HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-22-2135-2018Large-scale hydrological model river storage and discharge correction using a satellite altimetry-based discharge productLarge-scale hydrological model river storage and discharge
correctionEmeryCharlotte Mariecharlotte.emery@jpl.nasa.govhttps://orcid.org/0000-0002-3257-2017ParisAdrienhttps://orcid.org/0000-0003-2304-5132BiancamariaSylvainBooneAaronCalmantStéphaneGaramboisPierre-AndréSantos da SilvaJoecilaLEGOS, Université de Toulouse, CNES, CNRS, IRD, UPS, Toulouse, FranceGET, Université de Toulouse, UPS, CNRS, IRD, Toulouse, FranceLMI OCE IRD/UNB, Campus Darcy Ribeiro, Brasilia, BrazilMeteo France CNRS, CNRM UMR 3589, Toulouse, FranceICUBE – UMR 7357, Fluid Mechanics Team, INSA, Strasbourg, FranceCESTU, Universidade do Estado do Amazonas, Manaus, Brazilnow at: JPL, Pasadena, CA, USAnow at: CLS, Ramonville-Saint-Agne, FranceCharlotte Marie Emery (charlotte.emery@jpl.nasa.gov)6April20182242135216218August20179October201730January201827February2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://hess.copernicus.org/articles/22/2135/2018/hess-22-2135-2018.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/22/2135/2018/hess-22-2135-2018.pdf
Land surface models (LSMs) are widely used to study the continental
part of the water cycle. However, even though their accuracy is increasing,
inherent model uncertainties can not be avoided. In the meantime, remotely
sensed observations of the continental water cycle variables such as soil
moisture, lakes and river elevations are more frequent and accurate.
Therefore, those two different types of information can be combined, using
data assimilation techniques to reduce a model's uncertainties in its state
variables or/and in its input parameters. The objective of this study is to
present a data assimilation platform that assimilates into the large-scale
ISBA-CTRIP LSM a punctual river discharge product, derived from ENVISAT nadir
altimeter water elevation measurements and rating curves, over the whole
Amazon basin. To deal with the scale difference between the model and the
observation, the study also presents an initial development for a
localization treatment that allows one to limit the impact of observations to
areas close to the observation and in the same hydrological network. This
assimilation platform is based on the ensemble Kalman filter and can correct
either the CTRIP river water storage or the discharge. Root mean square
error (RMSE) compared to gauge discharges is globally reduced until 21 %
and at Óbidos, near the outlet, RMSE is reduced by up to 52 %
compared to ENVISAT-based discharge. Finally, it is shown that localization
improves results along the main tributaries.
Introduction
The continental part of the water cycle is commonly studied, at large scale,
with hydrological modelling. These models are generally issued from the
coupling of a land surface model (LSM) with a river routing model (RRM). The
LSM determines the water and energy budget at the surface by spreading
precipitations between the soil and the canopy. Meanwhile, the RRM transfers
water mass through the basin to the outlet and gives an estimate of river
discharge.
RRMs are mainly based on kinematic e.g. or sometimes
also diffusive wave models e.g.. Several RRMs
have been developed since the 1990s and they differ mainly in their modelling of the flow velocity
and the inclusion or not of groundwater and floodplain dynamics
.
The modelling of the river flow velocity is addressed in several ways in the
literature. and considered a uniform and
constant flow velocity over the entire basin in SWAM and TRIP RRMs,
respectively. Other studies rather use a spatially distributed, but still
constant in time, flow velocity based on topography and river channel
characteristics
. However,
most recent models rely on a time-varying and spatially distributed flow
velocity estimation based on the Manning formula , e.g.
, , and
. The first RRMs only modelled water flowing in the
river channel . Subsequent RRM
developments included the modelling of the groundwater inflow to the river
as well as the floodplains–river dynamics
. The routing network
is derived from a digital elevation model (DEM): some of them remain defined
on a regular mesh grid , while others use
an irregular discretization by sub-catchments, such as MGB-IPH
and CaMa Flood , or by river
reaches, such as RAPID . More information on some global
LSMs and their associated RRMs (with the corresponding references) could be
found, for example, in .
However, even if hydrological models become more and more accurate, inherent
model uncertainties are unavoidable. They originate from several sources:
simplification and lack of knowledge in the real physics, numerization and
discretization-induced errors and uncertainties in the input parameters and
forcing. All these uncertainties impact a model's outputs. In the worst case,
all those uncertainties could accumulate and result in the collapse of the
model. The model gives therefore an approximate view of the system's real
state.
Observations of the system can be used to calibrate and/or validate the model
and reduce its errors. These observations can be obtained from in situ or
remote techniques. In situ techniques mainly focus on measuring river water
elevations at a gauge station. Another important variable of interest in
river hydrology is the river discharge, which is sparsely measured compared
to water elevation. Based on river discharges and elevations measured at the
same time and at the same location, it is possible to build a rating curve
that represents the elevation–discharge relationship. This rating curve is
then applied to water elevation to set continuous discharge time series.
Institutions delivering in situ data provide mainly discharge. Even though in
situ measures are generally quite accurate with a high time sampling
(i.e. sub-daily), their main limitation is their local and spatially sparse
sampling over the river network. Furthermore, nowadays, remotely sensed data
from satellite missions are more and more available and provide useful
observations of rivers. The most straightforward and used instrument to
measure river water elevations is the nadir altimeter.
Altimeters were initially developed to measure ocean topography with
satellite missions GEOS-3 (1975–1978) and SEASAT launched in 1978
. Nadir altimetry consists in estimating water surface
elevation at the vertical (or at the nadir) of the satellite. It therefore
produces punctual water elevation observations along the satellite ground
track. These missions were followed by a long series of other missions:
GEOSAT (1985–1990), TOPEX-Poseidon (1992–2006), JASON-1 (2001–2013),
JASON-2 (2008–now), JASON-3 (2016–now), GFO (1998–2008),
ERS-1 (1991–2000), ERS-2 (1995–2003), ENVISAT (2002–2012),
SARAL (2013–now) and Sentinel-3 (2016–now). It is with TOPEX-Poseidon that
the use of nadir altimetry to monitor lakes
, floodplains
and rivers
developed widely.
However, the main limitations of nadir altimetry are their punctual
measurements (at the location where the satellite track crosses a river
stream) and their temporal sampling (from 10 to 35 days, depending on the
mission). Besides, in contrast to ocean surfaces, the signal over continental
surfaces is impacted by vegetation and topography surrounding the river.
Therefore, the purpose of this study is to combine model outputs and
altimetry-based products using data assimilation (DA) techniques, in order to
get more precise discharge estimates within the Amazon basin.
DA aims to improve model skills to forecast/simulate the physical system
evolution. To do so, DA techniques focus on either correcting the model's
input parameters (parameter estimation) or the model's outputs (state
estimation). State estimation (SE) consists in using observations to directly
correct the model output state. It is based on the assumption that the model
(and the observations) are known to be imperfect. So, SE aims at correcting
model outputs, whose errors result from all sources of uncertainties
previously described.
(a) The Amazon basin main tributaries and rivers with the
underlying topography from SRTM. (b) Schematic representation of the
ISBA-CTRIP system for a given grid cell. ISBA surface
runoff (QISBA,sur) flows into the river/surface reservoir S;
ISBA gravitational drainage (QISBA,sub) feeds the groundwater
reservoir G. The surface water is transferred from one cell to another
following the TRIP river routing network.
SE has been widely used in oceanography and meteorology
.
However, DA of remotely sensed observations to correct hydrological model
states is more recent. Moreover, it is more developed for LSMs than RRMs, as
shown for example by the global-scale Land Data Assimilation System of the
NASA Goddard Earth Observing System . This platform
assimilates simultaneously the SMOS soil moisture product, MODIS snow cover
extent fraction and integrated GRACE terrestrial water storage variations
into an ensemble Kalman filter (EnKF) to correct the states of several LSMs.
Other studies assimilate similar kinds of observations, along with in situ
data, into smaller-scale hydrological models
. As for RRMs, to the authors' knowledge,
there are few studies where remotely sensed and/or in situ data are
assimilated into global-scale RRMs. However, in the literature, there are
several studies that used assimilation techniques at smaller and local scales
with finer spatial resolution than global RRMs, using mostly in situ data
. For example, applied
the EnKF to correct soil, aquifer and surface water storage in a small river
in New Zealand (the Wairu). More particularly, they used gauge discharge data
from four gauges to correct water storages in the 380 sub-catchments dividing
the study zone. also used an EnKF over the Amazon
basin for three different experiments, which assimilate, first, ENVISAT water
surface anomalies from 212 virtual stations, and then discharge data from
109 gauges and finally remotely sensed discharges from 287 stations obtained
from . This study aimed at correcting discharge
estimated by the MGB-IPH hydrological model over more than
5000 sub-catchments comprising the Amazon River basin. Moreover, in two
different studies, and
assimilated for the Brahmaputra River in Asia
and the Zambezi River in Africa, respectively, using an extended Kalman
filter, ENVISAT water surface elevation measurements from six and
nine virtual stations in and
, respectively, to correct simulated water
volumes in 18 and 37 sub-catchments, respectively.
The objective of the present study is to investigate the contribution of
remotely sensed data, and in particular measurements derived from nadir
altimeters that provide local information, to improve a large-scale RRM via
DA. The scale difference between the observations and the model leads us to
also study the need to use localization methods within our DA framework. We
used an ensemble Kalman filter, to which we added a simple localization
module, to assimilate discharges derived from ENVISAT water surface elevation
measurements. These observations are used to correct the state of the
large-scale Total Runoff Integrated Pathways (TRIP, ) RRM
version included in the Surfaces Externalisées land surface modelling
platform (SurfEx,
) and developed at the Centre National de Recherches
en Météorologie (CNRM, France). This particular version is denoted by
the CTRIP acronym hereinafter. CTRIP is coupled with the
Interactions-Soil-Biosphere-Atmosphere (ISBA, ) LSM at
a resolution of 0.5∘× 0.5∘.
