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Gr rainfall intensity (#325)
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ENH: Gr rainfall intensity
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asjeb authored Sep 18, 2024
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29 changes: 29 additions & 0 deletions doc/source/_static/bib/references.bib
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Expand Up @@ -1295,5 +1295,34 @@ @book{monnier2021coursevariational
url = {https://www.math.univ-toulouse.fr/~jmonnie/Enseignement/CourseVDA.pdf}
}

@article{Astagneau_2022,
author = {Paul C. Astagneau and François Bourgin and Vazken Andréassian and Charles Perrin},
title = {Catchment response to intense rainfall: Evaluating modelling hypotheses},
journal = {Hydrological Processes},
volume = {26},
number = {e14676},
year = {2022},
doi = {10.1080/02626667.2021.1923720},

URL = {
https://doi.org/10.1002/hyp.14676
},
eprint = {
https://doi.org/10.1002/hyp.14676
}
}

@Article{hess-22-5317-2018,
AUTHOR = {Douinot, A. and Roux, H. and Garambois, P.-A. and Dartus, D.},
TITLE = {Using a multi-hypothesis framework to improve the understanding of flow dynamics during flash floods},
JOURNAL = {Hydrology and Earth System Sciences},
VOLUME = {22},
YEAR = {2018},
NUMBER = {10},
PAGES = {5317--5340},
URL = {https://hess.copernicus.org/articles/22/5317/2018/},
DOI = {10.5194/hess-22-5317-2018}
}

@Comment{jabref-meta: databaseType:bibtex;}
114 changes: 113 additions & 1 deletion doc/source/math_num_documentation/forward_structure.rst
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Expand Up @@ -351,8 +351,120 @@ Hydrological processes can be described at pixel scale in `smash` with one of th
Same as ``gr4`` transfer, see :ref:`GR4 Transfer <math_num_documentation.forward_structure.hydrological_module.gr4>`


.. _math_num_documentation.forward_structure.hydrological_module.gr6:
.. _math_num_documentation.forward_structure.hydrological_module.gr5_ri:

.. dropdown:: gr5_ri (Génie Rural 5 with rainfall intensity terms)
:animate: fade-in-slide-down

This hydrological module is derived from the model introduced in :cite:p:`Astagneau_2022`.


**Production**


In the classical gr production reservoir formulation, the instantaneous production rate is the ratio between the state and the capacity of the reservoir,
:math:`\eta = \left( \frac{h_p}{c_p} \right)^2`. The infiltration flux :math:p_s is obtained by temporal integration as follows:

.. math::
:nowrap:
\begin{eqnarray}
&p_s = \int_{t-\Delta t}^{t} (1 - \eta) dt \\
\end{eqnarray}
Assuming the neutralized rainfall :math:p_n constant over the current time step and thanks to analytically integrable function, the infiltration flux into the production reservoir is obtained:

.. math::
:nowrap:
\begin{eqnarray}
&p_s = & c_p \tanh\left(\frac{p_n}{c_p}\right) \frac{1 - \left( \frac{h_p}{c_p} \right)^2}{1 + \frac{h_p}{c_p} \tanh\left( \frac{p_n}{c_p} \right)} \\
\end{eqnarray}
To improve runoff production by a gr reservoir,
even with low production level in dry condition,
in the case of high rainfall intensity, :cite:p:`Astagneau_2022` suggests a modification
of the infiltration rate :math:p_s depending on rainfall intensity :math:p_n:
:math:`\eta = \left( 1 - \gamma \right) \left( \frac{h_p}{c_p} \right)^2 + \gamma` with :math:`\gamma = 1 - \exp(-p_n \times \alpha_1)`
and :math:`\alpha_1` in :math:`mm` per time unit.

