T = delta_func(Ta_val)\n",
    "\n",
    "    # compile the Ea values :\n",
    "    Gs_val = 1/Inv_PM_func(Fe_val,Fg_val,Fn_val,D_T,VPD_val,Ra_T) # R_s constant but stress factor in front of the PM evaluation\n",
    "    \n",
    "    \n",
    "    return(Gs_val)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "intensive-return",
   "metadata": {},
   "source": [
    "## Calibration algorithm"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "alternative-intranet",
   "metadata": {},
   "source": [
    "Use the global optimizer from the `scipy.optimize` package. Minimize the squared residual :\n",
    "\\begin{equation}\n",
    "\\Theta_{opt} = \\underset{\\theta_3, \\theta_4, g_s}{min} \\left( E_p(\\textbf{X}, \\theta_3, \\theta_4, g_s) - E_{obs})^2\\right)\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "still-metadata",
   "metadata": {},
   "outputs": [],
   "source": [
    "# def calibration(Data,model_run,bounds_shape, bounds = [(0.1,1),(0.001,0.1)], compute_VPD = False, linear = False):\n",
    "    \n",
    "#     if not linear:\n",
    "#         bounds.append(bounds_shape)\n",
    "        \n",
    "#         if compute_VPD:\n",
    "#             def residual(Coeff):\n",
    "#                 return((model_run(Data, Coeff[0], Coeff[1],Coeff[2], compute_VPD = True)-Data[\"Fe\"])**2).sum()\n",
    "#         else:\n",
    "#             def residual(Coeff):\n",
    "#                 return((model_run(Data, Coeff[0], Coeff[1],Coeff[2])-Data[\"Fe\"])**2).sum()\n",
    "#     else : \n",
    "#         if compute_VPD:\n",
    "#             def residual(Coeff):\n",
    "#                 return((model_run(Data, Coeff[0], Coeff[1],0, compute_VPD = True)-Data[\"Fe\"])**2).sum()\n",
    "#         else:\n",
    "#             def residual(Coeff):\n",
    "#                 return((model_run(Data, Coeff[0], Coeff[1],0)-Data[\"Fe\"])**2).sum()\n",
    "            \n",
    "#     coeff_opti = optimize.shgo(residual, bounds).x\n",
    "#     return(coeff_opti)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "floppy-rates",
   "metadata": {},
   "outputs": [],
   "source": [
    "def calibration(Data,model_run, bounds = [(0.1,1),(0.001,0.1),(-50,50)], compute_VPD = False):\n",
    "      \n",
    "    if compute_VPD:\n",
    "        def residual(Coeff):\n",
    "            return((model_run(Data, Coeff[0], Coeff[1],Coeff[2], compute_VPD = True)-Data[\"Fe\"])**2).sum()\n",
    "    else:\n",
    "        def residual(Coeff):\n",
    "            return((model_run(Data, Coeff[0], Coeff[1],Coeff[2])-Data[\"Fe\"])**2).sum()\n",
    "            \n",
    "    coeff_opti = optimize.shgo(residual, bounds).x\n",
    "    return(coeff_opti)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "sophisticated-canada",
   "metadata": {},
   "source": [
    "# Part III - Experiments"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "straight-springfield",
   "metadata": {},
   "source": [
    "## One site, one year\n",
    "Data from Howard Springs for the sole year 2016"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "enclosed-client",
   "metadata": {},
   "outputs": [
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       "Dimensions:                (latitude: 1, longitude: 1, time: 315551)\n",
       "Coordinates:\n",
       "  * time                   (time) datetime64[ns] 2002-01-01T00:30:00 ... 2019...\n",
       "  * latitude               (latitude) float64 -12.5\n",
       "  * longitude              (longitude) float64 131.2\n",
       "Data variables: (12/143)\n",
       "    Ah                     (time, latitude, longitude) float64 dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;\n",
       "    Ah_QCFlag              (time, latitude, longitude) float64 dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;\n",