In Sect. , we present the
study domain along with the ISBA-CTRIP model version and remotely sensed
product used in this study. Section provides
first a general presentation of the ensemble Kalman filter (EnKF) DA method.
Then we introduce the special features associated with the study and the
description of the assimilation strategy. Then, in
Sect. , we present results for a series of DA
experiments testing the ensemble generation strategy and the correction of
different state variables. Finally, Sect. discusses
these results and some perspectives. The last section gives the conclusions
and some perspectives of the study.
Study domain, model and data usedStudy domain: the Amazon basin
The study is focused on the Amazon River basin (see Fig. a).
It is the world's largest river in terms of averaged discharge
(2 × 105 m3 s-1) and drainage area
(6.15 × 106 km2). The discharge at its mouth represents
30 % of total freshwater inflow to the Atlantic Ocean
and its catchment area covers about 40 % of South
America. The river source is located in the Peruvian Andes and flows through
the Brazilian rainforest while receiving water from several important
tributaries: first, the Ucayali, the Japurá River, the Purus River and,
at Manaus, the Negro River (14 % of the total discharge). At this point,
the river has reached 56 % of its total discharge. From Manaus to its
mouth, it receives water from the Madeira River (17 % of the total
discharge), the Tapajós River and the Xingu River (11 % of the total
discharge all together) .
The Amazon basin's geology can be divided into three major morpho-structural
units: the western Andean Cordillera, the central Amazon trough and the
shields at the eastern part of the basin (Guiana shield to the north and the
Brazilian shield to the south). The northern and southern regions of the
basin are under a tropical climate with a dry and a wet season, but the
maximum rainfall season for the two parts occurs at different periods during
the year . This implies that annual peak discharge in
southern tributaries occurs a few months earlier than in northern
tributaries. Meanwhile, the central basin is under an equatorial climate
zone, implying high surface temperatures, air humidity and, especially,
precipitation. Thus, a vast floodplain along the mainstream is filled every
year, leading to the damping of discharge extremes.
ISBA-CTRIP modelModel presentation
The ISBA model is a relatively standard land
surface model (LSM) defined over a regular mesh grid at global scale. The
model's equations are solved for each grid cell separately from the others.
All grid cells are only correlated through the spatial patterns of
atmospheric (especially precipitation) and radiative inputs, vegetation cover
and soil composition. By taking into account the heterogeneity in
precipitation, topography and vegetation within each grid cell
and based on the force-restore method
, ISBA gives a diagnosis of the water and energy budgets
in each grid cell. Especially the ISBA-3L configuration
, used in and
, has been chosen for the present study. In this
version, the soil is divided into three layers: the superficial layer, the
root zone and the sub-root zone. Precipitation can either fall directly on
the soil surface or be intercepted by the canopy. The soil water content
varies with canopy dripping, surface infiltration, soil evaporation, plant
evapotranspiration, surface runoff and deep drainage (for more details, see
). Then, ISBA gives a diagnostic of
each water budget component, in particular the surface
runoff (QISBA,sur) and gravitational
drainage (QISBA,sub) which are the main inputs for CTRIP.
The CTRIP RRM is also defined over a regular mesh grid. In this study, it is
run at the same resolution as ISBA (0.5∘× 0.5∘).
CTRIP is dedicated to the lateral transfer of water from one cell to the
other, up to the continent–ocean interface following a river network
. The CTRIP version used in this
study is coupled with the ISBA LSM and was subsequently developed by
. It consists of a system of three
reservoirs (see Fig. b): the surface reservoir S (kg)
modelling the river, the groundwater reservoir G (kg) and the floodplain
reservoir F (kg).
Only the surface reservoir S sends water from cell to cell based on the
TRIP routing network. A cell can receive water from several upstream cells,
but sends water into a unique downstream cell based on a space and
time-varying flow velocity v(t) estimated with the Manning formula
. For any given cell, TRIP inputs are the TRIP outflow
from upstream cells Qin,TRIPS(t) and the ISBA surface
runoff for that cell QISBA,sur. Moreover, S receives water from
the groundwater reservoir G, QoutG(t), and can
exchange water mass with the floodplain F,
QoutF(t)-QinF(t).
G receives inflows from ISBA gravitational drainage QISBA,sub
and outflows to the river reservoir S. This outflow represents more a
delayed contribution of the gravitational drainage to the river than a real
groundwater dynamic.
The floodplain scheme activates when the water height in the river,
hS, exceeds a given critical bankful height Hc. Then,
part of the precipitation is intercepted by the
floodplain (PF(t)) and the water in the floodplain can either
evaporate (EF(t)) or infiltrate into the
soil (IF(t)). A detailed description of the floodplain scheme is
given in .
Model implementation over the Amazon
ISBA-CTRIP is run in offline mode. This implies that external atmospheric
data are needed to force the model. Here, the atmospheric data from the
Global Soil Wetness Projet 3 (GSWP3,
http://hydro.iis.u-tokyo.ac.jp/GSWP3) are used. The project consists
of three global-scale experiments with
the objective of investigating long-term changes in the energy–water–carbon
cycle components and their interactions. The 3-hourly resolution atmospheric
boundary conditions used in the present study were generated by dynamically
downscaling the global 2∘-resolution 20th Century Reanalysis
. This reanalysis assimilates several atmospheric
observations into the Climate Forecast System (CFS) operational model from
NCEP (National Centers for Environmental Prediction).
Map of hydro-geomorphological zones defined over the Amazon
basin.
For ISBA-CTRIP, the Amazon basin is composed of a total number of 2028 cells.
A sensitivity analysis (SA) of the ISBA-CTRIP has been conducted by
. In this analysis, the basin was divided into
nine hydro-geomorphological zones which are shown in
Fig. . These zones were designed to take into
account different components: (1) a hydrological component (the main course
is separated from the tributaries which have their own zones); and (2) a
geological component (the three major morpho-structural units are
distinguishable). The nine zones are the following: (1) the upstream Andean
part of the basin until the city of Iquitos, Peru; (2) the
mainstream from Iquitos to Óbidos; (3) the mainstream from Óbidos
to the river mouth; (4) left-bank tributaries from the Napo River to the
Japurá River;
(5) left-bank tributaries from the Japurá River to Óbidos, including
the Negro River and its drainage area; (6) right-bank tributaries from
Iquitos to the Purus River confluence at Anamã; (7) right-bank
tributaries from Anamã to Óbidos, including the Madeira River;
(8) right-bank tributaries exiting in zone 3, including the Tapajós River
and the Xingu River; and (9) left-bank tributaries exiting in zone 3. This
subdivision will be used within the DA platform.
ObservationsAltimetry-based discharge product
The altimetry-based discharge product used in this study is derived from
water surface elevations measured by the ENVISAT Radar Altimeter-2 altimeter
instrument at Virtual Station (VS). VS is computed where the altimeter track
crosses the river. The ENVISAT mission operated from September 2002 to
October 2010 on its nominal orbit, which has a 35-day repeat period and an
80 km inter-track distance at the Equator. The water surface elevations
measured over the Amazon basin were initially generated by
. The final product was referenced to the
EGM2008 geoid and the vertical precision ranged from
12 to 30–40 cm for most of the stations (and can reach several
metres for the worst stations).
Turning water surface elevation measures into an equivalent discharge
requires the use of elevation–discharge rating curves. The rating curves
used in this study have been built and validated by ,
using water surface elevations from ENVISAT
and
discharges simulated by hydrological–hydrodynamic model MGB-IPH (Model de
Grandes Bacias-Instituto de Pesquisas Hidráulicas,
). The model's original version, developed over
the Amazon River basin, consists of a large-scale distributed hydrological
model coupled with a hydrodynamic module that uses a simple storage scheme
for floodplains . The entire basin is divided into
5765 elementary catchments with an area varying between 100 and
5000 km2. A surface scheme is applied for each mini-basin to estimate
the main flows and a routing network is used to direct the flows from one
elementary catchment to another, down to the outlet. Two approaches are used
to estimate the river discharge: 1 – the Muskingum–Cunge method (MC) for
basin heads and small tributaries; 2 – the Saint-Venant equations (HD for
hydrodynamic) for the main stem and main tributaries. The DEM used is SRTM
, and parameters such as the river width and depth
are determined using geomorphological relationships calibrated over the
sub-basins . Moreover, the model version used to
determine the rating curves is the version developed by
, where gauge discharges are assimilated into the model
via an ensemble Kalman filter (EnKF, ), over the period
between 1998 and 2010. The assimilated discharges allow one to correct the
simulated discharges over both gauged and ungauged elementary catchments.
With a better estimation of discharges, also provide an
estimation of discharge uncertainty (modelled as a white noise) for each
elementary catchment.
The MGB-IPH discharges were used by as a baseline to
estimate the altimetric rating curves such that
∀j,∀i,∀talti,Qmgb,j(i)talti,i=ai×Halti,i-z0,ibi,
where
Halti,i is the altimetric water surface elevation at the
ith virtual station which is available at time talti,i,
Qmgb,j(i)(talti,i) is the discharge estimated
by the MGB-IPH model, at time talti,i, in the jth mini-basin
which contains the ith virtual station, and
ai, z0,i and bi are the rating curve parameters to be
determined. Those parameters are constant in time but vary from one virtual
station to the other.
To calculate those parameters at each virtual station, a global optimization
algorithm, the Shuffled Complex Evolution Metropolis developed by
, was used. It allowed determination of rating curves
for 767 ENVISAT virtual stations. More details about the rating curve
computation can be found in . Once rating curve
parameters are determined, altimetric water surface elevations are easily
converted into equivalent “altimetric discharges”. Moreover, the altimetric
discharges are provided with an estimation of their uncertainty including the
normalized deviation from the MGB discharge.
The quality assurance of the discharge product has been made by constraining
the rating curve coefficients within a physical range of values
. also conducted a sensitivity
analysis that showed a small sensitivity of the coefficient estimation to
their first guess value. The quality check was done by comparing the
satellite-derived discharge to the modelled discharge over a validation time
period distinct from the calibration period used to derive rating curves.
