"investigating the weir behavior of the PM equation to wind speed"

notebooks/Finished_project/simpler_model/latex_files/dry_season.tex

 \begin{tabular}{lrrrrrrrr}
 \toprule
-{} &  Varying\_gS &  Mod\_var\_gS &  Constant\_gS &  Mod\_cst\_gS &  PT\_model &  Bench\_PT &   fPAR\_PM &   Bench\_PM \\
+{} &  Varying\_gS &  Mod\_var\_gS &  Constant\_gS &  Mod\_cst\_gS &  PT\_model &  Bench\_PT &    fPAR\_PM &   Bench\_PM \\
 \midrule
-RMSE   &    0.613282 &    0.626179 &     0.728898 &    0.750989 &  0.635991 &  2.001012 &  2.367712 &   5.080072 \\
-RMedSE &    0.438843 &    0.364637 &     0.482288 &    0.520776 &  0.384338 &  1.306227 &  1.949292 &   4.380618 \\
-NS     &    0.746747 &    0.735983 &     0.642259 &    0.620246 &  0.727644 & -1.696088 & -2.774787 & -16.376981 \\
-RVE    &   -0.045086 &   -0.004417 &    -0.039894 &    0.069212 & -0.058347 & -0.613086 & -0.735152 &  -1.652722 \\
-MAE    &    0.482613 &    0.478669 &     0.573307 &    0.598198 &  0.496328 &  1.702450 &  2.084216 &   4.570540 \\
-bias   &   -0.124684 &   -0.012215 &    -0.110326 &    0.191402 & -0.161357 & -1.695466 & -2.033035 &  -4.570540 \\
+RMSE   &    0.931820 &    0.893647 &     2.796542 &    1.308901 &  0.635991 &  2.001012 &   4.892235 &   9.123506 \\
+RMedSE &    0.407479 &    0.461836 &     2.378363 &    0.990405 &  0.384338 &  1.306227 &   4.206196 &   7.780005 \\
+NS     &    0.415347 &    0.462268 &    -4.265958 &   -0.153581 &  0.727644 & -1.696088 & -15.115697 & -55.047735 \\
+RVE    &   -0.046600 &    0.068944 &     0.406716 &    0.336310 & -0.058347 & -0.613086 &  -1.601375 &  -2.999871 \\
+MAE    &    0.621935 &    0.649353 &     2.500643 &    1.043889 &  0.496328 &  1.702450 &   4.428542 &   8.296030 \\
+bias   &   -0.128870 &    0.190662 &     1.124757 &    0.930052 & -0.161357 & -1.695466 &  -4.428542 &  -8.296030 \\
 \bottomrule
 \end{tabular}

notebooks/Finished_project/simpler_model/latex_files/wet_season.tex

 \begin{tabular}{lrrrrrrrr}
 \toprule
-{} &  Varying\_gS &  Mod\_var\_gS &  Constant\_gS &  Mod\_cst\_gS &  PT\_model &  Bench\_PT &   fPAR\_PM &  Bench\_PM \\
+{} &  Varying\_gS &  Mod\_var\_gS &  Constant\_gS &  Mod\_cst\_gS &  PT\_model &  Bench\_PT &   fPAR\_PM &   Bench\_PM \\
 \midrule
-RMSE   &    0.770018 &    0.808084 &     0.801993 &    0.918294 &  0.831119 &  1.797729 &  1.936702 &  4.116962 \\
-RMedSE &    0.461744 &    0.448924 &     0.505759 &    0.609993 &  0.413534 &  1.265567 &  1.278348 &  2.545557 \\
-NS     &    0.800861 &    0.780686 &     0.783979 &    0.716784 &  0.768005 & -0.085430 & -0.259735 & -4.692554 \\
-RVE    &    0.009411 &    0.007280 &    -0.009784 &    0.022225 &  0.003059 & -0.373457 & -0.344705 & -0.896827 \\
-MAE    &    0.563843 &    0.597555 &     0.628617 &    0.720772 &  0.585899 &  1.541070 &  1.553741 &  3.389369 \\
-bias   &    0.034944 &    0.027033 &    -0.036332 &    0.082527 &  0.011358 & -1.386752 & -1.279987 & -3.330175 \\
+RMSE   &    1.159471 &    1.144177 &     2.855625 &    1.691557 &  0.831119 &  1.797729 &  4.221179 &   7.730995 \\
+RMedSE &    0.610701 &    0.656019 &     2.378363 &    1.358722 &  0.413534 &  1.265567 &  3.322006 &   5.591585 \\
+NS     &    0.548484 &    0.560317 &    -1.738771 &    0.038993 &  0.768005 & -0.085430 & -4.984405 & -19.073555 \\
+RVE    &   -0.131226 &   -0.062211 &    -0.036571 &    0.360602 &  0.003059 & -0.373457 & -0.986124 &  -1.817859 \\
+MAE    &    0.856273 &    0.882958 &     2.519780 &    1.411916 &  0.585899 &  1.541070 &  3.672916 &   6.750230 \\
+bias   &   -0.487281 &   -0.231006 &    -0.135799 &    1.339018 &  0.011358 & -1.386752 & -3.661761 &  -6.750230 \\
 \bottomrule
 \end{tabular}

