The knowledge of water storage volumes in catchments and in river networks
leading to river discharge is essential for the description of river
ecology, the prediction of floods and specifically for a sustainable
management of water resources in the context of climate change. Measurements
of mass variations by the GRACE gravity satellite or by ground-based
observations of river or groundwater level variations do not permit the
determination of the respective storage volumes, which could be considerably
bigger than the mass variations themselves.
For fully humid tropical conditions like the Amazon the relationship between
GRACE and river discharge is linear with a phase shift. This permits the hydraulic time constant to be determined and thus the total
drainable storage directly from observed runoff can be quantified, if the phase shift can be
interpreted as the river time lag. As a time lag can be described by a
storage cascade, a lumped conceptual model with cascaded storages for the
catchment and river network is set up here with individual hydraulic time
constants and mathematically solved by piecewise analytical solutions.
Tests of the scheme with synthetic recharge time series show that a
parameter optimization either versus mass anomalies or runoff reproduces the
time constants for both the catchment and the river network τC and τR in a unique way, and this then permits an individual
quantification of the respective storage volumes. The application to the
full Amazon basin leads to a very good fitting performance for total mass,
river runoff and their phasing (Nash–Sutcliffe for signals 0.96, for monthly
residuals 0.72). The calculated river network mass highly correlates (0.96
for signals, 0.76 for monthly residuals) with the observed flood area from
GIEMS and corresponds to observed flood volumes.
The fitting performance versus GRACE permits river runoff and
drainable storage volumes to be determined from recharge and GRACE exclusively, i.e. even for
ungauged catchments. An adjustment of the hydraulic time constants (τC, τR) on a training period facilitates a simple
determination of drainable storage volumes for other times directly from
measured river discharge and/or GRACE and thus a closure of data gaps
without the necessity of further model runs.
Introduction
In the context of water resources management and climate change there is an
ongoing discussion on how to assess available water resources, i.e. the
storage volumes which can be used for water supply in a dynamic way beyond
the limitations of sustainable extraction rates. The maximum average
extraction rate for a sustainable use of water resources is limited by the
long-term recharge of a catchment (Sophocleous 1997; Bredehoeft, 1997);
however, this rate-based definition of groundwater stress only allows an
assessment of water resources with respect to long-term sustainability and
does not permit short-term management in order to satisfy specific water
demands. Thus the knowledge of water resources involved in the water cycle
contributing to river discharge, such as parts of the groundwater or surface
water system, is essential.
Very little attention has so far been given to the quantification of the
storage volumes of renewable water resources participating in the dynamic
water cycle driven by precipitation P, actual evapotranspiration ETa
and river runoff R. The reason for this is seen in the problem that
observations of time-variant groundwater or river levels only permit the
estimation of volume changes but not absolute storage volumes, which could be
considerably bigger.
Natural systems consist of many different storage components like canopy,
snow/ice, surface, soil, unsaturated/saturated underground, drainage system
etc. Direct measurements of storage volumes from water or pressure levels
are problematic as they are based on assumptions and approximations. They
are based on point measurements and quite rare on large spatial scales
compared to the heterogeneity scale of the respective compartments. This
leads to large interpolation errors. In addition, the storage coefficients
for porous media describing the relationship between the measurable
groundwater heads or capillary pressure on the one hand, and storage volume
or absolute soil saturation on the other hand, are insufficiently known on
large scales. Remote sensing data have been limited to near-surface water
storage (open water bodies, soil) up until now and are thus of limited
benefit for the quantification of water storage with respect to accuracy and
coverage due to methodological constraints (Schlesinger, 2007).
In contrast to discharge-less basins and/or arid areas, which are nearly
exclusively driven by precipitation and evapotranspiration, the storage
dynamics of catchments draining into a river system allows the
hydraulically coupled storage compartments to be addressed via their contributions to river
discharge. These comprise groundwater, surface water, the river network and
temporarily inundated areas. All storages draining into the river system by
gravity are referred to as “drainable” storage here. So, aquifers or parts
of them not draining into the river system without an energy input are not
considered here.
River runoff R(t)=Q(t)/A (corresponding to river discharge Q(t) from the
related catchment area A) is driven by the storage height or mass density
MStorage=Vtot/A of all superposed hydraulically
coupled storages and is determined by their runoff–storage (R–S)
relationship. For time periods with no recharge or losses of water (as by
ETa), i.e. with no processes affecting MStorage other than river
discharge Q, a linear runoff–storage (R–S) relationship R(M)=M/τ leads to an exponential decrease in river discharge or streamflow Q(t)
depending on the related hydraulic time constant τ:
Q(t)=Q(t0)⋅e-⋅t-t0.τ
For this case the corresponding total drainable storage in terms of mass
density MStorage at any given time t0 can be determined by an
infinite temporal integration over river discharge Q(t) from the
corresponding catchment area A starting at time t0:
Mstorage(t0)=Vtot(t0)A=1A⋅∫t0∞Q(t)dt=Q(t0)A⋅∫t0∞e-⋅t-t0τdt=τ⋅Q(t0)A=τ⋅R(t0)
Contributions from several storage compartments (with individual time
constants) superpose, if they drain in parallel and if there is no feedback
from the river system. For this case, there is a wide range of time series
analysis methods (Tallaksen, 1995), which allow the flow
components to be separated into fast, medium or slow and the corresponding surface,
interflow or groundwater flow contributions according to their individual
time constants. Thus, measurements of the different time constants allow the drainable storage of the respective storage compartment and
the corresponding mean drainable storage to be determined from mean runoff R or recharge N:
M‾X=R‾⋅τX=N‾⋅τX.
On global scales the absolute storage volume of the drainable storages can
be determined from runoff time series directly, if there are distinct and
long enough periods of negligible or even negative recharge (actual
evapotranspiration ETa > precipitation) as it occurs in
seasonally dry regions (Niger, Mekong, some Amazon sub-catchments etc.).
From the purely exponential decrease in river discharge the time constant
can be determined directly from a curve fit, as shown in Fig. 1b for Amazon
sub-catchments. If the dry period is long enough the sequence of different
time constants taken from the discharge curve even permits a discrimination
between the fast response by overland flow and the slow response by the
groundwater system.