Discharge was also compared to some in situ gauges. Satellite-derived
discharge is of course heavily correlated with the model accuracy. Overall, a
comparison to 51 gauge measurements led to a mean Nash–Sutcliffe coefficient
of around 0.8 and a normalized root mean square error of around 10 % over
the validation period (Table 8 in ).
General framework of the DA method at a kth assimilation cycle.
The figure reads from top to bottom and from left to right. The three main
variables involved are the river initial storage, the river final storage and
the river discharge. M[k-1,k] is the model operator that maps
the initial river storage into final river storage, Zk is the
diagnostic operator and Hk is the observation operator that
maps simulated discharge into observed discharge for
assimilation.
Altimetric discharges have then to be compared to ISBA-CTRIP discharges.
However, while the virtual stations are irregularly distributed over the
entire basin, the model is defined over a coarse regular mesh grid of
0.5∘× 0.5∘. A preliminary treatment of the virtual
stations is applied to associate each ENVISAT virtual station with an
ISBA-CTRIP cell with respect to their localization and the drainage network.
The following algorithm has been used.
The CTRIP river network is compared to a realistic river system
(produced with GoogleEarth) to properly associate ISBA-CTRIP cells with a
given tributary in the basin.
Then, each virtual station is coupled with the closest ISBA-CTRIP cell
along the same tributary. It may be the cell containing the virtual station or
an adjacent cell according to the river network.
This algorithm allowed association of most of the virtual stations with a
unique CTRIP cell. However, some particular cases have been treated. First,
some virtual stations were located on tributaries too small to be represented
on the CTRIP river network. In this case, the virtual station was not
included in the study. Then, there were several very close virtual stations
associated with the same ISBA-CTRIP cell. In this second case, the virtual
stations with the lowest deviation from the MGB discharge were conserved.
Finally, over the 767 ENVISAT virtual stations initially available, 368
ENVISAT virtual stations were kept and associated with an ISBA-CTRIP cell.
Among them, 19 % (69 virtual stations out of 368) have been associated with
an adjacent cell.
In situ discharge product
At a national or basin scale, water agencies can share discharge time series,
such as the Agencia Nacional de Agua (ANA, hidroweb.ana.gov.br) in
Brazil for the Amazon River basin. For the present study, we retrieved
discharge time series from 145 ANA in situ stations over the entire basin.
These gauge discharges have then been used to evaluate the performances of
the DA (but they have not been assimilated into ISBA-CTRIP).
Method
The purpose of the SE DA is to correct model outputs using observations while
taking into consideration uncertainties in both the model and the
observations. In this work, as observed data correspond to discharge
estimates, we chose to correct model output variables such as discharge or
river storage. Indeed, following the results from the ISBA-CTRIP sensitivity
analysis (SA; ), discharge is mainly sensitive to
water inflow. Figure presents the general DA method
in the present study. The figure reads from top to bottom and from left to
right. Three types of state variables will be considered: the river initial
storage, the river final storage (which are both the main ISBA-CTRIP state
variables) and the river discharge that will all be compared to the observed
discharge. All three can be corrected through assimilation with specific
treatment that will be detailed in the following sections. The DA will use
several operators (in Fig. ,
M[k-1,k], Zk and Hk) that link
state variables with each other.
Data assimilation variables
The DA technique implemented in the present study is a sequential EnKF
. Here we shortly give the mathematical formalism used in
the rest of the paper and a brief description of the EnKF method.
First of all, the the term “assimilation window” used hereafter corresponds
to the period during which a complete assimilation cycle is conducted. It is
delineated by two consecutive observation times and will be denoted by
[k- 1, k]. From now on, the kth assimilation cycle will be the
cycle starting at time k- 1 and assimilating the available
observation(s) at time k.
Control variables
The vector xk∈Rnx is called the control
vector. It includes the nx uncertain variables to be estimated during
the kth DA cycle (within the time interval [k- 1, k]).
As stated before, control variables are prognostic or
diagnostic variables of the ISBA-CTRIP model. Prognostic variables are the
physical unknown in the differential equations' system that describes the
model's behaviour. Diagnostic variables are also physical variables, but they
are estimated from the prognostic variables. The choice of the control
variables determines the observation operator Hk that maps the
control variables into the observation space:
yk=Hkxk,
where yk are the control variables equivalent in the observation
space, also called model observations. They are then compared to the measured
observations yko (described in
Sect. ) during the DA step.
Unlike hydrodynamic models, which directly solve Saint-Venant equations and
for which discharge is a model state variable (or prognostic variable), the
hydrological model ISBA-CTRIP solve differential equations describing the
time evolution of water stock in the river (S), the groundwater (G) and
the floodplain (F). Then, water elevation and river discharge are
diagnostic variables derived from these prognostic variables. In CTRIP, river
discharge QoutS is computed as follows:
QoutS=L-1vSkgs-1,
with L (m) river section length, v the
flow velocity (estimated from the Manning formula) and S the surface water
mass.
Therefore, three types of variables can be considered as control variables in
the data assimilation scheme: the discharge QoutS (denoted Q
in the remaining of the study to simplify notation, which is a diagnostic
variable), the river final water stock Send (a prognostic
variable) or the river initial water stock 5also a prognostic variable).
Definition and complexity of the observation operator Hk, that
maps the control space into the observation space, depends on the nature of
the control variable. These three options are presented below.
Option 1: correcting ISBA-CTRIP discharges:
for this option, the control variables, gathered into the vector
xk, are the ISBA-CTRIP discharges Qi,k,
i= 1 …nx= 2028 (number of TRIP cells in the Amazon
basin) estimated for all 2028 cells in the TRIP Amazon basin, at the
assimilation cycle k.
The observation operator Hk resumes to a selection operator
Sk which selects the observed TRIP cells at the current
assimilation cycle:Hk=Sk.This operator is linear. The difficulty with this operator is that, once the
assimilation analysis is produced, it is necessary to convert the analysis
discharge Qi,ka, i= 1 …nx (i.e. the
corrected discharge obtained after assimilation) into the equivalent final
water stock Send,i,ka. Indeed, as already stated, in
ISBA-CTRIP, discharge is not a prognostic variable. Correction from the
assimilation step needs to be propagated to the model prognostic variables,
here the river final stock. Moreover, the analysis of the final water stock
Send,i,ka will be used as an initial condition for
the model run until the next assimilation cycle:
∀i= 1 …nx, ∀k,
Send,i,ka=Sini,i,k+1b.
However, the exact relationship linking discharge to the final river stock is
unknown.
A possible solution consists in inverting Eq. ().
Assuming that the discharge estimated by ISBA-CTRIP Qi,ka is
the instantaneous flow at the final time of the integration window,Qk,i,kgs-1a=L-1vSend,k,i,[kg]a⟺Qk,m3s-1a=ρ-1L-1vSend,k,i,[kg]a.We obtain that (for more details on this approximation, see
Appendix )Send,k,i,[kg]a≈ρLW2/5s-3/10n3/5Qk,i,kgs-1a3/5,with ρ (m3 kg-1) the water density, L (m) the river section
length, W (m) the river width, s (–) the riverbed slope and n (–) the
Manning coefficient in the riverbed. Then, for experiments with discharges as
control variables, the formula in Eq. () will be used to
convert corrected discharges into river stock and then propagate the model to
the next observation time.
Option 2: correcting ISBA-CTRIP final water stock:
for this option, the control variables, gathered into the vector
xk, are the ISBA-CTRIP final water stock Send,i,k,
i= 1 …nx estimated for all 2028 cells in the TRIP
Amazon basin, at the assimilation cycle k.
The computational cost for this option is the same as for the first option
but, now, the observation operator is defined asHk=Sk⋅Zk,where Zk is the operator (implicitly defined within TRIP) that
turns the river final stock Send,i,k into equivalent
discharge Qi,k. Even though Hk is not linear any more,
this option presents the advantage of correcting the river final stock
Sk,enda that can be directly used for the next
assimilation cycle and no additional uncertainties are introduced. However,
the corresponding analysis discharge Qi,ka is now unknown as
the explicit expression of Zk is also unknown. A potential
formula to determine Qi,ka can be deduced from
Eq. (). Such a formula will be necessary to make comparative
statistics to ENVISAT and in situ discharges and be able to evaluate the
assimilation performances.
Option 3: correcting ISBA-CTRIP initial water stock:
for this final option, the control variables, gathered into the vector
xk, are the ISBA-CTRIP initial water stock Sini,i,k,
i= 1 …nx estimated for all 2028 cells in the TRIP
Amazon basin, at the assimilation cycle k.
The discharge observations are used to correct the surface water stock at the
time prior to the observation time or, in other words, at the initial time of
the integrating window. Therefore, the observation operator is written as the
composition of the model operator M[k-1,k] with the observation
operator defined in Eq. ():Hk=Sk⋅Zk⋅M[k-1,k].This operator is highly non-linear as it contains the full model operator.
However, it is the only option where no uncertainties are added from the use
of an external formula to compute corrected discharge at the observation
time. Uncertainties in the stock–discharge relationship are only due to the
model uncertainties. Indeed, once the analysis initial water stock
Sini,k,ia is determined, the control variable update
must be propagated through the next assimilation time by re-integrating the
ISBA-CTRIP model over the assimilation window. The initial water stock
Send,k,ia and the analysis discharge
Qk,ia are automatically determined during this run.
Observation variables
In the framework of the state estimation, the observation variables, at a
given day within the Amazon basin, are the discharge estimates derived from
ENVISAT water surface elevations at the virtual stations associated with an
ISBA-CTRIP cell. The ENVISAT repeatability is 35 days, and therefore a given
virtual station will provide an observation every 35 days at best. During the
data assimilation experiments, all virtual stations will be used
simultaneously. Because of the ENVISAT orbit, the number of available
observations at a given day will vary between 0 and 15, and these
observations will be assimilated daily via the EnKF. Then, the observation
vector yko at the assimilation cycle k (equivalently,
at the simulation day k) is composed of the ny,k discharge measures
available at day k:
yko=yk,1o,yk,2o,…,yk,ny,ko,
where yk,jo, j= 1 …ny,k is the jth observation
among the ny,k at cycle k.