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       "    Ah_HMP_2m              (time, latitude, longitude) float64 dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;\n",
       "    Ah_HMP_2m_QCFlag       (time, latitude, longitude) float64 dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;\n",
       "    ...                     ...\n",
       "    Ws_SONIC_Av_QCFlag     (time, latitude, longitude) float64 dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;\n",
       "    ps                     (time, latitude, longitude) float64 dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;\n",
       "    ps_QCFlag              (time, latitude, longitude) float64 dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;\n",
       "    ustar                  (time, latitude, longitude) float64 dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;\n",
       "    ustar_QCFlag           (time, latitude, longitude) float64 dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;\n",
       "    crs                    float64 -2.147e+09\n",
       "Attributes: (12/50)\n",
       "    BulkDensity:              1500\n",
       "    FgDepth:                  0.08\n",
       "    OrganicContent:           0.01\n",
       "    PythonVersion:            2.7.16 |Anaconda, Inc.| (default, Mar 14 2019, ...\n",
       "    QC_version:               PyFluxPro V1.0.1\n",
       "    SwsDefault:               0.10\n",
       "    ...                       ...\n",
       "    title:                    Flux tower data set from the Howard Springs sit...\n",
       "    tower_height:             23m\n",
       "    vegetation:               Woody savanna\n",
       "    xl_datemode:              0\n",
       "    xl_filename:              E:/My Dropbox/Dropbox/Data_flux_data/Site data ...\n",
       "    xl_moddatetime:           2020-01-21 11:59:06</pre><div class='xr-wrap' hidden><div class='xr-header'><div class='xr-obj-type'>xarray.Dataset</div></div><ul class='xr-sections'><li class='xr-section-item'><input id='section-db8e72e8-1a85-4afd-b59b-b5659ae716fc' class='xr-section-summary-in' type='checkbox' disabled ><label for='section-db8e72e8-1a85-4afd-b59b-b5659ae716fc' class='xr-section-summary'  title='Expand/collapse section'>Dimensions:</label><div class='xr-section-inline-details'><ul class='xr-dim-list'><li><span class='xr-has-index'>latitude</span>: 1</li><li><span class='xr-has-index'>longitude</span>: 1</li><li><span class='xr-has-index'>time</span>: 315551</li></ul></div><div class='xr-section-details'></div></li><li class='xr-section-item'><input id='section-735bfdcc-d898-40ae-a85e-717456dcae99' class='xr-section-summary-in' type='checkbox'  checked><label for='section-735bfdcc-d898-40ae-a85e-717456dcae99' class='xr-section-summary' >Coordinates: <span>(3)</span></label><div class='xr-section-inline-details'></div><div class='xr-section-details'><ul class='xr-var-list'><li class='xr-var-item'><div class='xr-var-name'><span class='xr-has-index'>time</span></div><div class='xr-var-dims'>(time)</div><div class='xr-var-dtype'>datetime64[ns]</div><div class='xr-var-preview xr-preview'>2002-01-01T00:30:00 ... 2019-12-...</div><input id='attrs-e909b7a9-7c2f-479b-bb55-83517622b04a' class='xr-var-attrs-in' type='checkbox' ><label for='attrs-e909b7a9-7c2f-479b-bb55-83517622b04a' title='Show/Hide attributes'><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg></label><input id='data-d0b8dc36-860b-4692-8f23-a2018c91a183' class='xr-var-data-in' type='checkbox'><label for='data-d0b8dc36-860b-4692-8f23-a2018c91a183' title='Show/Hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-var-attrs'><dl class='xr-attrs'><dt><span>long_name :</span></dt><dd>time</dd><dt><span>standard_name :</span></dt><dd>time</dd></dl></div><div class='xr-var-data'><pre>array([&#x27;2002-01-01T00:30:00.000000000&#x27;, &#x27;2002-01-01T01:00:00.000000000&#x27;,\n",