(a)R–S diagram with counter clockwise hysteresis for mean monthly observed runoff Ro versus GRACE dM for fully humid catchments including a phase adaption for the Amazon upstream from Obidos (Riegger and Tourian, 2014).
(b)R–S diagram with clockwise hysteresis for mean monthly observed runoff Ro versus GRACE anomaly dM for seasonally dry catchments in the Amazon basin (Riegger and Tourian, 2014).
(c) Mean monthly runoff R and recharge N for fully humid catchments in the Amazon basin (log scale for R).
(d) Mean monthly runoff R and recharge N for seasonally dry catchments in the Amazon basin (log scale for R) including exponential fittings for runoff Rsim.
Catchments with permanent input, i.e. no periods of negligible recharge,
however, do not show an exponential behaviour for discharge. For these cases
the hydraulic time constant cannot be taken from discharge dynamics
directly, but has to be estimated by hydrological models. These are intended to
describe the large number of storages distributed over the catchment by the
assumed processes and calibrate the involved parameters by their respective
superposed flows versus the observed river discharge. The main difficulties
in verifying large or global-scale hydrological models or land surface
models (GHMs or LSMs) consist of the quantification of local individual
storage volumes and related flows by local ground-based measurements. Thus,
even though distributed hydrological models very much support an
understanding of processes in the water cycle, the limitation of the
calibration versus river discharge exclusively introduces an ambiguity in
the impact of contributing processes and the related storages and flows.
Since 2013 GRACE observations of the time-variable gravity field provide
monthly distributions of mass density on large spatial scales > ∼200 000 km2 (Tapley et al., 2004). However, as the water
storage in different compartments (snow, ice, vegetation, soil, surface water,
ground water etc.) superposes with all other terrestrial (geophysical)
masses, only the time-variant part of the GRACE signal can be used to
quantify the terrestrial water storage (TWS) anomalies (monthly mass signals
minus long-term average), but not the related absolute storage volumes.
Nevertheless, this for the first time permits a direct comparison of
measured TWS and observed river runoff Ro. Surprisingly some GHMs showed a considerable phase shift between
measured mass anomalies by GRACE and river discharge as well as between
calculated and measured runoff and an underestimation of mass signal
amplitudes (Güntner et al., 2007; Chen et al., 2007; Schmidt et al., 2008; Werth et al., 2009; Werth and Güntner, 2010) even though they comprise a
large number of storages like soil, surface water, groundwater etc. This is
emphasized by Scanlon et al. (2019), who for tropical basins recognize the
main cause of the discrepancies in insufficient storage capacity and lack of
surface water inundation.
The direct comparison of GRACE anomalies and river runoff on large spatial
and monthly timescales by Riegger and Tourian (2014) revealed that measured
runoff–storage (R–S) diagrams show hysteresis curves of distinct form and
extent (Fig. 1a, b), which are characteristic for different climatic conditions
(like fully humid, seasonally dry or boreal) and can be explained by
considering recharge and runoff properties (Fig. 1c, d).
Thus for example, catchments in fully humid conditions (like the full Amazon
basin upstream from Obidos (295 in Fig. 1a) and some of its catchments like upstream
Manacapuru (501 in Fig. 1c)) with a permanent input, i.e. only positive recharge
(Fig. 1c), show a counterclockwise hysteresis (Fig. 1a). If this can be fully
described by a positive phase shift, river runoff and storage behave like a
linear time-invariant (LTI) system (Riegger and Tourian, 2014), i.e. the R–S
relationship is linear, if the phase shift is adapted as shown in Fig. 1a.
For this case the hysteresis can be purely assigned to a time lag. Once the
phase shift is adapted the slope in the R–S diagram corresponds to the
hydraulic time constant via τ=M/R. The time constant and the reasonable
assumption of a proportional R–S relationship (no runoff for empty storage)
then facilitates the quantification of the drainable storage (Eq. 3), i.e.
the volume related to the hydraulically coupled storage compartment, which
drains by gravity.
In contrast, catchments with distinct periods of zero or negative recharge
(like Niger, Mekong or Rio Branco (504), Rio Jurua (506) in the Amazon basin;
Fig. 1b) show a clockwise hysteresis in the R–S diagram and a form which
is determined by an increase in mass and runoff during wet periods, a
decrease in mass and runoff with different slopes corresponding to different
time constants and a possible mass loss without a related runoff (by
negative recharge (Fig. 1d) by evapotranspiraton from the soil zone) during
dry periods. This type of hysteresis is determined by storage changes not
connected with river discharge (uncoupled) and cannot be explained by a time
lag as it is not causal.
The consequence from the above discussion is that the determination of the
hydraulic time constant and thus the drainable storage is only possible for
catchments for which the hysteresis is fully explained by a positive phase
shift; i.e. uncoupled storages are either negligible or can be separated from
GRACE mass by other means (as shown below for boreal regions).
Based on this method, Tourian et al. (2018) apply an adaption of the phase
shift using a Hilbert transform in order to determine the hydraulic time
constants and the total drainable water storage for the sub-catchments of
the Amazon basin without a consideration of the form of the R–S hysteresis.
To be sure, this leads to reasonable results for the sub-catchments with
permanent input (Fig. 1a, c) for which the time-dependent uncoupled storage
is negligible. However, for Rio Branco (504) or Rio Jurua (506) this
condition is not fulfilled as the hysteresis is determined by mass changes
in the uncoupled storage and by runoff with different time constants
(Fig. 1b, d). For these catchments the exclusive adjustment of the phase
shift leads to negative time lags – which are not physical – and as a
consequence to misleading time constants and thus to considerable errors in
the determination of drainable storage volumes.
The accurate description of the R–S hysteresis of a catchment and its river
network is the prerequisite for an accurate description of the system
dynamics and the related storage volumes on the land masses (canopy, soil,
overland flow, saturated/unsaturated underground) and in the river
network.