Measurement errors ϵkm come from errors in
ENVISAT water surface elevations, errors in MGB discharges and uncertainties
in the rating curve parameters used to turn water surface elevation into
discharge. , and
noticed that the concavity of the elevation–discharge
relationship implies that the higher a water elevation, the more uncertain
the corresponding discharge. Therefore, the observation error standard
deviation σk,jo, associated with the jth observation at
cycle k, is defined with respect to the instantaneous discharge measure
yk,jo, i.e.:
σk,jo=ηjo×yk,jo,j=1…ny,k,
where ηjo∈ [0, 1] is a constant depending
on the virtual station, such that ηjo models the relative error.
The observation error standard deviation σk,jo is then a
fraction of the instantaneous discharge. For each virtual station, the value
of ηjo depends on, first, the deviation from the MGB discharge,
noted σjmgb [%] and determined by .
As MGB discharges were used to determine ENVISAT discharges from ENVISAT
water elevations, σjmgb represents the deviation between
the two discharges data. Second, to take into account that MGB discharge is
not perfect (in other words, to take into account some deviation from the
real discharge), 0.05 is added to σjmgb and the sum is
rounding up to the nearest whole number, giving
ηjmgb=Eσjmgb+0.05,
where the function E is the ceiling function. Finally,
ηjo is equal to the first multiple of 5, above
ηjmgb. At the end, ηjo varies from 0.10
to 0.35 over the entire basin and is constant in time. Besides, the
representativeness error ϵkr induced when
a virtual station is associated with cells of the coarse TRIP mesh grid is
neglected here.
Moreover, for a given assimilation cycle and also between different cycles,
the observations are considered uncorrelated in space and time. The
observation error covariance matrix at cycle kRk is then a
diagonal definite positive square matrix.
The ensemble Kalman filter (EnKF) for ISBA-CTRIP state estimationThe EnKF sequential estimation
In the EnKF framework, the model and observation operator are not linear.
Therefore, the main idea is to use stochastic ensembles to represent the
control variables PDFs along with the error models
. First, the background control variables
xkb are stochastically represented by an ensemble of ne members:
Xe,kb=xkb,[1]xkb,[2]…xkb,ne.
Each member is estimated separately through the EnKF prediction step. For
each control variable case (see Sect. ), each member
of the control ensemble Xe,kb is estimated
by integrating the model operator from the corresponding analysis member at
the previous assimilation cycle, while adding external uncertainties (see
Sect. ):
∀l=1…ne,xkb,[l]=M[k-1,k]xk-1a,[l].
Then, the background control ensemble must be compared to the observations.
Depending on the control variables nature, the model operator is already
included (option 3) or not (option 1 and 2) within the observation operator.
Besides, following , an additional noise
ϵko is added to the observation vector
yko to avoid ensemble collapse. The observation ensemble
thus obtained is noted:
Ye,ko=yko,[1]yko,[2]…yko,ne.
Finally, the EnKF analysis step is applied to each member of the ensemble such that
∀l=1…ne,xka,[l]=xkb,[l]+Kk,eyko,[l]-Hkxkb,[l],
where the direct non-linear observation operator is applied to convert the
control variables into equivalent model observations.
The particularity of the EnKF is that the Kalman gain (Ke,k) is
stochastically estimated from the different control and model observation
ensemble, as follows:
Ke,k=PHTe,kHPHTe,k+Rk-1.
Localization of the error covariance matrices
In the framework of state estimation, the sampling error can introduce
artificial correlations into the background/analysis error covariance
matrices, and generate spurious correlations between two distant grid cells
in the mesh . The ensemble size ne is limited
by computational constraints. Therefore, before the EnKF analysis step, a
numerical processing of the matrices [PHT]e,k and
[HPHT]e,k is necessary to suppress spurious
correlations that can potentially degrade the analysis. Localization methods
are designed to reduce these problems.
There exist two types of localization techniques
. The first one is called
B-localization. It is based on explicitly modifying the background error
covariance matrix Pe,kb. It consists in
multiplying the matrix Pe,kb by a
correlation matrix generated from a radial function, namely a function of the
two/three spatial dimensions which monotonously decrease with the distance
between control variables
. This
modified matrix replaces Pe,kb in the
calculation of the Kalman gain matrix Ke,k. The other
common localization technique is called R-localization or local analysis.
This one consists in performing the analysis step in characteristic sub-spaces of the overall
problem space.
However, all these localization techniques described above have been
developed for atmospheric modelling where problems are in two or three
dimensions. The use of localization in hydrology is more limited. Several
studies exist to improve subsurface flow modelling
, but these approaches have a
dimensionality close to atmospheric approaches, as they take place in a
continuous medium in two or three dimensions. Other studies using
localization allow estimation of better model parameters, still continuously
defined in two or three dimensions .
The localization method used with the CTRIP river routing model is of the
B-localization type. However, it can not be simply defined on a
two-dimensional radial function. Indeed, the river flow is along several
one-dimensional flow directions, modelled by the routing network. The
localization technique must consider the routing network to decorrelate
adjacent cells on the mesh grid but located in two different sub-catchments.
Nevertheless, along a same flow direction, the correlation between two
distinct cells depends on the distance between the two cells. Then, for each
assimilation cycle, the localization consists in a localization mask
delimiting an influence area for each observation. These influence areas
gather a limited number of neighbouring downstream and upstream cells around
the observed cell with respect to the river routing network. We chose a fixed
localization scale for simplicity and as a first step in the feasibility
study of the development of a localization method for a hydrology application.
To determine the number of cells defining the influence area, the basin
subdivision into nine hydro-geomorphological zones is used with a mean flow
velocity for each zone. The influence area, for a given observed cell, is
given by the criteria below. For an influence area of size p cells, the
area is composed of the following.
The observed cell.
The p downstream cells according to the river routing network.
all the cells upstream the observed one covering p upstream levels.
However, the going up stops when the hydro-geomorphological zone is different
from the one of the observed cell.
The number of cells within the influence area depends on the mean flow
velocities (averaged over a year of simulation) in the zone in which the
considered cell is situated. Those mean velocities are calculated from the
free run simulation, namely the ISBA-CTRIP simulation realized without any
assimilation step. The ISBA-CTRIP resolution is 0.5∘× 0.5∘,
or approximately 50 km × 50 km. Given the
river meanderings, the minimal covered distance through a cell is of 50 km.
Furthermore, by comparing the free run discharge to in situ and ENVISAT
discharges, it seems that the free run underestimate discharge (and so the
flow velocity). Consequently, to fix the number p of cells defining the
influence area in each hydro-geomorphological zone, the following steps have
been performed:
the mean velocity for the cells into a given zone is converted into an
equivalent distance in km,
the maximal distance within the zone is kept and rounded to the closest
higher multiple of 50,
the number p determining the size of the influence area is the number
of cells covered by the maximal rounded distance, knowing that 50 km = 1 cell.
The number of cells into the influence area is presented, for each zone, in
the Table .
The final localization mask is presented into a matrix of size
nx×ny,k (with nx, the number of control variables, and ny,k, the
number of observation variables, at the assimilation cycle k) containing
only 0 and 1. The localization mask S has the same dimension as
the matrix [PHT]e,k. So, the localization matrix is
term-to-term multiplied (sign “×” in
Eq. ) to the error covariance matrix
[PHT]e,k such as in and :
xa=xb+S×PbHTHS×PbHT+R-1yo-Hxb.
To then extend the localization to the error covariance matrix
[HPHT]e,k, the lines in [PHT]e,k
corresponding to the observed cells are extracted to form the second
localization matrix. This second matrix is also term-to-term multiplied to
[HPHT]e,k. This localization step is applied in each
assimilation cycle with respect to the activated ENVISAT virtual stations at
the current assimilation cycle.
Generating the ensembles
The background error covariance matrices [PHT]e,k et
[HPHT]e,k are estimated from the control variable ensemble
using the definition suggested by
, and . To get a large
ensemble, while maintaining a reasonable computational time, the ensemble
size ne has been set to a hundred members. Details on how they are
exactly calculated are given in
Appendix . These matrices have a
nx×ny,k and ny,k×ny,k size, respectively. The elements
in the error covariance matrices, depend only on the definition of the
background ensemble stored in the control matrix Xe,kb and
the parameter uncertainties taken into consideration to generate
H(Xe,kb). In the framework of state estimation, we
choose to consider uncertainty into the initial condition and uncertainty
into the precipitation forcing .
Perturbation of the initial condition:
the vector containing initial surface reservoir storage for each nx= 2028
CTRIP cell at the assimilation cycle k is called ck. To ease
the notations, we will omit, as much as possible, the assimilation cycle k
subscript, knowing that, for all randomly perturbed variables and constants,
a new ensemble is generated at each cycle.
We used the Amazon basin division into ns= 9 hydrogeomorphological zones
(see Fig. ). Initial conditions are perturbed by
applying a multiplying factor over each zone ηsc,[l] such that
∀s=1…ns,∀l=1…ne,cs,k[l]=ηic,[l]⋅cs,k,where cs,k is the reduction of the initial condition
ck to the only zone s. For the perturbation to vary from one
member to another, the value ηsc,[l] is the
realization of a Gaussian distribution, different for each member
[l]= 1 …ne and for each hydrogeomorphological
zone s. The Gaussian distributions used have a mean value of 1 and a
standard deviation of σηs,kc whose values
are detailed in Table .
The ηic,[l] values depend on the assimilation cycle k
and on the hydrogeomorphological zone in which the ith cell is.
Firstly, a more important perturbation is applied to cells situated on
the river mainstream (zones 2 and 3), as we assume that the uncertainties are
more important in those zones. Indeed, discharges in these zones are the
highest of the entire basin. Besides, several cells are confluence cells and
are subject to backwater effects. As ISBA-CTRIP does not model the backwater
effects, the water stock uncertainties in these cells are increased.