       "       &#x27;2002-01-01T01:30:00.000000000&#x27;, ..., &#x27;2019-12-31T22:30:00.000000000&#x27;,\n",
       "       &#x27;2019-12-31T23:00:00.000000000&#x27;, &#x27;2019-12-31T23:30:00.000000000&#x27;],\n",
       "      dtype=&#x27;datetime64[ns]&#x27;)</pre></div></li><li class='xr-var-item'><div class='xr-var-name'><span class='xr-has-index'>latitude</span></div><div class='xr-var-dims'>(latitude)</div><div class='xr-var-dtype'>float64</div><div class='xr-var-preview xr-preview'>-12.5</div><input id='attrs-9022e8e3-06a4-425d-ab87-210967d6ee99' class='xr-var-attrs-in' type='checkbox' ><label for='attrs-9022e8e3-06a4-425d-ab87-210967d6ee99' title='Show/Hide attributes'><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg></label><input id='data-aaa7138d-ff66-4f6c-afdb-28e8eaa1e488' class='xr-var-data-in' type='checkbox'><label for='data-aaa7138d-ff66-4f6c-afdb-28e8eaa1e488' title='Show/Hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-var-attrs'><dl class='xr-attrs'><dt><span>long_name :</span></dt><dd>latitude</dd><dt><span>standard_name :</span></dt><dd>latitude</dd><dt><span>units :</span></dt><dd>degrees north</dd></dl></div><div class='xr-var-data'><pre>array([-12.4952])</pre></div></li><li class='xr-var-item'><div class='xr-var-name'><span class='xr-has-index'>longitude</span></div><div class='xr-var-dims'>(longitude)</div><div class='xr-var-dtype'>float64</div><div class='xr-var-preview xr-preview'>131.2</div><input id='attrs-9828723a-5509-41a3-9e5b-d1115f61a91e' class='xr-var-attrs-in' type='checkbox' ><label for='attrs-9828723a-5509-41a3-9e5b-d1115f61a91e' title='Show/Hide attributes'><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg></label><input id='data-c2f44b3e-ad0c-444e-9e0a-b97da2b977fd' class='xr-var-data-in' type='checkbox'><label for='data-c2f44b3e-ad0c-444e-9e0a-b97da2b977fd' title='Show/Hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-var-attrs'><dl class='xr-attrs'><dt><span>long_name :</span></dt><dd>longitude</dd><dt><span>standard_name :</span></dt><dd>longitude</dd><dt><span>units :</span></dt><dd>degrees east</dd></dl></div><div class='xr-var-data'><pre>array([131.15005])</pre></div></li></ul></div></li><li class='xr-section-item'><input id='section-0897fbc6-a86a-4aa3-9bdc-3ddefe5be0cb' class='xr-section-summary-in' type='checkbox'  ><label for='section-0897fbc6-a86a-4aa3-9bdc-3ddefe5be0cb' class='xr-section-summary' >Data variables: <span>(143)</span></label><div class='xr-section-inline-details'></div><div class='xr-section-details'><ul class='xr-var-list'><li class='xr-var-item'><div class='xr-var-name'><span>Ah</span></div><div class='xr-var-dims'>(time, latitude, longitude)</div><div class='xr-var-dtype'>float64</div><div class='xr-var-preview xr-preview'>dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;</div><input id='attrs-b50521a1-e554-44e0-a79b-4e3cb2b2992e' class='xr-var-attrs-in' type='checkbox' ><label for='attrs-b50521a1-e554-44e0-a79b-4e3cb2b2992e' title='Show/Hide attributes'><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg></label><input id='data-cf1cb414-1cfd-4271-8aeb-70d72bb79512' class='xr-var-data-in' type='checkbox'><label for='data-cf1cb414-1cfd-4271-8aeb-70d72bb79512' title='Show/Hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-var-attrs'><dl class='xr-attrs'><dt><span>coverage_L3 :</span></dt><dd>85</dd><dt><span>coverage_L4 :</span></dt><dd>100</dd><dt><span>description_L4 :</span></dt><dd>Gap filled using aws, access, era5, </dd><dt><span>group_name :</span></dt><dd></dd><dt><span>height :</span></dt><dd>23m</dd><dt><span>instrument :</span></dt><dd>HMP45C</dd><dt><span>long_name :</span></dt><dd>Absolute humidity</dd><dt><span>serial_number :</span></dt><dd></dd><dt><span>standard_name :</span></dt><dd>not defined</dd><dt><span>units :</span></dt><dd>g/m3</dd><dt><span>valid_range :</span></dt><dd>1,35</dd></dl></div><div class='xr-var-data'><table>\n",