Recent developments in river routing schemes of global hydrologic models
with a hydrodynamic modelling of the flow in the river network system have
successfully dealt with the description of phase shifts generated by the
time lag in the river network (Paiva et al., 2013; Luo et al., 2017;
Siqueira et al., 2018). Getirana et al. (2017a) emphasize the importance of
integrating an adequate river routing schemes not only for an improved phase
agreement with observed river discharge but also for an appropriate fit of
the total mass amplitude to GRACE by the inclusion of the corresponding
river network storage. Yet a hydrodynamic modelling of a complete river
network system for the determination of the river network time lag and
storage means a huge modelling effort (Getirana et al., 2017b).
A far more simple approach is presented by Riegger and Tourian (2014),
describing the system by macroscopic variables, summarizing all coupled
storage compartments on landmasses and in the river network, and analogously
all uncoupled storage compartments in one respective single storage by their
effect on the R–S relationship. The intention of such a “top-down” or
lumped approach is to integrate the catchment-scale water balance and
describe the system by large-scale variables and parameters, which are
directly measurable or adjustable. For this purpose recharge based on
moisture flux divergence or catchment water balance using GRACE can be used,
which is quite accurate, yet limited to global scales (see below). Thus,
opposite to distributed hydrological models which are based on spatially or temporally distributed data (for hydrometeorological input, local storage
conditions in vegetation, soil and underground) and a detailed description
of internal processes – which cannot be verified locally at present – this
“top-down” approach uses measured catchment-scale input, storages and
runoff. Where necessary and possible catchment-scale parameters are used to
separate coupled and uncoupled storages (like MODIS snow coverage for boreal
regions; Riegger and Tourian, 2014). In addition the time lag between
storage and river discharge need not be explicitly described by an excessive
routing scheme. Instead the related phase shift can be adapted by
mathematical methods. This leads to a description of the system behaviour
with high accuracy (Nash–Sutcliffe efficiency of 0.97 for the whole Amazon basin) by an adaption of
only two parameters, the hydraulic timescale and the phase shift, even
though the physical cause of the phase shift is not addressed explicitly.
A disadvantage of the above approaches (Riegger and Tourian, 2014; Tourian
et al., 2018) is that it does not permit the individual
drainable storage volumes on landmasses and in the river network to be quantified separately,
but only the total drainable volume of the catchment. The information
contained in the phase shift or time lag is not used for a quantification of
the river network storage volume. Yet, as observations of inundated areas in
river networks such as those from the GIEMS “Global Inundation Extent from
Multi-Satellites” project Prigent et al. (2007); Papa et al. (2008); Papa
et al. (2013) and hydrodynamic models of the river network (Paiva et al., 2013, Getirana et al., 2017b; Siqueira et al., 2018) indicate a
considerable contribution of river network storage corresponding to a non
negligible time lag, the river network storage must be considered in the
integration of the total catchment water balance. As a sequence of storages
(cascaded storages) leads to a time lag (i.e. a phase shift; Nash, 1957) and
storages draining in parallel (as for overland and groundwater flow) just
lead to a superposition (with no time lag), a storage cascade is considered
as an appropriate description to account for a time lag.
This paper explores the accuracy and uniqueness of a lumped, top-down
approach called a “cascaded storage” approach, which is based on the integration of
given recharge in the water balance and utilizes a cascade of a catchment
storage and a river network storage for a simple description of the observed
time lag and the individual storage volumes. This permits a description of the system with a minimum number of macroscopic observation data and an adaption
of only two parameters, the hydraulic time constants of the catchment and
the river network. These time constants then could be used for nowcasts or
even forecasts (within the time lag) of river discharge and/or drainable
storage volumes directly from measurements without the need for further
modelling.
The paper is structured as follows: Sect. 2 presents the mathematical
framework of piecewise analytical solutions of the water balance equation
for a cascade of catchment and river network storages. It also contains the
description of observables, which permit the comparison of calculated and
measured values. The “single storage” approach is handled as the specific case
for a negligible river network time constant. In Sect. 3, the properties
of the cascaded storage approach and its impact on the performance of the
parameter optimization are described for synthetic recharge data and
compared to the single storage approach. Based on the cascaded storage
approach a fully data-driven approach is presented which permits a
simplified determination of the drainable storage volumes directly from
measurements without the need for further model runs. In Sect. 4 the
approach is applied to data from the Amazon basin and evaluated versus
measurements of GRACE mass, river runoff and flood area from GIEMS. The
results are compared to GHM/LSM studies. In Sect. 5 the approach and its
performance and limitation is discussed. Possible future investigations in
order to overcome some of its limitations are sketched. Conclusions are
drawn in Sect. 6.
Mathematical framework
In order to investigate the impact of a non negligible river water storage
on the time lag in the river system, the water balance of the total system
comprising both the catchment and river network storage has to be
considered. A conceptual model corresponding to a Nash cascade (Nash, 1957),
called a cascaded storage approach here, is set up with individual time
constants for the different storages and with the following properties:
Surface water and shallow groundwater storages on the land mass which are
draining into the river network and are being fed by recharge are summarized
to a so-called “catchment” storage MC with time constant τC. Overland and groundwater flow from the land masses are summarized to a “Catchment” runoff RR.
River runoff (river discharge per catchment area), which addresses
hydraulically the flow in the river channel network including inundated
areas, is determined by its hydraulic time constant τR. The
respective river network storage MR is assumed to be instantaneously
distributed within the river network system. Internal routing effects, which
might lead to an additional delay in streamflow response, are not
considered.
Any possible hydraulic feedback from the river to the catchment system is
assumed to be negligible.
Temporal variations of uncoupled storage compartments like soil or open
water bodies are considered as negligible.
These conditions are chosen for the sake of conceptual and mathematical
simplicity. It has to be emphasized here that for a general applicability on
a global coverage several coupled storages with different time constants and
different uncoupled storage compartments with their respective time
dependency have to be considered, of course. For fully tropical climatic
conditions with permanent recharge, however (as for the full Amazon basin),
variations in the soil water storage are negligible and the different
dynamics of overland and groundwater flow cannot be distinguished. Thus,
applications of this first approach are limited to catchments for which the
hysteresis can be fully described by a time lag; i.e. no impacts of other
coupled or uncoupled storages exist.
The abbreviations used throughout the paper are described in Appendix Table A1).