Secondly, at the first assimilation cycle, the initial condition before
perturbation c1 is identical for every member. At this particular
cycle, the ensemble variance after perturbation depends only on the perturbation
method presented in Eq. () while, for the other assimilation
cycles, the successive previous assimilation cycles introduced an additional
variability between members, before the perturbation step in Eq. ().
Therefore, the initial condition variance is more important at the second
assimilation cycle and after. Then, to generate a larger variability at the
first assimilation cycle, the standard deviation σηs,kc
of the variable ηic,[l] is more important at the first cycle
than for the others.
Perturbation of precipitations:
another source of uncertainties for the generation of the ensemble
Hk(Xeb) lies in the precipitation
fields. Atmospheric forcing comes from the GSWP3 product (see
Sect. ). Precipitation forcing
F has been perturbed using the procedure presented by .
The ensemble of perturbed precipitation fields
F̃e is defined such thatF̃e=F̃[1],F̃[2],…F̃ne=φp[1]⋅F,φp[2]⋅F,…φpne⋅F,where
F is the two-dimensional field of precipitation forcing
before perturbation (with a time step of 3 h),
F̃[l], for l= 1 …ne, is the
lth perturbed precipitation field,
φp[l], for l= 1 …ne, is the
lth multiplying uniformly distributed field of F to generate
F̃[l]. More details on how the fields
φp[l] have been generated are given in
Appendix .
Constant values used to generate the background control ensemble
Xe,kb, the observation ensemble
Ye,ko and the model observation ensemble
H(Xe,kb). k is the assimilation
index and s is the basin zone index.
Presentation of the different state estimation experiments. The
“SE” acronym stands for “state estimation”, indexes “1”, “2” or
“3” are to differentiate the control variables (“1”: initial river
storage, “2”: final river storage and “3”: discharge) and the suffixes
“direct”, “diag” and “local” indicate the localization scheme
(“direct”: without localization, “diag”: diagonal error covariance
matrices and “local”: with localization).
Exp. nameControl variableLocalization schemeSE1-directinitial storageNo – [PHT]e,k and [HPHT]e,k defined in Eqs. ()–()SE1-diaginitial storageNo – diagonal [PHT]e,k and [HPHT]e,kSE1-localinitial storageYes – see Sect. SE2-localfinal storageYes – see Sect. SE3-localdischargeYes – see Sect. Assimilation diagnostics
During the assimilation experiment, it is necessary to quantify the
assimilation performances. The quality of the assimilation will be evaluated
in a given cell i by estimating the root mean square error (RMSE) between
the simulated discharge Qi* and the observed discharge
Qi†, giving
RMSEi*,†=1K†∑k=1K†Qi,k*-Qi,k†2m3s-1.K† represents the total number of available observed discharges
Qi† at the studied cell for the study period. The “*”
superscript can be either the superscript “f” for the free run (without
assimilation) or the superscript “a” for the analysis run (after
assimilation). The “†” superscript can be either “o” for the
observed ENVISAT discharge or “situ” for the gauge discharge.
Based on this definition, the assimilation performance will be estimated at
each cell with the normalized RMSE (RMSEn) defined by
RMSEni*,†=100×RMSEi*,†Qi,•†‾,[-],
where Qi,•†‾ corresponds to the mean of
Qi,k† averaged over the available time steps k.
Also, to evaluate the global performance of the assimilation over the entire
basin, a global RMSEn (RMSEnglobal) will be determined by
RMSEnglobal*,†=100×1∑i=1I†wi∑i=1I†wi⋅RMSEni†,[-],
where I† is the total number of stations and wi a weighting
constant depending on the maximal discharge at the ith station
(maxk(Qi,.†)) and the maximal discharge over the basin
(maxi,k(Q.,.†)) such that
wi=log10maxkQi,.†log10maxi,kQ.,.†.
With this weighting, cells along the mainstream and the largest tributaries
(with the highest discharges) have more importance in the global statistics
than cells located in basin heads.
Besides, the analysis run is available as an ensemble. The statistics will
then be estimated for each member of the ensemble and the mean (see
Eq. ) of the ensemble will be presented, such as
RMSEnia,†‾=1ne∑l=1neRMSEnia,[l],†,
where RMSEnia,[l],† is the normalized root mean square
deviation of the lth member of the analysis discharge ensemble.
Assimilation strategy
The state estimation experiments have the objective of testing the different
control variables described in Sect. . Also, another
objective is to test, validate and criticize the localization mask introduced
in Sect. . In the following, experiment names
using the localization will have the “-local” suffix, ones without any
localization will have the “-direct” suffix and ones with no correlation
between cells will have the “-diag” suffix. The objective of this study is
to determine the best strategy for assimilating ENVISAT discharges into the
ISBA-CTRIP model using the EnKF to correct the model state variables. The
different experiments are presented in Table . After
analysing these five elementary simulations over a single year, a last
experiment will be run over the entire ENVISAT observing period (from
September 2002 to June 2010), based on the best configurations.
For all the DA experiments, the observation errors are those described in
Sect. , and the model errors are those presented in
Sect. . Moreover, each experiment in
Table lasts 365 assimilation cycles of 1 day (so 1 year of
assimilation) from 1 January to 31 December 2009. We chose this period as it
overlaps with another altimetry mission (namely JASON-2), and future works
may include comparison of the two datasets' contribution. The numerous
ensemble ISBA-CTRIP simulations were realized with the CALMIP
high-performance computation platform
(https://www.calmip.univ-toulouse.fr/spip/) with the EOS supercomputer.
In the SE1-direct experiment, ENVISAT discharges are assimilated to correct
the initial surface reservoir storage in TRIP (and inherently TRIP simulated
discharges). For this first experiment, a classical EnKF, without any
localization treatment of the error covariance matrices
[PHT]e,k and [HPHT]e,k,
is conducted. This first experiment will be compared to the two next
experiments, SE1-diag and SE1-local, which will highlight the contributions
and/or limitations of the chosen localization approach. Finally, the last two
experiments, SE2-local and SE3-local, will test the other possible control
variables and the reliability of the operator Zk. More
particularly, the SE2-local experiment is based on control vector option 2
(see Sect. ) that assimilates discharges to correct
the final rive storage, and SE3-local is based on control vector option 1
(see Sect. ) that assimilates discharges to directly
correct the ISBA-CTRIP discharges.
ResultsFree run performances
The current section briefly presents the model performance without
assimilation called the free run. As all in situ and ENVISAT VS have been
associated with a unique ISBA-CTRIP cell, it is possible to compare observed
discharge at these stations to corresponding ISBA-CTRIP simulated discharge.
To begin with, a sample of 12 in situ stations, spread over the entire basin
(over the mainstream and the main tributaries), is selected. The location and
the name of these stations are represented in
Fig. . Figure compares
ISBA-CTRIP free run discharges to in situ and ENVISAT discharges at the
12 stations over 1 year of simulation (year 2009). From this comparison the
following observations can be drawn.
Over the majority of cells where there are both an in situ station and a virtual
station, the two discharge time series are similar (but not identical; see
Fig. , panels 1–3,
5–6, 8–9, and 12). These results are due to the fact that gauge discharges
were directly assimilated into the MGB-IPH hydrological model to correct the
MGB-IPH estimated discharges . Then, those same
estimated discharges were used to calculate parameters of the rating curves
between ENVISAT water elevations and MGB discharges .
Even though these rating curves have been derived from a model that
assimilated in situ data, ENVISAT-derived discharges depend essentially on
the remotely sensed water surface elevation variations .
Therefore, ENVISAT discharges remain independent enough of in situ data.
A strong difference between the in situ and ENVISAT discharges could
indicate either that the rating curve parameters were not correctly estimated
or that in situ/ENVISAT/MGB-IPH discharges have strong errors. As an example,
see Fig. , panel 11, at Itaituba, where the gauge
discharge is discontinuous and is even equal to 0 for some dates. Another
example is the gauge discharge at Manicoré, in
Fig. , panel 10, .
Finally, in most cases, the free run discharge is quite different
from the observed discharge. At downstream mainstream stations (at Manacapuru
and Óbidos in Fig. , panels 2 and 3), the
ISBA-CTRIP model is not able to reproduce flooding occurring between June and
August. Therefore, in the free run, the discharge peak occurs earlier in the
year and the discharge variations in this period are faster than the observed
discharge variations. Similarly, at most of the right-bank tributary
stations, the free run discharge peak is higher than the observed discharge
peak (see Fig. , panels 7–12). However, the
seasonal cycle is well reproduced for all these stations. These results
illustrate the necessity of conducting the DA experiments.
Map of the 12 in situ stations used to evaluate assimilation
performance: (1) São Paulo de Olivenca (Solimões), (2) Manacapuru
(Solimões), (3) Óbidos (Amazonas), (4) Ipiranga (Putumayo/Icá),
(5) Serrinha (Negro), (6) Uaicás (Branco), (7) Porto Seguro (Jutaí),
(8) Santos Dumont (Juruá), (9) Lábrea (Purus), (10) Manicoré
(Madeira), (11) Itaituba (Tapajós), and (12) Boa Sorte
(Xingu).
Comparison between the ISBA-CTRIP free run (blue line),
ENVISAT-derived observed discharges (green markers) and ANA gauge discharges
(black dots) over the year 2009. For each panel, the x-axis represents time
(in days) and the y-axis represents discharge
(in m3 s-1).
Then, Fig. displays the global performances of the free
run. For each ENVISAT virtual station (see Fig. a) and each
in situ station (see Fig. b), the RMSEn (defined in
Eq. ) between the simulated and observed discharges is
calculated and its value is indicated by a colour at the location of the
station over the basin. The results are similar between ENVISAT and gauge
discharges, confirming good concordance between the two
discharge datasets. RMSEn shows important deviations in basin heads on most of the
tributaries as well as at the confluence between right-bank tributaries and
the mainstream. Apart from the confluence and basin heads, the largest
tributaries, such as the Negro and the Madeira, are well represented.