notebooks/Finished_project/simpler_model/latex_files/whole_year.tex

 \begin{tabular}{lrrrrrrrr}
 \toprule
-{} &  Varying\_gS &  Mod\_var\_gS &  Constant\_gS &  Mod\_cst\_gS &  PT\_model &  Bench\_PT &   fPAR\_PM &  Bench\_PM \\
+{} &  Varying\_gS &  Mod\_var\_gS &  Constant\_gS &  Mod\_cst\_gS &  PT\_model &  Bench\_PT &   fPAR\_PM &   Bench\_PM \\
 \midrule
-RMSE   &    0.867409 &    0.931554 &     0.924532 &    1.038747 &  0.899593 &  1.845079 &  1.789343 &  3.849381 \\
-RMedSE &    0.471161 &    0.514767 &     0.540893 &    0.648825 &  0.439765 &  1.392651 &  1.189700 &  2.635833 \\
-NS     &    0.709874 &    0.665378 &     0.670403 &    0.583937 &  0.687945 & -0.312710 & -0.234599 & -4.713745 \\
-RVE    &    0.024020 &    0.013361 &    -0.002139 &    0.024325 &  0.021555 & -0.364515 & -0.270482 & -0.801912 \\
-MAE    &    0.614918 &    0.676997 &     0.692477 &    0.799959 &  0.619385 &  1.600261 &  1.406583 &  3.199727 \\
-bias   &    0.092765 &    0.051602 &    -0.008259 &    0.093944 &  0.083244 & -1.407761 & -1.044603 & -3.096990 \\
+RMSE   &    1.283445 &    1.273254 &     2.945861 &    1.762784 &  0.899593 &  1.845079 &  3.956166 &   7.332395 \\
+RMedSE &    0.678966 &    0.767610 &     2.438221 &    1.470380 &  0.439765 &  1.392651 &  3.034375 &   5.581212 \\
+NS     &    0.364825 &    0.374872 &    -2.346290 &   -0.198222 &  0.687945 & -0.312710 & -5.035150 & -19.731502 \\
+RVE    &   -0.153913 &   -0.086192 &    -0.130060 &    0.369599 &  0.021555 & -0.364515 & -0.869089 &  -1.660680 \\
+MAE    &    0.957122 &    0.988365 &     2.571213 &    1.505232 &  0.619385 &  1.600261 &  3.390706 &   6.414795 \\
+bias   &   -0.594413 &   -0.332875 &    -0.502292 &    1.427393 &  0.083244 & -1.407761 & -3.356427 &  -6.413556 \\
 \bottomrule
 \end{tabular}