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       "</table></div></li><li class='xr-var-item'><div class='xr-var-name'><span>Ah_HMP_23m</span></div><div class='xr-var-dims'>(time, latitude, longitude)</div><div class='xr-var-dtype'>float64</div><div class='xr-var-preview xr-preview'>dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;</div><input id='attrs-48625366-2a03-49d1-9842-17550e29421f' class='xr-var-attrs-in' type='checkbox' ><label for='attrs-48625366-2a03-49d1-9842-17550e29421f' title='Show/Hide attributes'><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg></label><input id='data-e80d11c5-746a-4d68-8fc5-f8cc61f4c0d2' class='xr-var-data-in' type='checkbox'><label for='data-e80d11c5-746a-4d68-8fc5-f8cc61f4c0d2' title='Show/Hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-var-attrs'><dl class='xr-attrs'><dt><span>coverage_L3 :</span></dt><dd>73</dd><dt><span>coverage_L4 :</span></dt><dd>73</dd><dt><span>group_name :</span></dt><dd></dd><dt><span>height :</span></dt><dd>23m</dd><dt><span>instrument :</span></dt><dd>HMP45C</dd><dt><span>long_name :</span></dt><dd>Absolute humidity</dd><dt><span>serial_number :</span></dt><dd></dd><dt><span>standard_name :</span></dt><dd>not defined</dd><dt><span>units :</span></dt><dd>g/m3</dd><dt><span>valid_range :</span></dt><dd>5,30</dd></dl></div><div class='xr-var-data'><table>\n",
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       "</table></div></li><li class='xr-var-item'><div class='xr-var-name'><span>Ah_HMP_23m_QCFlag</span></div><div class='xr-var-dims'>(time, latitude, longitude)</div><div class='xr-var-dtype'>float64</div><div class='xr-var-preview xr-preview'>dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;</div><input id='attrs-0721c284-6c24-473a-b098-796bcdc64daa' class='xr-var-attrs-in' type='checkbox' ><label for='attrs-0721c284-6c24-473a-b098-796bcdc64daa' title='Show/Hide attributes'><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg></label><input id='data-ccfa0360-4c8e-497d-9f2a-d9030fdce828' class='xr-var-data-in' type='checkbox'><label for='data-ccfa0360-4c8e-497d-9f2a-d9030fdce828' title='Show/Hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-var-attrs'><dl class='xr-attrs'><dt><span>long_name :</span></dt><dd>Ah_HMP_23mQC flag</dd><dt><span>units :</span></dt><dd>none</dd></dl></div><div class='xr-var-data'><table>\n",
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       "</table></div></li><li class='xr-var-item'><div class='xr-var-name'><span>Ah_HMP_2m</span></div><div class='xr-var-dims'>(time, latitude, longitude)</div><div class='xr-var-dtype'>float64</div><div class='xr-var-preview xr-preview'>dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;</div><input id='attrs-5ce38ae6-c50b-493f-8598-d6c403c3a2f1' class='xr-var-attrs-in' type='checkbox' ><label for='attrs-5ce38ae6-c50b-493f-8598-d6c403c3a2f1' title='Show/Hide attributes'><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg></label><input id='data-ac9960ae-e08c-4192-94e9-2f5c9bf45dfd' class='xr-var-data-in' type='checkbox'><label for='data-ac9960ae-e08c-4192-94e9-2f5c9bf45dfd' title='Show/Hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-var-attrs'><dl class='xr-attrs'><dt><span>coverage_L3 :</span></dt><dd>66</dd><dt><span>coverage_L4 :</span></dt><dd>66</dd><dt><span>group_name :</span></dt><dd></dd><dt><span>height :</span></dt><dd>2m</dd><dt><span>instrument :</span></dt><dd>HMP45C</dd><dt><span>long_name :</span></dt><dd>Absolute humidity</dd><dt><span>serial_number :</span></dt><dd></dd><dt><span>standard_name :</span></dt><dd>not defined</dd><dt><span>units :</span></dt><dd>g/m3</dd><dt><span>valid_range :</span></dt><dd>2,30</dd></dl></div><div class='xr-var-data'><table>\n",