Concerning global statistics (see Eq. ),
RMSEglobalf,o is equal to 71.12 % compared to ENVISAT
discharges and RMSEglobalf,simu is equal to 68.96 %
compared to gauge discharges. These deviations are likely due to atmospheric
forcing, parametrization and modelling errors, especially floodplain
parametrization. The DA experiments will focus on correcting the model
outputs which result from those uncertainties.
RMSEn for the free run simulation compared to the ENVISAT
discharges (a) and the gauge discharges (b).
Analysis RMSEn for the SE1-direct experiment with respect to
(a) the ENVISAT discharge and (b) the gauge
discharge.
Evaluation of the localization method
The first series of experiments assimilates ENVISAT discharges to correct the
ISBA-CTRIP initial river stock (see the three first rows in
Table ). They differ on the definition of the background
error covariance matrices [PHT]e,k and
[HPHT]e,k. The SE1-direct experiment uses the complete
stochastic matrix defined in Eqs. () and ().
In the SE1-diag experiment, these matrices are processed such that covariance
between two different CTRIP cells is set to 0 if the two variables are
situated in two different CTRIP cells. Lastly, SE1-local is based on the
localized version of the matrices presented in
Sect. . So,
Table displays the global RMSEn (see the
definition in Eq. ) for the three experiments compared
to the free run global statistics. From
Table , we can see that the RMSE between the
free run discharge and both the ENVISAT and gauge discharges is reduced for
all experiments, showing that the data assimilation platform is working
correctly. The SE1-diag experiment gives the worst results when compared to
both the ENVISAT discharge and the gauge discharge. Then, compared to ENVISAT
discharges, SE1-local gives the best results by reducing the global RMSEn by
more than 56 % (49 % for SE1-direct), while SE1-direct presents slightly
better global statistics than SE1-local when compared to gauge discharges
(RMSEn is reduced by 16.5 % for SE1-direct and by 15.25 % for SE1-local).
Overall, the global statistics are more reduced when compared to ENVISAT
discharges than to gauge discharges. This is due to the fact that gauge
discharges are not directly assimilated, unlike ENVISAT discharges. The next
subsections present and analyse in more detail results from each experiment.
Global statistics for experiments with different localization
schemes.
Figure displays the mean analysis RMSEn
(RMSEnia,†‾ defined in
Eq. ) for each ENVISAT virtual station
(Fig. a) and for each in situ station
(Fig. b). First of all, results between the ENVISAT
RMSEn and the in situ RMSEn are similar, due to the similarity between
ENVISAT and gauge discharge time series at most stations. According to
Fig. a, the assimilation worked quite well along the
mainstream and the main left-bank tributaries, namely the Negro River, the
Japurá and the Icá, with several stations where RMSEn is below
20 %. The assimilation performances are more moderate over right-bank
tributaries, where RMSEn is mostly between 20 and 60 %. Over the entire
basin, RMSEn remains high in all basin heads, along small tributaries and
also at most confluences; see for example the Jutaí–Solimões
confluence (RMSEn above 60 %), the
Purus–Solimões/Madeira–Solimões/Tapajós–Amazon confluences
(RMSEn above 40 %) or the Xingu–Amazon confluence (RMSEn above 80 %).
Figure compares the mean analysis discharge in
red line at the 12 stations previously introduced in
Sect. . For most stations, we can see
that the mean analysis discharges recovers a seasonal cycle closer to the
observations than the free run. It is especially true for stations along the
mainstream, namely Sao Paulo de Olivenca, Manacapuru and Óbidos
(Fig. , panels 1–3). Also, for stations along
right-bank tributaries (Fig. , panels 7–12), the
analysis seasonal discharge peak is lowered compared to the free run seasonal
discharge peak and fits better the observations. This shows the good
functioning of the assimilation platform.
SE1-direct ensemble mean analysis discharge (red line) compared to
the free run discharge (blue line), the ENVISAT observed discharges (green
markers) and the measured gauge discharges (black dots) over the year 2009.
For each panel, the x-axis represents time (in days) and the y-axis
represents discharge (in m3 s-1).
(a) Same as Fig. , panel 5 but
only over the 35 first days of simulation. (b) Location of all
active ENVISAT VS on the 25th day of the assimilation (yellow circles)
compared to the location of the Serrinha stations (red
circle).
Nevertheless, mean analysis discharge
for all displayed stations presents a chaotic behaviour with numerous local
minima and maxima. We can assume that this behaviour is present for all CTRIP
cells in the basin. Moreover, for a given cell, most of these sudden
variations are asynchronous with ENVISAT observation dates for this cell. For
example, at Serrinha in the left panel in
Fig. , an ENVISAT observation is available on
the 4th day of the 35-day repeat period when big off-peaks appear on the
25th day of the same 35-day repeat period. The right panel in
Fig. displays the Serrinha station (red
circle) with all ENVISAT observations available during the 25th day (yellow
circles). On inspection of the contribution of all these observations to the
analysis control variable at Serrinha (not shown here), we find that it is
observation number 4 that has the highest impact on the analysis (and not
observation number 5, as could be expected). This observation 4, located on a
very small Negro tributary, has a low discharge value and is responsible for
the low corrected discharge at Serrinha after the assimilation step.
These abrupt variations are completely artificial and directly result from
the assimilation processing. Indeed, for days with unrealistic
peaks/off-peaks, there are multiple ENVISAT observations available on the
basin, which impact many cells all over the basin, even if they are located
on other sub-catchments or tributaries. This is due to the construction of
the error covariance matrices [PHT]e,k and
[HPHT]e,k. As these matrices are generated from the ensemble
with a limited number of members, some spurious elements may appear in the
matrices and link two cells that are very distant in the basin or even
situated on different sub-basin. This first experiment highlights the
necessity to treat the error covariance matrices to limit such spurious elements.
SE1-diag results
In the SE1-diag experiment, the error covariance matrices are forced to be
diagonal. The objective of such processing on the error covariance matrices
is to limit the impact of a given observation only to the observed cell.
According to Table , the assimilation
experiment allowed one to reduce the global ENVISAT, an in situ RMSE, when
compared to the free run. However, among all three experiments, it is the one
which gives the worst global performances. In this experiment, the chaotic
behaviour of the mean analysis discharge is not present any more (not shown
here). Nevertheless, the mean analysis discharge remains close to the free
run discharge, except for regular peaks/off-peaks at an observation time when
it is closer to the observed discharge. Therefore, the information brought by
only one local observation is not enough. With the localization (see next
section), whose results are presented in the following section, the
information of several neighbouring VS is used and should constrain the
analysis discharge more.
SE1-local results
SE1-local uses the localization treatment presented in
Sect. . Figure
displays the RMSEn evolution from the SE1-direct to SE1-local experiments for
both ENVISAT and gauge discharge. Green colours indicate that the SE1-local
experiment reduced the RMSEn compared to the SE1-direct experiment, while
yellow to red colours indicate that the SE1-local experiment increased them.
The RMSEn is mostly improved over the entire basin and more particularly
along major right-bank tributaries. However, the RMSEn is generally degraded
along the mainstream. These maps show the good performances of the
localization method over tributaries. Now, it appears that, compared to gauge
discharges (see Fig. b), the SE1-direct experiment
gives better results, especially along the mainstream. Indeed,
Table details the local
RMSEnia,† at ENVISAT/in situ stations located along
the mainstream and confirms that SE1-direct gives better results. As the
global RMSEn is defined as the mean of the RMSEn weighted by the maximum
discharge at the station (see Eq. ), this explains why
the SE1-direct experiment gives a better global RMSEn compared to gauge
discharge.
Analysis RMSEn difference between SE1-direct and the SE1-local
experiment with respect to (a) the ENVISAT discharge and
(b) gauge discharge. Negative RMSEn differences (green colours) mean
that the results of the SE1-local experiment are better than the SE1-direct
results at the given CTRIP cell. Positive RMSEn differences (yellow, orange
and red colours) mean that the results of the SE1-direct experiment are
better that the SE1-local results at the given CTRIP.
Then, Fig. displays the mean analysis discharge
for SE1-local compared to the free run discharge, with corresponding ENVISAT
discharge and gauge discharge, at the 12 in situ stations already used in
Figs. and . Except for
stations along the mainstream (Fig. , panels 1–3)
and also at Boa Sorte (Fig. , panel 12), the
analysis discharge shows less sharp variations. From these results, we can
say that the localization scheme is necessary and improves the assimilation.
Discussions on the localization scheme
The localization mask has been built to avoid the effect of spurious
correlations between distant cells or ones situated on different sub-basins.
The current localization scheme meets this constraint. Indeed, results for
the SE1-local experiment are globally improved compared to the previous experiments.
Nevertheless, along the mainstream, the initial experiment without
localization gives better results. We can interpret that by the fact that
discharge along the mainstream integrates hydrological processes from all the
upstream basins. So when, in the SE1-local experiment, we limit the impact of
the observation to only close cells, we suppress part of the information
brought by distant cells to mainstream cells.
Local RMSEnia,situ‾ for the SE1-direct and
SE1-local experiments at in situ stations along the mainstream (from the most
upstream to the most downstream).
StationRMSEnia,situ‾SE1-directSE1-localTamishiyacu100.9792.54Tabatinga22.2438.81São Paulo de Olivenca21.7734.86Itapéua17.2028.58Manacapuru18.1530.38Jatuarana19.7231.01Óbidos16.6028.11
Therefore, the current localization mask should be improved. The main
difficulty here is to determine the size of the influence area for each
observation. Currently, this size is predetermined and is constant in time
according to averaged flow velocity. A potential development is to consider
an influence area size that can vary in time, according to the hydrological
season (high-flow/low-flow season). For example, during high-flow season, the
flow velocity is higher so is the size of the influence area. Thus, the error
covariance matrices would depend on the river time and space dynamic (as if
there were defined from a well-sampled and significant ensemble).