The total system behaviour is described by two balance equations, one for
catchment storage (Eq. 4) and one for river storage (Eq. 6).
Catchment storage:
∂∂tMC(t)=N(t)-RC(t)=N(t)-1τC⋅MC(t)
with
RC(t)=1τC⋅MC(t).
River storage:
∂∂tMR(t)=RC(t)-RR(t)=RC(t)-1τR⋅MR(t)
with
RR(t)=1τR⋅MR(t),
with a proportional R–S relationship for hydraulically coupled storages. N
denotes the recharge as input, RC the catchment runoff from the
catchment storage MC, which cannot be measured directly on large
spatial scales, and RR the river runoff from the river network storage
MR, which can be measured at discharge gauging stations.
The water balance equation, Eq. (4), for the catchment is generally solved
by the following:
MC(t-t0)=MC(t0)⋅e-t-t0τC+∫t0tN(w)⋅ew-tτC⋅dw,
where MC(t0) is the initial condition and N(t) the time-dependent
recharge.
For recharge N(t) being given with a certain temporal resolution in time
units or by periods of piecewise constant values and arbitrary length
(stress periods) the recharge time series can be described as follows:
N(t)=∑i=0n-1Ni+1⋅γi+1(t)withγi+1(t)=01fort∉t,t1+1t∈ti,ti+1foreachintervalti,ti+1
For calculation convenience Eq. (8) can be solved successively for each
stress period using the values at the end of the last period as the starting
value, which leads to the piecewise analytical solution for catchment mass
for a time t∈ti,ti+1 in stress period i+1:
Mi+1C(t-ti)=MiC(ti)⋅e-t-tiτC+Ni+1⋅τC⋅1-e-t-tiτC.
The respective catchment runoff RC based on Eq. (5) and MC from
Eq. (10) is used as input for the river network water balance, Eq. (6), and
leads to the general solution for the river network storage MR:
MR(t-t0)=MR(t0)⋅e-t-t0τR+∫t0tRC(u)⋅eu-tτR⋅du,
and the iterative solutions for time t∈ti,ti+1
in stress period i+1:
Mi+1R(t-ti)=MiR(ti)⋅e-t-tiτR+Ni+1⋅τR⋅1-e-t-tiτR+MiC(ti)-Ni+1⋅τC⋅τRτC-τR⋅e-t-tiτC-e-t-tiτR.
The total mass MT is then given by the following:
MiT=MiC+MiR.
The mixed term in Eq. (12) and thus the total mass are commutative in (τC, τR) and show a singularity at τC=τR with an asymptotic value. For τR > τC solutions also exist with analogous values in total mass MT for
MR > MC.
It has to be emphasized here that the piecewise analytical solutions for
time periods of constant recharge provide a mathematical solution for an
arbitrary temporal resolution without numerical limitations. Finite difference solutions are limited by stability criteria
(ti+1-ti) < τ and accuracy criteria
(ti+1-ti) < τ/10 for the smallest τ. Analytical
solutions facilitate an exact calculation of the response of the river
network during the time interval of constant recharge (though the time
constant of the river network could be much shorter than the time interval
or the time constant of the catchment). Thus the very high temporal
discretization, which otherwise would be needed using a finite difference
scheme, is avoided.
The observables related to measurements by GRACE and discharge from gauging
stations are the total mass anomaly dMT and the river runoff RR.
GRACE observations with acceptable error are still limited to monthly
resolution. Discharge as well as some of the meteorological variables like
precipitation, evapotranspiration or moisture flux divergence are often
measured in daily values, and some of the products are measured in monthly values. For an
optimal adaption to the monthly resolution of GRACE products, the approach
presented here is based on monthly values but could also be applied to daily
data without problems.
The mass values used in the calculations here are assigned to the interval
boundaries while the values for monthly recharge and measured runoff are
constant over the interval and temporally assigned to the centre of the
interval. Thus, for a comparison of the calculated mass and runoff values
versus the observed monthly values of GRACE and discharge, the calculated
values have to be averaged over the interval. As the dynamics follow an
exponential behaviour the mean values cannot be taken from arithmetic
averages at the interval boundaries but instead from an integral average
over the interval.
The mean storage mass for MX is given for each interval ti,ti+1 by the following:
M‾i+1X=1ti+1-ti∫titi+1Mi+1X(t-ti)⋅dt,
leading to mean runoff
R‾X(t)=1τX⋅M‾X(t),
i.e. mean catchment mass and runoff:
M‾i+1C=(MiC-Ni+1⋅τC)⋅τC(ti+1-ti)1-e-ti+1-tiτC+Ni+1⋅τC
and
R‾iC=1τC⋅M‾iC
and mean river mass and runoff:
M‾i+1R=(MiR-Ni+1⋅τR)⋅τR(ti+1-ti)⋅1-e-ti+1-tiτR+Ni+1⋅τR+MiC-Ni+1⋅τC(ti+1-ti)⋅τRτC-τR⋅τR⋅e-ti+1-tiτR-τC⋅e-ti+1-tiτC+τC-τR.
The observables, which allow a comparison to measured data, are as follows.
Average river runoff:R‾iR=1τR⋅M‾iR,corresponding to measured monthly runoff.
– Average total mass:M‾iT=M‾iC+M‾iR,corresponding to monthly GRACE data.
The equations Eqs. (10)–(20) are self-consistent, i.e. the
corresponding balance equations are fulfilled with the following:
Mi+1T(t-ti)-MiT(t-ti)+R‾i+1R(t-ti)=Ni+1.
For the single storage approach the above piecewise analytical solutions of
the cascaded storage approach (Eqs. 8–21) are used for τR≪τC (here τR=10-3 months). For this case the river network mass is negligible
compared to the catchment mass.