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       "</table></div></li><li class='xr-var-item'><div class='xr-var-name'><span>Ah_HMP_2m_QCFlag</span></div><div class='xr-var-dims'>(time, latitude, longitude)</div><div class='xr-var-dtype'>float64</div><div class='xr-var-preview xr-preview'>dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;</div><input id='attrs-2f9719a6-5645-44d3-95e6-35a9c40b978a' class='xr-var-attrs-in' type='checkbox' ><label for='attrs-2f9719a6-5645-44d3-95e6-35a9c40b978a' title='Show/Hide attributes'><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg></label><input id='data-dde90926-c413-4960-848a-4e36d7bf2fc7' class='xr-var-data-in' type='checkbox'><label for='data-dde90926-c413-4960-848a-4e36d7bf2fc7' title='Show/Hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-var-attrs'><dl class='xr-attrs'><dt><span>long_name :</span></dt><dd>Ah_HMP_2mQC flag</dd><dt><span>units :</span></dt><dd>none</dd></dl></div><div class='xr-var-data'><table>\n",
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       "</table></div></li><li class='xr-var-item'><div class='xr-var-name'><span>Ah_IRGA_Av</span></div><div class='xr-var-dims'>(time, latitude, longitude)</div><div class='xr-var-dtype'>float64</div><div class='xr-var-preview xr-preview'>dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;</div><input id='attrs-9ac86f51-33b3-4ab0-96b2-973c5591d348' class='xr-var-attrs-in' type='checkbox' ><label for='attrs-9ac86f51-33b3-4ab0-96b2-973c5591d348' title='Show/Hide attributes'><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg></label><input id='data-ff39f65c-1702-4eff-96a7-8efa53e4806b' class='xr-var-data-in' type='checkbox'><label for='data-ff39f65c-1702-4eff-96a7-8efa53e4806b' title='Show/Hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-var-attrs'><dl class='xr-attrs'><dt><span>coverage_L3 :</span></dt><dd>76</dd><dt><span>coverage_L4 :</span></dt><dd>76</dd><dt><span>group_name :</span></dt><dd></dd><dt><span>height :</span></dt><dd>23m</dd><dt><span>instrument :</span></dt><dd>Li-7500</dd><dt><span>long_name :</span></dt><dd>Absolute humidity</dd><dt><span>serial_number :</span></dt><dd></dd><dt><span>standard_name :</span></dt><dd>not defined</dd><dt><span>units :</span></dt><dd>g/m3</dd><dt><span>valid_range :</span></dt><dd>2,35</dd></dl></div><div class='xr-var-data'><table>\n",
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       "</table></div></li><li class='xr-var-item'><div class='xr-var-name'><span>Ah_IRGA_Av_QCFlag</span></div><div class='xr-var-dims'>(time, latitude, longitude)</div><div class='xr-var-dtype'>float64</div><div class='xr-var-preview xr-preview'>dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;</div><input id='attrs-9ceace67-bf97-4872-a0fc-7899afd2f06d' class='xr-var-attrs-in' type='checkbox' ><label for='attrs-9ceace67-bf97-4872-a0fc-7899afd2f06d' title='Show/Hide attributes'><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg></label><input id='data-8f5d830a-2c39-452f-87cc-693ea0f34125' class='xr-var-data-in' type='checkbox'><label for='data-8f5d830a-2c39-452f-87cc-693ea0f34125' title='Show/Hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-var-attrs'><dl class='xr-attrs'><dt><span>long_name :</span></dt><dd>Ah_IRGA_AvQC flag</dd><dt><span>units :</span></dt><dd>none</dd></dl></div><div class='xr-var-data'><table>\n",