.. math::
:nowrap:
\begin{eqnarray}
&p_s& &=& &\int_{t-\Delta t}^{t} (1 - \eta) dt\\
&& &=& &\int_{t-\Delta t}^{t} \left(1 - (1-\gamma) \left(\frac{h_p}{c_p} \right)^2 \right) dt - \int_{t-\Delta t}^{t} \gamma dt\\
&& &=& &\left[ \frac{ c_p }{ \sqrt{1-\gamma} } \tanh \left( \frac{\sqrt{1-\gamma} \ h_p}{c_p} \right) \right]_{t-\Delta t}^t - \gamma \Delta t
\end{eqnarray}
We denote :math:`\lambda := \sqrt{1 - \gamma}`, then

.. math::
:nowrap:
\begin{eqnarray}
\tanh \left( \lambda \frac{h_p + p_n}{c_p} \right) - \tanh\left( \lambda \frac{h_p}{c_p} \right) &=&
\tanh \left( \lambda \frac{p_n}{c_p} \right) \left(1 - \tanh \left( \lambda \frac{h_p + p_n}{c_p} \right) \tanh \left( \lambda \frac{h_p}{c_p} \right) \right) \\
&=& \tanh \left( \lambda \frac{p_n}{c_p} \right) \left(1 - \frac{ \tanh \left( \lambda \frac{h_p}{c_p} \right) + \tanh \left( \lambda \frac{p_n}{c_p} \right) } { 1 + \tanh \left( \lambda \frac{h_p}{c_p} \right) \tanh \left( \lambda \frac{p_n}{c_p} \right) } \tanh \left( \lambda \frac{h_p}{c_p} \right) \right) \\
&\sim& \tanh \left( \lambda \frac{p_n}{c_p} \right) \left(1 - \frac{ \lambda \frac{h_p}{c_p} + \tanh \left( \lambda \frac{p_n}{c_p} \right) } { 1 + \lambda \frac{h_p}{c_p} \tanh \left( \lambda \frac{p_n}{c_p} \right) } \lambda \frac{h_p}{c_p} \right) \\
&=& \tanh \left( \lambda \frac{p_n}{c_p} \right) \frac{1 - \left( \lambda \frac{h_p}{c_p} \right)^2}{1 + \lambda \frac{h_p}{c_p} \tanh \left( \lambda \frac{p_n}{c_p} \right)}
\end{eqnarray}
Thus

.. math::
:nowrap:
\begin{eqnarray}
p_s &=& \frac{c_p}{\lambda} \tanh \left( \lambda \frac{p_n}{c_p} \right) \frac{1 - \left( \lambda \frac{h_p}{c_p} \right)^2}{1 + \lambda \frac{h_p}{c_p} \tanh \left( \lambda \frac{p_n}{c_p} \right)} - \gamma \Delta t
\end{eqnarray}
.. note::

Note that if :math:`\alpha_1 = 0`, we return to the general writting of the instantaneous production rate.


**Transfer**

In context of high rainfall intensities triggering flash flood responses, it is crucial to account for fast dynamics related to surface/hypodermic runoff
and slower responses due to delayed/deeper flows (e.g. :cite:p:`hess-22-5317-2018`).
Following :cite:p:`Astagneau_2022` for a lumped GR model, we introduce at pixel scale in `smash` a function to modify the partitioning between fast
and slower transfert branches depending on rainfall intensity of the current time step only (small pixel size):

.. math::
:nowrap:
\begin{eqnarray}
&p_{rr}& =& (1 - Q_9)(p_r + p_{erc}) + l_{exc}\\
&p_{rd}& =& Q_9(p_r + p_{erc}) \\
&Q_9& =& 0.9 \tanh(\alpha_2 p_n)^2 + 0.1
\end{eqnarray}
with :math:`\alpha_2` in :math:`mm` per time unit.


.. note::

If :math:`\alpha_2 = 0`, we return to the ``gr-4/gr-5`` writting of the transfer.
If :math:`\alpha_2 = \alpha_1 = 0`, it is equivalent to ``gr-5`` structure.



