Skip to content
simpler_model.ipynb 1.74 MiB
Newer Older
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000
{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "numerous-london",
   "metadata": {},
   "source": [
    "# **Location of the stress factor in potential evapo-transpiration models**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "local-referral",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Part I - Methodology \n",
    "\n",
    "## <u> Motivation </u> \n",
    "\n",
    "### Theoretical background\n",
    "\n",
    "In the literature, this scaling of the potential evapo-transpiration is commonly found with simpler radiation-temperature models such as the Priestley and Taylor model instead of the Penman-Monteith one. Recall the expression of the Priestley and Taylor model: \n",
    "\n",
    "\\begin{equation}\\label{eq_PT}\n",
    "    \\lambda E_{p,PT} = \\alpha \\frac{\\Delta}{\\Delta + \\gamma}(Rn - G)\n",
    "\\end{equation}\n",
    "\n",
    "The Priestley and Taylor is not physicaly based as can be the Penman-Monteith model. Then we aim at investigating the differences obtained when using such a model. \n",
    "\n",
    "This notebook compares different newly implemented models : \n",
    "* Priestley and Taylor model with a stress factor : \n",
    "\\begin{align}\n",
    "    E_{a, PT}  = f_{PAR}.S(\\theta).E_{p,PT}(\\textbf{X})\n",
    "\\end{align}\n",
    "\n",
    "* Modified varying surface conductance Penman-Monteith model :\n",
    "\\begin{align}\n",
    "    E_{a, var, PM}  = f_{PAR}.E_{p,PM mod}(\\textbf{X}, S(\\theta))\n",
    "\\end{align}\n",
    "\n",
    "* Modified constant surface conductance Penman-Monteith model :\n",
    "\\begin{align}\n",
    "    E_{a, cst, PM}  = f_{PAR}.S(\\theta).E_{p,PM mod}(\\textbf{X})\n",
    "\\end{align}\n",
    "\n",
    "The modified version of the Penman-Monteith equation takes into account the double sided exchange of sensible heat. This model was developped at the leaf scale but is tested here at the canopy scale in the framework of the big leaf model (*Schymanski and Or, 2017*). The analytical expression of the modified expression is henceforth: \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",
    "### Modelling experiements\n",
    "\n",
    "Different experiments are carried out to compare the different models and assess their behavior: \n",
    "1. \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": "excellent-wilson",
   "metadata": {},
   "source": [
    "# Part II - Functions set up"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "tired-tampa",
   "metadata": {},
   "source": [
    "## Importing relevant packages"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "after-banks",
   "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": "productive-protection",
   "metadata": {},
   "source": [
    "## Path of the different files (pre-defined python functions, sympy equations, sympy variables)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "improved-transcription",
   "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": "sacred-prison",
   "metadata": {},
   "source": [
    "## Importing the sympy variables and equations defined in the theory.ipynb notebook"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "biological-timing",
   "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": "indonesian-cabin",
   "metadata": {},
   "source": [
    "## Importing the performance assessment functions defined in the analysis_function.py file"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "grave-transsexual",
   "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": "devoted-forward",
   "metadata": {},
   "source": [
    "## Data import, preprocess and shape for the computations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "toxic-combination",
   "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": "frank-venue",
   "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": "concrete-power",
   "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": "arbitrary-bradford",
   "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": "latest-private",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\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": "horizontal-journalism",
   "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": "cognitive-discovery",
   "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": "starting-kernel",
   "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": "immediate-combine",
   "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\",\"Fe_QCFlag\",\"Fn_QCFlag\",\"Fg_QCFlag\",\"Ws_QCFlag\",\"Sws_QCFlag\",\"Ta_QCFlag\",\"ustar_QCFlag\",\"RH_QCFlag\", \"VPD_QCFlag\"]]\n",
    "    \n",
    "    # convert to dataframe\n",
    "    df = df.to_dataframe()\n",
    "    \n",
    "    # aggregate following the rule stated in freq\n",
    "    df = df.groupby([pd.Grouper(level = \"latitude\"), pd.Grouper(level = \"longitude\"), pd.Grouper(level = \"time\", freq = Freq)]).mean()\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": "perfect-portable",
   "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": "immediate-virus",
   "metadata": {},
   "source": [
    "### Water stress functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "western-click",
   "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": "danish-forge",
   "metadata": {},
   "source": [
    "### Soil water potential"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "finite-album",
   "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": "horizontal-station",
   "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": "extensive-miller",
   "metadata": {},
   "source": [
    "### Penman-Monteith"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "romance-liberal",
   "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": "hispanic-recording",
   "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": "jewish-stopping",
   "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": "neither-excuse",
   "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": "loved-concept",
   "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": "architectural-forestry",
   "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": "several-adelaide",
   "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": 20,
   "id": "exotic-radiation",
   "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": 21,
   "id": "defined-patch",
   "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",