Impact of the chosen control variables
In the second series of experiments, all of them uses the localization scheme
(see Sect. ) to correct different types of
state variables. After assimilating ENVISAT discharge to correct river
initial storage (SE1-local experiment), we are testing in a second experiment
the assimilation of ENVISAT discharge to correct river final storage
(SE2-local) and, in a last experiment, the assimilation of ENVISAT discharge
to directly correct river discharge (SE3-direct). These two other experiments
need to use an empirical relationship (see Eq. ) linking
simulated river final storage to simulated discharge. For the SE2-local
experiment, the formula is used to convert analysis final rive storage to
discharge. Indeed, experiment statistics are based on discharge and, when
correcting the final river storage, we do not have an equivalent discharge.
For the SE3-local experiment, the formula is used during the assimilation
steps to convert the analysis discharge into an equivalent river storage to
propagate in time the corrected discharge.
Global statistics for experiments with different types of control
variables.
SE1-local ensemble mean analysis discharge (red line) compared to
the free run discharge (blue line), the ENVISAT observed discharges (green
markers) and the measured gauge discharges (black dots) over the year 2009.
For each panel, the x-axis represents time (in days) and the y-axis
represents discharge (in m3 s-1).
Analysis RMSEn differences between SE1-local and SE2-local
experiments with respect to (a) the ENVISAT discharge and
(b) gauge discharge.
Analysis RMSEn differences between SE1-local and SE3-local
experiments with respect to (a) the ENVISAT discharge and
(b) gauge discharge.
Table displays the global RMSEn for the three
experiments compared to the free run global statistics. For all experiments,
the assimilation enables to improve the RMSEn compared to the free run. Also,
compared to both ENVISAT and gauge discharge, SE3-local experiment (discharge
is the control variable) gives the best results, followed by SE1-local
experiment (initial river storage is the control variable). Finally, it is
SE2-local experiment (final river storage is the control variable) that gives
the worst results, even if it is still improving the RMSEn compared to the free run.
Figures –
display, for each ENVISAT (Figs. a
and a) and for each in situ stations
(Figs. b and b), mean RMSEn difference (in percent)
between SE1-local experiment and SE2-local
(Fig. ) and between SE1-local experiment
and SE3-local experiment (Fig. ).
Figure shows a slight increase of the RMSEn
in SE2-local experiment globally over the Amazon basin except for some basin
heads. Also, the upstream part of the mainstream is more degraded (RMSEn
increased of more than 60 %). These degraded results imply that the
assimilation of discharges may not be adapted to correct the final river
stock. However, we need to keep in mind that the analysis discharges are
determined from the analysis final river stock using Eq. (). The
bad SE2-local experiment results can either be due to bad assimilation
results or to an unadapted formula to convert the final river storage into
discharge. On the other hand, SE3-local experiment gives better general
results. As in Table and
Fig. , the SE3-local experiment shows a
global improvement of the RMSEn compared to the SE1-local experiment (apart
from a few cells upstream the Amazon mainstream). Indeed, even if
Eq. () is still used to convert the analysis discharges back into
river stock, it is used within the assimilation experiment (and not
afterwards as for the SE2-local experiment). Therefore, the formula
uncertainties are accounted for within the EnKF. Also, as the observed
discharges are directly used to correct the simulated discharge, it appears
logical that the assimilation gives better results as the link between the
observed and the simulated variables is immediate.
Discussions
From the different approaches tested in this paper, it appears that there is
no one specific configuration that gives the best results for all rivers,
when compared to both ENVISAT and gauge discharges. In contrast, the most
effective configuration depends on the size and location of the rivers. Along
the river mainstream (the Solimões and the Amazon in
Fig. a), the SE1-direct experiment clearly gives the best
results (see the three first rows in Table ).
When the contribution of observations on tributaries is suppressed with the
localization, the assimilation is less effective along the mainstream cells
(see panels 1 to 3 in Fig. for SE1-direct and
compare to the same panels in Fig. for SE1-local).
This could be due to the fact that discharge along the mainstream is the
result of hydrological processes from the entire drainage area. So, using all
available observations helps the EnKF to correct the most efficiently
discharge on the mainstream. However, it is different for cells along
tributaries. As presented in Table , the
localization method improves assimilation results for most cells along
tributaries compared to the SE1-direct experiment. Along these cells, the
localization allows the impact of observation from different sub-basins to be
suppressed, especially the ones that are not connected to these cells.
Finally, the comparison between the two experiments with localization
(i.e. SE1-diag and SE1-local) shows that the local experiment (SE1-local)
performs better than SE1-diag. This result was expected, as the localization
mask is more realistic in SE1-local, because there is more than one cell
impacted by the correction from one observation, in contrast to SE1-diag.
Nevertheless, among all experiments (see Table ), the one
producing the best results globally is SE3-local, where the localization
method is used to directly correct the discharge. Therefore, the SE3-local
configuration is used for an 8-year experiment, from 25 September 2002 (first
date with an ENVISAT observation of the study
domain) to 24 September 2010 (last date with an ENVISAT observation). At the
basin scale, RMSEn between model outputs and gauge discharges is reduced by
27.11 % (it decreases from 96.71 to 70.49 %) and RMSEn between model
outputs and ENVISAT discharges is reduced by 63.28 % (it decreases from
75.10 to 27.58 %). RMSEnglobalf,situ is high, because
a large fraction of in situ stations (25 out of 108) are situated along very
small tributaries or at basin heads, where the local RMSEnf,situ
is largely over 100 %. These very high RMSEnf,situ have
a huge impact on the global RMSEnglobalf,situ (despite
the weighting used to calculate RMSEnglobalf,situ). If
the statistics are computed using only cells with a RMSEnf,situ
below 100 %, we find that the global
RMSEnglobalf,situ is reduced by 14.66 % (it goes from
49.80 to 42.50 %) and the RMSEnglobalf,o is reduced
by 50.21 % (it goes from 51.74 to 25.76 %). This shows the limitation of
this assimilation scheme, as ISBA-CTRIP resolution (roughly 50 km by 50 km)
does not simulate basin heads well (rivers are too small to be correctly
represented in a coarse grid).
Statistics between analysis and in situ stations for the different
assimilation experiments. Italic values indicate the best result among the three
experiments for all tested gages.
StatisticsRMSEni*,situ‾FreeSE1-directSE1-localSE3-localrun1. São Paulo de Olivenca49.6721.7734.8640.982. Manacapuru36.0118.1530.3830.763. Óbidos34.6516.6028.1128.334. Ipiranga34.6335.4332.7533.545. Serrinha20.6526.8223.9928.246. Uaicás79.5351.2851.3348.927. Porto Seguro44.1646.3239.8138.938. Santos Dumont28.3735.5427.5533.639. Lábrea50.6240.3940.3339.8610. Manicoré72.3635.0454.3861.9611. Itaituba43.3366.4347.5046.2312. Boa Sorte149.3858.99112.5882.75
Figure displays, for the 12 in situ
stations (see Fig. for their locations)
already used in Figs. and , the mean analysis discharge over the whole
experiment time period (red line), which is compared to the free run
discharge at the station (blue line), the ENVISAT discharge (green markers)
and the gauge discharge (black markers). Overall, analysis discharge is quite
close to observed discharges (ENVISAT and in situ).
SE3-local ensemble mean analysis discharge (red line) compared to
the free run discharge (blue line), the ENVISAT observed discharges (green
markers) and the measured gauge discharges (black dots) from
25 September 2002 and for 8 years. In each panel, the x-axis represents time (in
days) and the y-axis represents discharge
(in m3 s-1).
However, despite the use of the localization, the analysis discharge keeps
presenting a quite chaotic behaviour: more particularly at Sao Paulo de
Olivenca (Fig. , panel 1), Manacapuru
(Fig. , panel 2) and during high flow
seasons along right-bank tributaries
(Fig. , panels 7 to 12). This shows the
limit of assimilating 35-day repeat period ENVISAT observations. If no data
are missing at a given VS, it means that there will be, at most, 11 available
observations during 1 year. Moreover, in a state estimation context, only the
model output state is corrected and not the model parametrization or, in our
set-up, forcings. Therefore, if the model is not constrained by direct or
neighbouring observations, it naturally goes back to free run discharge. The
performance of the assimilation, with respect to the daily in situ data, is
therefore often limited by the low ENVISAT observation frequency. In future
works, it will be interesting to study the assimilation of similar data with
a higher frequency, such as the JASON-2 altimeter data (which has a 10-day
repeat period but a coarser spatial sampling).
Conclusions and perspectives
This study presents, over the Amazon basin, the assimilation of a
satellite-derived discharge product into a large-scale hydrological model to
correct its state variables. The remotely sensed discharge data are derived
from the ENVISAT nadir altimeter and are assimilated into the ISBA-CTRIP
model using an ensemble Kalman filter. Five experiments were carried out over
the year 2009. For all experiments, the assimilations were able to reduce the
modelling errors compared to both observed and gauge discharges.
The first experiments tested different definition of the background error
covariance matrices, where the influence of a given observation is either
reduced to the only observed cell (SE1-diag), or limited to a few close cells
on the hydrological network (SE1-local), or not limited and can potentially
impact the entire basin (SE1-direct). Results showed that the complete
stochastic matrices gave the best results along the mainstream and the
localization treatment appeared necessary along the tributaries. The need for
the localization is explained by the spurious elements in the error
covariance matrix due to the limited ensemble size and the methodology used
to generate it.
The last tests compared the corrections of different state variables: the
river initial storage (SE1-local), or the river final storage (SE2-local), or
the river discharge (SE3-local). The main difficulty with these different
types of variables is, on the one hand, the relationship between the control
and the observed variables (gathered in the observation operator) and, on the
other hand, the reciprocal relationship to generate inputs for the next DA
cycle. Results showed that correcting river discharge gives the best global
results over the entire basin, as the link between the observed and corrected
variables is the most straightforward. Therefore, the final experiment
(SE3-local-long) uses the SE3-local configuration over the whole ENVISAT
observation period (from September 2002 to 2010) and confirms the possibility
of using such low-resolution remotely sensed data in a large-scale model.