Properties and optimization performance
For the evaluation of the parameter optimization performance of the cascaded
storage approach an example with synthetic recharge as input is
investigated. This permits the quantification of the uniqueness and accuracy
of the parameter estimation undisturbed by noise. It also facilitates the
discrimination of errors in the calculation scheme itself and impacts
arising from undescribed processes when compared to real-world data. For an
application to GRACE measurements the main question is if and why the time
constants τC and τR can be determined independently by
an optimization versus anomalies in total mass and/or river runoff. Thus, in
order to understand the optimization results with respect to uniqueness the
general properties of the approach are presented and discussed first. For
the synthetic case a recharge time series of sinusoidal form with a period
of 12 arbitrary time units and length units with an amplitude and mean value
of 1 is used as the driving force, and the calculation is run until
equilibrium is reached. The example in Fig. 2 shows the effect of a non
negligible river network time constant τR=2.5 time units for a
catchment time constant τC=3 time units, which leads to an
increase in total mass MT(t)=MC(t)+MR(t) with respect to the average level and signal
amplitude and to a phase shift between total mass MT and river mass
MR, i.e. the corresponding river runoff RR.
Time series of recharge N, catchment mass MC, river network mass
MR and total masses MT for the synthetic case at equilibrium.
In order to describe the general behaviour of the mass and runoff time
series and their dependence on τC and τR, their
properties are summarized here in the form of statistical values for the
synthetic case with the sinusoidal recharge in equilibrium. This helps our understanding of why unique values for the time constants are achieved in the
parameter optimization process. The values of time constants τC
and τR used for the statistical description cover a wide range
from 0.1 to 100 time units and are combined independently.
Catchment and river mass
Based on the mean mass values, Eqs. (14), (16), (18), of each stress period
the long-term averages for the storage compartments are given by the following:
22aM‾C=N‾⋅τC,22bM‾R=N‾⋅τR,22cM‾T=N‾⋅(τC+τR).
For τR≪τC (here τR=10-3) the river network mass is negligible and the
solution corresponds to a single storage approach. For a non negligible
river network storage the given average values for total mass MT mean
that the effective “total” time constant is given by the sum of the
catchment and river time constants τT=τC+τR, which means that the total mass MT observed
by GRACE is bigger than the mass MC calculated for the catchments
alone. However, Eq. (22c) cannot be used for the determination of τT=τC+τR from GRACE
measurements directly as GRACE only provides mass anomalies.
The relative signal amplitudes (standard deviations normalized with those of
the respective input) of both the catchment mass MC or river mass
MR show the same functional form σMC/σN∼σMR/σRC= SD(MC) /N for the respective time constants
τC or τR (Fig. 3, τR=10-3)
with a monotonous increase to an asymptotic value σMC/σN∼σMR/σRC=2, which is reached at about one full period of the
input. The superposition of the signal amplitudes for the observable total
mass MT(t)=MCC(t)+MR(t)
leads to a complex behaviour for σMT/σN(τC, τR) (Fig. 3), if the river time constant τR is not negligible (τR=10-3) and
especially if it gets close to τC.
Signal amplitudes of total mass normalized by recharge: σMT/σN versus total mass time constant τT=τC+τR for different river
time constants τR.
Catchment and river runoff
The calculated long-term averages of the runoff contributions RC and
RR correspond to the ones of the water balance equations, Eqs. (4), (6),
given by the mean recharge and thus are not dependent on the time constants.
R‾R(t)=R‾C(t)=N‾
Thus, an observed long-term average of runoff does not permit the
determination of the time constant and hence the storage volume (Eq. 22).
The relative signal amplitudes of both catchment and river runoff
(normalized with the respective input σRC/σN and σRR/σRC) show the same
functional form corresponding to a single storage approach (Fig. 4, τR=10-3) and decrease monotonously with the
respective time constants τC and τR to an asymptotic
zero. However, the signal amplitude of the observable river runoff σRR/σN(τC+τR), normalized with recharge N, shows a deviations for different τC and τR with the same τC+τR
(Fig. 4).
Signal amplitudes in standard deviations for river runoff normalized by recharge: σRR/σN versus total mass time constant
τT=τC+τR for
combinations in (τC, τR).
Both observables, total mass and river runoff, show a non unique behaviour
with respect to combinations in (τC, τR) for the same
τT=τC+τR and
considerable deviations from the single storage approach (τR=10-3). Measurements of the signal amplitudes thus only
provide coarse estimates of the total time constant τT, yet do
not permit distinction between τR and τC and between
catchment and river network storage.
However, so far, only the signal amplitudes are examined, but not the
specific properties of the time series, i.e. the dynamic response to input
signals in form and phase. The convolution in the solution of the balance
equation, Eqs. (8) and (11), leads to a different phasing with respect to the
input N(t), which can be utilized for a separation of the respective time
constants.
Phasing
For the synthetic example with a sinusoidal recharge time series N(t) as
input the phasing ω of the different response signals is determined
by the fit of a sinusoidal function (Fig. 5). This facilitates the easy
determination of the phasing and thus the relative phase shift Δω between the signals. Masses and the related runoffs are in phase
for the same storage compartments (Eq. 15). For a negligible river network
time constant (τR=10-3) river runoff RR is
in phase with the catchment storage MC.
Phasing of river network mass with respect to recharge time series
displayed versus τC for different τR.
The functional form of the phasing ωMC for the catchment
mass MC or the corresponding runoff RC relative to recharge
N(t) (Fig. 5) can be empirically described by the monotonous function:
ωMC(τC)=ωmax1-e-τCλ,
with the empirical parameters ωmax=2.8 and
λ=2.7 and an error ε < ∼2 % relative to the maximum.
As the catchment runoff RC with the phasing ωMC
serves as input into the river system, the phasing of the river system with
respect to catchment runoff RC, which has the same functional form
as Eq. (24), is added on top of it (Fig. 5). The resulting phasing of the
river network storage or river runoff is thus given by a superposition in
the following form:
ωRR(τC,τR)=ωmax1-e-τCλ+ωmax1-e-τRλ
for any combination (τC, τR) and with the same
empirical parameters as in Eq. (24).
As total mass MT(t)=MC(t)+MR(t) is the superposition of the signals with the
respective amplitudes and phasing, the phasing of total mass MT(t) is
situated between catchment and the river system mass according to τR. This means that for non negligible river network mass (τR>0) a phase shift between total mass (GRACE)
and observed river discharge and also between total mass and modelled
catchment mass must occur. The phasing of total mass MT(t) for all
combinations (τC, τR) (Fig. 6) shows the same functional
form as ωMC and ωMR (Eqs. 24, 25) if displayed
versus the total time constant τT=τC+τR.