.. _math_num_documentation.forward_structure.hydrological_module.gr6:

.. dropdown:: gr6 (Génie Rural 6)
:animate: fade-in-slide-down
Expand Down
34 changes: 25 additions & 9 deletions smash/_constant.py
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Expand Up @@ -64,10 +64,12 @@ def get_neurons_from_hydrological_module(hydrological_module: str, hidden_neuron

HYDROLOGICAL_MODULE = [
"gr4",
"gr4_ri",
"gr4_mlp",
"gr4_ode",
"gr4_ode_mlp",
"gr5",
"gr5_ri",
"gr6",
"grc",
"grd",
Expand Down Expand Up @@ -97,10 +99,12 @@ def get_neurons_from_hydrological_module(hydrological_module: str, hidden_neuron
HYDROLOGICAL_MODULE,
[
["ci", "cp", "ct", "kexc"], # % gr4
["ci", "cp", "ct", "alpha1", "alpha2", "kexc"], # % gr4_ri
["ci", "cp", "ct", "kexc"], # % gr4_mlp
["ci", "cp", "ct", "kexc"], # % gr4_ode
["ci", "cp", "ct", "kexc"], # % gr4_ode_mlp
["ci", "cp", "ct", "kexc", "aexc"], # % gr5
["ci", "cp", "ct", "alpha1", "alpha2", "kexc", "aexc"], # % gr5_ri
["ci", "cp", "ct", "be", "kexc", "aexc"], # % gr6
["ci", "cp", "ct", "cl", "kexc"], # % grc
["cp", "ct"], # % grd
Expand Down Expand Up @@ -139,10 +143,12 @@ def get_neurons_from_hydrological_module(hydrological_module: str, hidden_neuron
HYDROLOGICAL_MODULE,
[
["hi", "hp", "ht"], # % gr4
["hi", "hp", "ht"], # % gr4_ri
["hi", "hp", "ht"], # % gr4_mlp
["hi", "hp", "ht"], # % gr4_ode
["hi", "hp", "ht"], # % gr4_ode_mlp
["hi", "hp", "ht"], # % gr5
["hi", "hp", "ht"], # % gr5_ri
["hi", "hp", "ht", "he"], # % gr6
["hi", "hp", "ht", "hl"], # % grc
["hp", "ht"], # % grd
Expand Down Expand Up @@ -183,10 +189,12 @@ def get_neurons_from_hydrological_module(hydrological_module: str, hidden_neuron
HYDROLOGICAL_MODULE,
[
["pn", "en", "pr", "perc", "lexc", "prr", "prd", "qr", "qd", "qt"], # % gr4
["pn", "en", "pr", "perc", "lexc", "prr", "prd", "qr", "qd", "qt"], # % gr4-ri
["pn", "en", "pr", "perc", "lexc", "prr", "prd", "qr", "qd", "qt"], # % gr4_mlp
["pn", "en", "lexc", "qt"], # % gr4_ode
["pn", "en", "lexc", "qt"], # % gr4_ode_mlp
["pn", "en", "pr", "perc", "lexc", "prr", "prd", "qr", "qd", "qt"], # % gr5
["pn", "en", "pr", "perc", "lexc", "prr", "prd", "qr", "qd", "qt"], # % gr5-ri
["pn", "en", "pr", "perc", "lexc", "prr", "prd", "pre", "qr", "qd", "qe", "qt"], # % gr6
["pn", "en", "pr", "perc", "lexc", "prr", "prd", "prl", "qr", "qd", "ql", "qt"], # % grc
["ei", "pn", "en", "pr", "perc", "prr", "qr", "qt"], # % grd
Expand Down Expand Up @@ -250,13 +258,15 @@ def get_neurons_from_hydrological_module(hydrological_module: str, hidden_neuron