These experiments offer several perspectives. First, the localization
treatment could be improved by combining the three tested approaches
according to the cell's position on the river: discharge correction for cells
along the mainstream should be impacted by all upriver observations, while
correction for cells on tributaries should be impacted only by close
observations along the same sub-catchment. Moreover, the size of the area of
influence for a given observations could also vary in time according to the
season (high flow/low flow). With ulterior developments of the localization
method, new challenges may appear such as the risk of imbalance, already
studied in the field atmospheric DA e.g.. The
analysis of imbalance may need to be considered in future works.
A main limitation of assimilating ENVISAT data is their low repeat period
(one observation every 35 days, at best). Indeed, corrected discharges often
present strong sudden variations between unobserved and observed dates, as
the model goes back to its free run when it is not constrained by an
observation. However, there are other satellite altimetry missions with
different repeat periods, for example JASON-2 (10-day repeat period from
June 2008 to October 2016), JASON-3 (10-day repeat period, launched in
January 2016), Sentinel-3A (27-day repeat period, launched in February 2016)
or Sentinel-3B (27-day repeat period, which should be launched in 2018).
Also, the incoming SWOT (Surface Water and Ocean Topography, launch scheduled
for 2021) wide-swath altimetry mission will provide a remotely sensed
discharge product. SWOT will have a 21-day repeat period, with an almost
global spatial coverage thanks to its two 50 km swaths. All these data could
be combined with ENVISAT data (during the overlapping period) within the
assimilation scheme to have a denser network of observation over the study
domain, to get a better estimate of discharge (similar to a reanalysis) over
a multi-decadal time frame .
To improve these DA results, several aspects could be investigated. For
example, one could study whether a more realistic ensemble method generation
could be helpful. In the present study, only the model initial condition and
the precipitation forcing are perturbed to generate the background forecast
ensemble. More uncertainties in this ensemble could be added by also
perturbing CTRIP parameters and/or ISBA outputs. Another DA aspect to look
into is the potential use of a smoothing data assimilation algorithm, such as
the ensemble Kalman smoother . A smoother could
help to have less “variability” in the corrected discharge. Finally, the
assimilation scheme presented in this study could be applied to other river
basins in the world, as ISBA-CTRIP is a global LSM. However, more work is
needed to apply the DA platform at a global scale.
The CTRIP code is open source and is available as a
part of the surface modelling platform
called SURFEX, which can be downloaded at
http://www.cnrm-game-meteo.fr/surfex/. SURFEX is updated approximately
every 3 to 6 months and the CTRIP version presented in this paper is from
SURFEX version 7.3. If more frequent updates are needed, please follow the
procedure to obtain a SVN or Git account in order to access real-time
modifications of the code (see the instructions at the previous link). The
ISBA-CTRIP model is coupled to the DA codes via the OpenPalm coupler
available at http://www.cerfacs.fr/globc/PALM_WEB/. To get the DA
routines coupled to ISBA-CTRIP with OpenPalm, please directly contact
C. Emery (charlotte.emery@jpl.nasa.gov) or S. Biancamaria
(sylvain.biancamaria@legos.obs-mip.fr). To obtain the GSWP3 forcings, please
refer to the following url:
http://search.diasjp.net/en/dataset/GSWP3_EXP1_Forcing
(10.20783/DIAS.501). The ENVISAT-based discharge data are available by
contacting A. Paris (aparis@cls.fr). Finally, the in situ discharges used to
validate the results are available on the ANA website
(http://hidroweb.ana.gov.br/default.asp).
Equations to compute river storage from discharge using the Manning formula
This Appendix provides more details and the approximation used to derive
Eq. (). This equation allows conversion of simulated discharges
Qi,ka to equivalent final river storage
Send,k,ia using the Manning formula. We chose to
invert Eq. (). Assuming that the discharge estimated
by ISBA-CTRIP Qi,ka is the instantaneous flow at the final
time of the integration window,
Qk,i,kgs-1a=L-1vSend,k,i,[kg]a⟺Qk,m3s-1a=ρ-1L-1vSend,k,i,[kg]a.
To ease the notations, we will skip the units in the following equations
knowing that discharges are expressed in m3 s-1 and
water stock in kg. Then, ∀k, ∀i:
Qk,ia=L-1ρ-1s1/2n-1WhSW+2h-S2/3Send,k,ia.Wesuppose:W≫hSQk,ia≈L-1ρ-1s1/2n-1h2/3Send,k,ia.YetS=ρLWhS,soQk,ia≈L-5/3ρ-5/3W2/3s1/2n-1Send,k,ia5/3.
Finally giving Eq. ():
Send,k,ia≈ρLW2/5s-3/10n3/5Qk,ia3/5,
with ρ (m3 kg-1) the water density, L (m) the river section
length, W (m) the river width, s (–) the riverbed slope and
n (s m-1/3) the Manning coefficient in the riverbed. Then, for
experiments with discharges as control variables, the formula in
Eq. () will be used to turn back corrected discharges into
river stock and propagate the model up to the
next observation time.
Definition of error covariance matrices
The background error covariance matrices [PHT]e,k
and [HPHT]e,k are estimated from the definition
suggested by , and
such that
PHTe,k=ne-1-1Xe,kb-X•,kb‾⋅1neTHXe,kb-HX•,kb‾⋅1neTT,
and
HPHTe,k=ne-1-1HXe,kb-HX•,kb‾⋅1neTHXe,kb-HX•,kb‾⋅1neTT,
with Xe,kb the control matrix containing the ne control
vector xkb,[l], l= 1 …ne from the background
ensemble such that
Xe,kb=xkb,[1]…xkb,Ne
and H(Xe,kb) the same control matrix
expressed in the observation space such that
HXe,kb=Hxkb,[1]…Hxkb,ne.X•,kb‾ and H(X•,kb)‾ are the ensemble sample
expectations of the control matrix Xe,kb and its mapping on
the observation space H(Xe,kb) respectively such that
X•,kb‾=1ne∑l=1nexkb,[l]HX•,kb‾=1ne∑l=1neH(xkb,[l]).
The vectors' dimensions are nx and ny,k, respectively, and
1ne is a vector of size ne containing
only 1 s.
Perturbations of precipitations
The ensemble of perturbed precipitation
fields F̃e is defined such that
F̃e=F̃[1],F̃[2],…F̃ne=φp[1]⋅F,φp[2]⋅F,…φpne.F,
where
F is the two-dimensional field of precipitation forcing before
perturbation (with a time step of 3 h),
F̃[l], for l= 1 …ne,
is the lth perturbed precipitation field,
φp[l], for l= 1 …ne,
is the lth multiplying uniformly distributed field of F to
generate F̃[l].
The precipitation field F is then perturbed by applying a random
multiplying field such that
φp[l]=1-ηF,[l]+2UF[l]⋅ηF,[l],
where
UF[l] is random field following a uniform law
between 0 and 1,
ηF,[l] is a scalar representing the relative error of
the precipitations.
Therefore, φp[l] is a random field following a uniform
law between (1 -ηF,[l]) and (1 +ηF,[l]).
The precipitation relative error ηF quantifies the
uncertainties in the precipitation intensity. The variable
ηF is different for each member of the ensemble and follows a
Gaussian law with expectation ηF‾= 30 % and
standard deviation σηF= 0.1 %
.
The fields UF[l], for
l= 1 …ne, allow one to introduce a time and
space correlation in the precipitation error and are generated with the
algorithm presented in . This algorithm generates
two-dimensional Gaussian random fields S[l] with a zero mean
and a space-correlation length of e-1. These Gaussian random fields are
turned into uniform random fields by applying the complementary error
function erfc():
UF,k′[l]=12erfcSk′[l]2,
where k′ is the atmospheric forcing proper time step, equal to 3 h in
ISBA-CTRIP, and shorter than the ISBA-CTRIP output time step, equal to 24 h.
The space PDF of UF[l] decreases
of e-1 when the distance is equal to the
space-correlation length τx (here, the letter x exceptionally
denotes the spacial dimension). For the simulations, τx is fixed to
1.0∘ and is invariant from one
member to another and from one assimilation cycle to another.
For the time correlation, the parameter ϑ[l]ϑ[l]=1-Δk′τk[l]
determines the time correlation length
between the different fields Sk′[l]. It is concretely
generated by combining the random field from the previous time step
Sk′-1[l] and an auxiliary random field
Wk′[l] with the same properties such that
Sk′[l]=ϑ[l]Sk′-1[l]+1-(ϑ[l])2Wk′[l],
with Δk′= 3 h the forcing time step and τk[l] the
time constant characterizing ϑ[l]. ϑ[l]= 0
generates a white noise (which means a perturbation uncorrelated in time),
while ϑ[l]= 1 makes the perturbation constant in time. The
variable τk takes a different value for each member as it follows a
Gaussian law with an expectation equal to τk‾= 12 h
(or 43 200 s) and a standard deviation of στk= 3 h
(or 10 800 s). These values are chosen so that the time correlation has
effects during an assimilation window of 1 day. All variables used to
generate the ensembles with their value are summarized in
Table .
Constant values used to perturb the precipitation fields. k is the
assimilation index.
The authors declare that they have no conflict of interest.
Acknowledgements
This work has been performed using HPC resources from CALMIP
(grant 2016-P1408). The GSWP3 team is acknowledged for letting the authors
use their different forcing fields. This work was supported by the CNES,
through a grant from the Terre-Océan-Surfaces
Continentales-Atmosphère (TOSCA) committee affiliated with the project
entitled “Towards an improved understanding of the global hydrological cycle
using SWOT measurements”. The European Space Agency (ESA) is also thanked
for providing to the scientific community observations from the RA2 altimeter
embarked on ENVISAT. Charlotte Marie Emery was supported by a CNES/région
Midi-Pyrénées PhD grant. This work was done as a private venture and
not in the author's capacity as an employee of the Jet Propulsion Laboratory,
California Institute of Technology. Edited
by: Albrecht Weerts Reviewed by: Rolf Hut and one anonymous
referee
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