Phasing of total mass versus total time constant τT=τC+τR.
It can be approximated by the fitting function MT fit:
ωMT(τC,τR)=ωmax1-e-τC+τRλ,
with the empirical parameters ωmax=2.95 and
λT=3.2.
The phase shift between GRACE total mass and river runoff is thus given by the following:
Δω(τC,τR)=ωRR-ωMT=ωmax(1-e-τCλ)+ωmax(1-e-τRλ)-ωmax(1-e-τC+τRλ).
The empirical phase shift Δω from Eq. (27) corresponds to the
one determined by a phase adaption Δωadapt (Eqs. 38 and 39) of total mass and runoff within <∼5 % (see
Supplement). This in principle allows for a determination of τC
and τR separately from the adapted phase shift Δωadapt and the total mass time constant τT=τC+τR according to Eq. (27). However,
errors introduced by the linear interpolation used for the adaption of the
phase shift lead to a much lower accuracy than the parameter estimation via
the time series.
Parameter estimation
The analytical solutions for synthetic recharge time series permit the
evaluation of the uniqueness and accuracy of the parameter optimization for
given observables independent from limitations in the accuracy of numerical
schemes and independent from noise in real-world data sets. For given
combinations (τC, τR) the analytical solutions are
used as synthetic measurements and are fitted with the same algorithm in
order to retrieve the fit parameters (τC,
τR).
As the total mass MT (Eq. 20), and the phasing, (Eqs. 25–27), are
commutative in (τC, τR), either of the data ranges τR<τC or τR>τC has to be used for a unique optimization. This is
realized via an additional constraint in the optimization. For the
discussion here the condition τR<τC is used, which hydrologically reflects the more frequent situations
that the inundation volume is smaller than the catchment storage, but the
results can also be applied to τR>τC, which might be the case in flat areas with a dense river network
(such as the Amazon), which typically leads to temporarily inundated areas.
As absolute signal values are not relevant for the determination of the time
constant from runoff or not available for GRACE data, the optimization
versus the respective time series is based on signal amplitudes and the
phasing. Thus, for a unique determination of (τC,
τR) the following conditions have to be fulfilled.
Optimization versus runoff:
28σRR/σN(τ^C,τ^R)=σRR/σN(τC+τR)29ωRR(τ^C,τ^R)=ωmax1-e-τCλ+ωmax1-e-τRλ.
Optimization versus mass anomalies:
30σMT/σN(τ^C,τ^R)=σMT/σN(τC,τR)31ω^MT(τ^C,τ^R)=ωMT(τC+τR)=ωmax1-e-τC+τRλ.
With the constraints τR<τC or
τR>τC there is only one
(τC, τR) fulfilling
the respective conditions, thus leading to unique solutions. The
optimization delivers RMSE errors for the time series in the range 10-8-10-7 and estimated time constants (τC,
τR) with a relative error ε(τX)/τX which does not depend on absolute
values of (τC, τR) but on their ratio τR/τC (Fig. 7).
Relative error ε(τX)/τX of the time constants τC and τR for the cascaded storage approach with respect to optimizations
versus total mass MT or versus river runoff RR.
For the synthetic case relative errors ε(τX)/τX are very small (∼10-7 at τR/τC∼0) and show an
exponential increase to a maximum of ∼1 % at τR∼τC. The error for τR<τC is analogous to τR>τC and equal for an optimization versus
runoff or mass anomalies.
For catchments showing a phase shift between total mass and runoff the
description of the system by a single storage approach (τR=10-3) leads to a considerably higher relative error
ε(τX)/τX in the
estimated time constant τC∼(τC+τR) and thus also in drainable
storage volume. It follows a power function and corresponds to ε<10 % for τC<3 and ε>40 % for
τC>6. For this case the
optimization versus river runoff or mass anomalies leads to different total
time constants (relative difference ε>7 %
for τC>5). Even though this
might look like an acceptable result for τC<3, there are still inevitable deviations in signal
amplitudes (10 %–20 %) and phasing between the modelled and measured signals
for both total mass and river runoff time series.
It can be summarized that in contrast to the single storage approach the
cascaded storage approach permits the determination of both time constants
(τC, τR) independently in a unique, highly
accurate way for optimizations with respect to either total mass anomalies
or river runoff. However, it has to be mentioned that even though the
theoretical error in time constants remains below 1 % for τR∼τC, the ambiguity for τR<τC or τR>τC cannot be solved without further information on the volume
of the river network.
Fully data-driven determination of drainable storage volumes
For the case that river discharge is available for a sufficient period of
time the cascaded storage approach facilitates a simple determination of the
drainable water storage volumes both for the catchment and for the river
network directly from observations without the necessity of new model runs.
The two time constants (τC, τR) adapted during
a training period permit the quantification of the drainable water storage volumes
MT, MC and MR at other times directly from observations of
GRACE mass anomalies and river discharge. With a simple numerical adaption
of the phase shift Δω resulting from the time constants
(τC, τR) according to Eq. (27) a quite accurate
determination of the total drainable storage volume from measured river
discharge exclusively (or the other way round, of runoff from GRACE mass
anomalies) is possible.
These can be determined by the following calculations.
Long-term averages of drainable storage volumes from observed runoffRo.
32M‾simC=τC⋅R‾o,33M‾simR=τR⋅R‾o,34M‾simT=τT⋅R‾owithR‾o=N‾andτT=τC+τR
according to Eqs. (22a, 22b, 22c) and (23).
Time series of drainable storage volumes from GRACE and observed runoffRowithout the need for a phase adaption.MsimR(t)=τR⋅Ro(t),
(NSS0.961, NSR0.576, corrR0.859 vs. MR from Eq. 18),
MsimT(t)=dMT(t)+M‾T=GRACE(t)+M‾T=GRACE(t)+τT⋅N‾⋅=GRACE(t)+τT⋅R‾0,
(NSS0.973, NSR0.751, corrR 0.901 vs. MT from Eq. 20),
MsimC(t)=MT(t)-MR(t)=GRACE(t)+τT⋅R‾o-τR⋅Ro(t),
(NSS0.906, NSR-0.065,
corrR0.607 vs. MC from Eq. 16).