RR_PARAMETERS = [
"kmlt", # % ssn
"ci", # % (gr4, gr4_mlp, gr4_ode, gr4_ode_mlp, gr5, gr6, grc)
"cp", # % (gr4, gr4_mlp, gr4_ode, gr4_ode_mlp, gr5, gr6, grc, grd)
"ct", # % (gr4, gr4_mlp, gr4_ode, gr4_ode_mlp, gr5, gr6, grc, grd)
"ci", # % (gr4, gr4_ri, gr4_mlp, gr4_ode, gr4_ode_mlp, gr5, gr5_ri, grc, gr6)
"cp", # % (gr4, gr4_ri, gr4_mlp, gr4_ode, gr4_ode_mlp, gr5, gr5_ri, grc, gr6, grd)
"ct", # % (gr4, gr4_ri, gr4_mlp, gr4_ode, gr4_ode_mlp, gr5, gr5_ri, grc, gr6, grd)
"alpha1", # % (gr4_ri, gr5_ri)
"alpha2", # % (gr4_ri, gr5_ri)
"cl", # % grc
"be", # % gr6
"kexc", # % (gr4, gr4_mlp, gr4_ode, gr4_ode_mlp, gr5, gr6, grc)
"aexc", # % (gr5, gr6)
"be", # % (gr6)
"kexc", # % (gr4, gr4_ri, gr4_mlp, gr4_ode, gr4_ode_mlp, gr5, gr5_ri, grc, gr6)
"aexc", # % (gr5, gr5_ri, gr6)
"ca", # % loieau
"cc", # % loieau
"kb", # % loieau
Expand All @@ -276,9 +286,9 @@ def get_neurons_from_hydrological_module(hydrological_module: str, hidden_neuron

RR_STATES = [
"hs", # % ssn
"hi", # % (gr4, gr4_mlp, gr4_ode, gr4_ode_mlp, gr5, gr6, grc)
"hp", # % (gr4, gr4_mlp, gr4_ode, gr4_ode_mlp, gr5, gr6, grc, grd)
"ht", # % (gr4, gr4_mlp, gr4_ode, gr4_ode_mlp, gr5, gr6, grc, grd)
"hi", # % (gr4, gr4_ri, gr4_mlp, gr4_ode, gr4_ode_mlp, gr5, gr5_ri, grc, gr6)
"hp", # % (gr4, gr4_ri, gr4_mlp, gr4_ode, gr4_ode_mlp, gr5, gr5_ri, grc, gr6, grd)
"ht", # % (gr4, gr4_ri, gr4_mlp, gr4_ode, gr4_ode_mlp, gr5, gr5_ri, grc, gr6, grd)
"hl", # % grc
"he", # % gr6
"ha", # % loieau
Expand All @@ -302,6 +312,8 @@ def get_neurons_from_hydrological_module(hydrological_module: str, hidden_neuron
(0, np.inf), # % ci
(0, np.inf), # % cp
(0, np.inf), # % ct
(0, np.inf), # % alpha1
(0, np.inf), # % alpha2
(0, np.inf), # % cl
(0, np.inf), # % be
(-np.inf, np.inf), # % kexc
Expand Down Expand Up @@ -361,6 +373,8 @@ def get_neurons_from_hydrological_module(hydrological_module: str, hidden_neuron
1e-6, # % ci
200, # % cp
500, # % ct
3e-4, # % alpha1
1e-3, # % alpha2
500, # % cl
10, # % be
0, # % kexc
Expand Down Expand Up @@ -418,6 +432,8 @@ def get_neurons_from_hydrological_module(hydrological_module: str, hidden_neuron
(1e-6, 1e2), # % ci
(1e-6, 1e3), # % cp
(1e-6, 1e3), # % ct
(1e-6, 5e-4), # % alpha1
(1e-5, 1.0), # % alpha2
(1e-6, 1e3), # % cl
(1e-3, 20), # % be
(-50, 50), # % kexc
Expand Down
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