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       "</table></div></li><li class='xr-var-item'><div class='xr-var-name'><span>Ah_IRGA_Sd</span></div><div class='xr-var-dims'>(time, latitude, longitude)</div><div class='xr-var-dtype'>float64</div><div class='xr-var-preview xr-preview'>dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;</div><input id='attrs-09a3e34b-0137-4f0b-9443-a0ea8770194b' class='xr-var-attrs-in' type='checkbox' ><label for='attrs-09a3e34b-0137-4f0b-9443-a0ea8770194b' title='Show/Hide attributes'><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg></label><input id='data-613190a7-216f-4456-b44d-1d659e13574e' class='xr-var-data-in' type='checkbox'><label for='data-613190a7-216f-4456-b44d-1d659e13574e' title='Show/Hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-var-attrs'><dl class='xr-attrs'><dt><span>coverage_L3 :</span></dt><dd>84</dd><dt><span>coverage_L4 :</span></dt><dd>84</dd><dt><span>group_name :</span></dt><dd></dd><dt><span>height :</span></dt><dd>not defined</dd><dt><span>instrument :</span></dt><dd>not defined</dd><dt><span>long_name :</span></dt><dd>Absolute humidity</dd><dt><span>serial_number :</span></dt><dd>not defined</dd><dt><span>standard_name :</span></dt><dd>not defined</dd><dt><span>units :</span></dt><dd>g/m3</dd><dt><span>valid_range :</span></dt><dd>0,20</dd></dl></div><div class='xr-var-data'><table>\n",
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       "</table></div></li><li class='xr-var-item'><div class='xr-var-name'><span>Ah_IRGA_Sd_QCFlag</span></div><div class='xr-var-dims'>(time, latitude, longitude)</div><div class='xr-var-dtype'>float64</div><div class='xr-var-preview xr-preview'>dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;</div><input id='attrs-64e7799c-0654-45e3-9984-3766003212c1' class='xr-var-attrs-in' type='checkbox' ><label for='attrs-64e7799c-0654-45e3-9984-3766003212c1' title='Show/Hide attributes'><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg></label><input id='data-0a3f67db-21ec-419f-83bc-ec5da93f9696' class='xr-var-data-in' type='checkbox'><label for='data-0a3f67db-21ec-419f-83bc-ec5da93f9696' title='Show/Hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-var-attrs'><dl class='xr-attrs'><dt><span>long_name :</span></dt><dd>Ah_IRGA_SdQC flag</dd><dt><span>units :</span></dt><dd>none</dd></dl></div><div class='xr-var-data'><table>\n",
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       "</table></div></li><li class='xr-var-item'><div class='xr-var-name'><span>Ah_IRGA_Vr</span></div><div class='xr-var-dims'>(time, latitude, longitude)</div><div class='xr-var-dtype'>float64</div><div class='xr-var-preview xr-preview'>dask.array&lt;chunksize=(315551, 1, 1), meta=np.ndarray&gt;</div><input id='attrs-d7528b3d-76c9-4a55-944e-f37e07504569' class='xr-var-attrs-in' type='checkbox' ><label for='attrs-d7528b3d-76c9-4a55-944e-f37e07504569' title='Show/Hide attributes'><svg class='icon xr-icon-file-text2'><use xlink:href='#icon-file-text2'></use></svg></label><input id='data-8353346a-79f3-4456-9b32-cd5b88e84eb9' class='xr-var-data-in' type='checkbox'><label for='data-8353346a-79f3-4456-9b32-cd5b88e84eb9' title='Show/Hide data repr'><svg class='icon xr-icon-database'><use xlink:href='#icon-database'></use></svg></label><div class='xr-var-attrs'><dl class='xr-attrs'><dt><span>coverage_L3 :</span></dt><dd>82</dd><dt><span>coverage_L4 :</span></dt><dd>82</dd><dt><span>group_name :</span></dt><dd></dd><dt><span>height :</span></dt><dd>23m</dd><dt><span>instrument :</span></dt><dd>Li-7500</dd><dt><span>long_name :</span></dt><dd>Absolute humidity</dd><dt><span>serial_number :</span></dt><dd></dd><dt><span>standard_name :</span></dt><dd>not defined</dd><dt><span>units :</span></dt><dd>(g/m3)2</dd><dt><span>valid_range :</span></dt><dd>0,400</dd></dl></div><div class='xr-var-data'><table>\n",