The simplified calculations directly based on observations lead to accurate
equivalences to the fully calculated time series of total storage and the river
network storage volumes MT and MR and to a reasonable description
of the catchment storage volumes MC.
Time series of total drainable storage volumesMT, directly from
observed runoff Roor simulated river runoffRsimRfrom GRACE with a numerical phase adaption ofΔω.
Use of the phase shift Δωadapt adapted between GRACE
and observed river runoff by a linear temporal interpolation (Riegger and
Tourian, 2014) permits a simple description of river runoff directly from
GRACE (Eq. 38) or of total drainable water storage MT directly from
observed runoff (Eq. 39) and corresponds to Δω from Eq. (27)
within <∼10 %. Both lead to very similar fitting
performances.
RsimR(ti)=1τT⋅(1-Δω)⋅GRACE(ti)+Δω⋅GRACE(ti-1)+τT⋅R‾o
(NSS0.943, NSR0.698, corrR0.864 vs. measured Ro,)
MsimT(ti)=τT⋅(1-Δω)⋅Ro(ti)+Δω⋅Ro(ti+1)
(NSS0.946, NSR0.483, corrR0.859 vs. GRACE)
For the representativeness of the fitting performance the fully
data-driven approach (Eqs. 35–39) is compared to the respective masses and
runoff from the cascaded storage approach applied to the Amazon basin (see
below) and not to synthetic data. The related calculations are accessible in
the Microsoft Excel workbook provided in the Supplement.
This performance means that the determination of the two time constants
(τC, τR) by the cascaded storage approach
during a sufficient training period facilitates a simple quantification of
drainable storage volume or runoff time series directly from measured river
discharge or GRACE anomalies. This provides a possibility to close data gaps
in river discharge or GRACE directly from measurements with high accuracy.
Application to the Amazon basin
The R–S diagram of the full Amazon basin shows a hysteresis (Fig. 1a)
corresponding to a phase shift, which can be interpreted as the time lag of
river discharge. The Amazon basin upstream from Obidos is situated in a fully
humid tropic environment with permanent, yet variable recharge and is large
enough (4 704 394 km2) for low noise levels in the signals of GRACE and
moisture flux divergence. With permanent recharge, flow contributions from
overland flow and groundwater cannot be distinguished in the discharge
curve. Also, on a spatial average over the full Amazon basin, with permanent
recharge the uncoupled storages (like soil water storage, open water bodies
etc.) are not time variant, i.e. there is no dry-out effect. Any
contribution from time-dependent, uncoupled storages could be recognized in
the R–S diagram as it would appear as a hysteresis, which does not
correspond to a time lag, or by the respective deviations in the scatter
plots of calculated versus measured runoff or storage volumes (see
the Supplement). This is not the case.
Generally recharge from different approaches and products can serve as input
to the system, such as the following:
N(t)=P(t)-ET(t),
from the hydrometeorological products precipitation P and actual
evapotranspiration ETa.
from the terrestrial water balance with monthly temporal derivatives of
GRACE measurements and measured river runoff Ro of the basin.
Here recharge (mm month-1) is taken either from the water balance (Eq. 42), or
from moisture flux divergence (Eq. 41), provided by ERA-INTERIM of ECMWF and
processed by the Institute of Meteorology and Climate Research, Garmisch,
Germany. For GRACE mass anomalies, data from GeoForschungsZentrum GFZ Potsdam
Release 5 are used in millimetre equivalent water height. Both are handled as
described in detail in Riegger and Tourian (2014). Their spatial resolution
limits the application of the approach to global scales ≫200000 km2. River discharge is taken from the ORE HYBAM
project (http://ore-hybam.org, last access: April 2016) and converted to runoff
(mm month-1) by normalization with the basin area. For a comparison of the calculated river network storage with observations from the Global
Inundation Extent from Multi-Satellite (GIEMS; Prigent et al., 2001), flood
area (km2) is used. As GRACE mass anomalies are most accurate for a
monthly time resolution at present, the other data sets are aggregated to a
monthly resolution as well. For the parameter optimization, time series of
river runoff and GRACE mass anomalies are used for the time period from
January 2004 until January 2009. Monthly runoff and the storage volume of
the basin and river network are calculated for Amazon based on different
recharge products here and optimized either versus runoff or GRACE mass
anomalies. The results calculated with recharge from the terrestrial water
balance optimized versus GRACE are shown in Figs. 8–10 for both (a) the
monthly signal and (b) the monthly residual (monthly value minus mean
monthly value) for January 2003–2009.
Time series of river runoff RmR for the Amazon and optimization versus GRACE for the signal (a) for the
residual (b).
Time series of total mass anomalies dMT for the Amazon and
optimization versus GRACE for the signal (a) for the residual (b).
Time series of river network storage MmR and inundated
area from GIEMS for the Amazon and optimization versus GRACE for the signal (a) for the residual (b).
The calculated river runoff RR, total mass anomaly dMT and river
network mass MR fit very well with the measured river runoff, GRACE and
the flooded area from GIEMS both with respect to the signal and the
de-seasonalized monthly residual.
The cascaded storage approach reproduces the phase shift between measured
runoff Ro and total mass dMT from GRACE. The
calculated river network mass MR of about 50 % of the total mass
MT for Amazon is proportional to observed runoff Ro without any
phase shift (Fig. 11).
R–S relationships for observed runoff Ro versus the mass
anomalies of GRACE, calculated total mass dMT and river network mass
dMR.
Calculated hydraulic time constants, mean values and signal amplitudes for
the absolute storage volumes are provided in Table 2 for the full Amazon
basin upstream from Obidos. In addition the performance of optimizations either
versus river runoff (column A) or versus GRACE (column B) and for different
recharge products (column D, E) is displayed. This shows that the
optimization versus different references leads to a very similar results
while the fitting performance for the two recharge products (columns A, B
and D, E) is quite different. For recharge from water balance (Eq. 42) the
resulting time constants and thus the storage masses differ in a range of
∼5 % for the different references while they vary
∼10 % for recharge from moisture flux divergence.