notebooks/theory/pyFile_storage/theory_equation.py

+# -*- coding: utf-8 -*-
+#
+# This file is for use with essm.
+# Copyright (C) 2021 ETH Zurich, Swiss Data Science Center.
+#
+# essm is free software; you can redistribute it
+# and/or modify it under the terms of the GNU General Public License as
+# published by the Free Software Foundation; either version 2 of the
+# License, or (at your option) any later version.
+#
+# essm is distributed in the hope that it will be
+# useful, but WITHOUT ANY WARRANTY; without even the implied warranty of
+# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+# General Public License for more details.
+#
+# You should have received a copy of the GNU General Public License
+# along with essm; if not, write to the
+# Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston,
+# MA 02111-1307, USA.
+"""Equations defined in notebooks/theory/theory.ipynb and dependencies."""
+
+from __future__ import division
+
+from __main__ import (
+    T0, VPD, Delta, E, G, L, Mw, Pvs, R, Rn, S, S_mvg, T, alpha, alpha_PT, c1,
+    c_p, e_a, e_s, g_a, g_s, gamma, h, m, n, r_a, r_s, rho_a, theta, theta_1,
+    theta_2, theta_3, theta_4, theta_res, theta_sat
+)
+from essm import Eq
+from essm.equations import Equation
+from sympy import Abs, Eq, Piecewise, exp, log
+
+
+class eq_m_n(Equation):
+    """Relation between n and m parameters in MVG model"""
+
+    expr = Eq(m, 1 - 1 / n)
+
+
+class eq_MVG_neg_case(Equation):
+    """Mualem-Van Genuchtem model if h < 0"""
+
+    expr = Eq(
+        theta,
+        theta_res + ((-theta_res + theta_sat) / (Abs(alpha * h) ** n + 1)) ** m
+    )
+
+
+class eq_MVG(Equation):
+    """Complet Van Genuchtem model"""
+
+    expr = Eq(
+        theta,
+        Piecewise((theta_sat, h > 0), (
+            theta_res + ((-theta_res + theta_sat) /
+                         (Abs(alpha * h) ** n + 1)) ** m, True
+        ))
+    )
+
+
+class eq_sat_degree(Equation):
+    """Relative Saturation degree equation"""
+
+    expr = Eq(S_mvg, (theta - theta_res) / (-theta_res + theta_sat))
+
+
+class eq_MVG_h(Equation):
+    """Mualem Van Genuchten model, h as function of theta"""
+
+    expr = Eq(h, (-1 + S_mvg ** (-1 / m)) ** (1 / n) / alpha)
+
+
+class eq_h_FC(Equation):
+    """Compute the field capacity out of the soil properties"""
+
+    expr = Eq(h, ((n - 1) / n) ** ((1 - 2 * n) / n) / alpha)
+
+
+class eq_theta_4_3(Equation):
+    """Equation valide between theta_4 and theta_3"""
+
+    expr = Eq(
+        theta, theta / (theta_3 - theta_4) + theta_4 / (-theta_3 + theta_4)
+    )
+
+
+class eq_theta_2_1(Equation):
+    """Equation valide between theta_2 and theta_1"""
+
+    expr = Eq(
+        theta, theta / (-theta_1 + theta_2) + theta_1 / (theta_1 - theta_2)
+    )
+
+
+class eq_water_stress_simple(Equation):
+    """Simple water stress factor function (h_3 independant of T_0)"""
+
+    expr = Eq(
+        S,
+        Piecewise((0, theta < theta_4), (
+            theta / (theta_3 - theta_4) + theta_4 /
+            (-theta_3 + theta_4), theta < theta_3
+        ), (1, theta < theta_2), (
+            theta / (-theta_1 + theta_2) + theta_1 /
+            (theta_1 - theta_2), theta < theta_1
+        ), (0, True))
+    )
+
+
+class eq_Pvs_T(Equation):
+    """Saturation vapour pressure, Slide 8"""
+
+    expr = Eq(Pvs, c1 * exp(L * Mw * (1 / T0 - 1 / T) / R))
+
+
+class eq_Delta(Equation):
+    """Slope of d.Pvas(T)/d.T """
+
+    expr = Eq(
+        Delta,
+        L * Mw * c1 * exp(L * Mw * (1 / T0 - 1 / T) / R) / (R * T ** 2)
+    )
+
+
+class eq_PT(Equation):
+    """Priestley-Taylor equation for evaporation flux"""
+
+    expr = Eq(E, Delta * alpha_PT * (-G + Rn) / (L * (Delta + gamma)))
+
+
+class eq_PM(Equation):
+    """Penman-Monteith equation"""
+
+    expr = Eq(
+        E, (Delta * (-G + Rn) + c_p * rho_a * (-e_a + e_s) / r_a) /
+        (L * (Delta + gamma * (1 + r_s / r_a)))
+    )
+
+
+class eq_PM_VPD(Equation):
+    """Penman-Monteith equation using VPD"""
+
+    expr = Eq(
+        E, (Delta * (-G + Rn) + VPD * c_p * rho_a / r_a) /
+        (L * (Delta + gamma * (1 + r_s / r_a)))
+    )
+
+
+class eq_PM_g(Equation):
+    """Penman-Monteith equation using stomatal conductance"""
+
+    expr = Eq(
+        E, (Delta * (-G + Rn) + VPD * c_p * g_a * rho_a) /
+        (L * (Delta + gamma * (g_a / g_s + 1)))
+    )
+
+
+class eq_PM_inv(Equation):
+    """Inverse PM equation for inverse modelling of surface resistance"""
+
+    expr = Eq(
+        r_s, (
+            -Delta * E * L * r_a - Delta * G * r_a + Delta * Rn * r_a -
+            E * L * gamma * r_a + VPD * c_p * rho_a
+        ) / (E * L * gamma)
+    )
+
+
+__all__ = (
+    'eq_m_n',
+    'eq_MVG_neg_case',
+    'eq_MVG',
+    'eq_sat_degree',
+    'eq_MVG_h',
+    'eq_h_FC',
+    'eq_theta_4_3',
+    'eq_theta_2_1',
+    'eq_water_stress_simple',
+    'eq_Pvs_T',
+    'eq_Delta',
+    'eq_PT',
+    'eq_PM',
+    'eq_PM_VPD',
+    'eq_PM_g',
+    'eq_PM_inv',
+)