In order to illustrate the benefits of the cascaded versus a single storage
approach even in the fitting quality, results for a fixed τR=10-3, which correspond to a single storage, are shown
(column C, F) for different recharge products. With the single storage
approach – besides the much worse fitting performance – the resulting time
constant τT=τC+τR
is overestimated (corresponding to the investigations in Sect. 3) and the
modelled signal amplitude is about 20 % less than that measured from
GRACE. In addition a non negligible phase shift remains between the modelled
runoff and measured discharge.
The statistical characteristics are listed for calculated river
runoff RR, total mass MT, basin mass MC, river network
mass MR, observed river runoff Ro, GRACE mass anomalies and
flood areas from GIEMS using the following: root mean square error (RMSE) of simulated
versus measured values, Nash–Sutcliffe coefficient of the signal (NSS)
(simulated values versus long-term mean of measured values), Nash–Sutcliffe coefficient of monthly residuals (NSR) (simulated values versus monthly
mean of measured), correlation of simulated versus measured
signals (corrS), correlation of simulated versus measured monthly
residuals (corrR). Avg and SD are the long-term mean and standard deviations. The
prefix “d” is used for anomalies related to the long-term mean.
Results are compared for the different optimization references: runoff
(Ro) or GRACE for recharge from water balance (R+dM/dt) (A, B) and for
atmospheric input (-divQ) (D, E). The cascaded storage approach is compared to the single storage approach in (C, F).
The cascaded storage approach with recharge from the water balance (Eq. 42) leads to high-accuracy fits between calculated and measured river
runoff and total storage mass for the signals (NSSRR-Ro=0.96, NSS dMT- GRACE =0.98) and for the
residuals (NSRRR-Ro= 0.74, NSR dMT- GRACE =0.74; the respective calculations are available in the Excel workbook provided
in the Supplement).
The comparison of the water budgets for 14 different GHMs and LSMs (Getirana et
al., 2014) for the Amazon basin permits the results of the
cascaded storage approach to be sorted into those of the GHMs and LSMs (Fig. 14 of Getirana et al., 2014). With a Nash–Sutcliffe coefficient, NSR (R= with
respect to the mean seasonal cycle), of 0.74 and a correlation corrR=0.90 compared to an NSR of 0.58 and a correlation corrR=0.84 for
the best LSM, the cascaded storage approach outperforms the GHMs/LSMs for the Amazon basin upstream of Obidos. Even the fully data-driven approach (Eq. 39) leads to a performance comparable to the best GHM–LSM models tested by Getirana et al. (2014) with an NSR of
0.483 and a correlation corrR of 0.859 for simulated mass anomaly versus GRACE. The related calculations are accessible in the
Excel workbook provided in the Supplement.
This is partly seen as the result of the simplicity of the lumped approach
averaging out errors that emanate from the large number of different
processes described by the GHMs and LSMs. However, the main reason for the
better performance is seen in the quality of recharge data taken from the
water balance using GRACE and river runoff, as the use of moisture flux
divergence for this purpose leads to much worse performance.
The calculated river network mass MR of the Amazon varies in the range
of 40 %–65% of total mass MT with an average of ∼50 %,
corresponding to the values found by Paiva et al. (2013) and Papa et al. (2013) or ∼41 % by Getirana et al. (2017a). The correlation
versus the observed flood area from GIEMS is higher for the calculated river
network mass MR (0.96 for the signal and 0.76 for the monthly residual)
than for GRACE (0.92 and 0.65 respectively). The consistency of the
calculated river network mass (and the corresponding observed river runoff
RR=MRτR-1=0.742MR) with the flood
areas is seen much more clearly in the phasing (Fig. 12), which shows a clear
phase shift for GRACE versus GIEMS (see also Papa et al., 2008), yet none
for calculated MR. As Getirana et al. (2017a) already emphasized, only
an appropriate description of the river network storage permits a correct
description of the total storage in amplitude and phasing for a comparison
with GRACE.
GRACE mass, calculated river network mass dMR and observed
river runoff dRo versus flood area dGIEMS, all displayed as anomalies
(please consider dMR=dRRτR=1.53dRR).
Discussion
Distributed hydrological models use a lot of detailed local information in
order to address a large number of involved processes for each grid cell. In
this way they provide a spatially distributed and a very detailed
composition of the involved storages and flows. However, it is very
difficult to discriminate the respective processes locally with the
consequence that only their superposition can be compared to measurable data
like river discharge. This creates a kind of ambiguity between the different
contributions, thus losing some of the benefits of a detailed description.
As has also been pointed out by Getirana et al. (2017a), for a comparison
of the superposed storages to GRACE anomalies the river network storage
changes have to be quantified as well, as only the total storage changes are
measured by GRACE. This means that an appropriate description of the river
network storage and the time lag is an inevitable prerequisite for an
appropriate adaption of model parameters. A hydrodynamic modelling of the
river network facilitates the quantification of its storage to be sure, yet
it involves a real computational challenge.
The cascaded storage approach permits the quantification of the drainable storage volumes for both land masses and river network directly from GRACE and measured river discharge for gauged catchments. For ungauged catchments for which GRACE and recharge data can be used, it provides good estimates for the respective storages and also for the otherwise unknown river discharge. This is achieved by adapting only two parameters, the time constants. Neither
detailed information on local vegetation, surface, unsaturated/saturated zone, etc., and related flow processes nor a hydrodynamic modelling with detailed hydraulic information of the river network on river roughness, cross section, gradient or backwater effects is needed.
At present, the cascaded storage approach is limited to climatic and
physiographic conditions for which the hysteresis is completely explained by
a time lag, i.e. that no impacts of uncoupled storage components are visible
in the R–S diagram. For global coverage the cascaded storage approach has
to be extended by an explicit integration of coupled and uncoupled storage
compartments to account for other regional climatic and physiographic
conditions. The uncoupled storage components then have to be quantified
either by their absolute storage volume or by their relative contribution to
total storage.
As Riegger and Tourian (2014) have shown for boreal catchments, this can be
done by means of remote sensing and a conceptional description. Boreal
catchments are temporarily dominated by snow, leading to a huge hysteresis
due to a superposition of masses from fully coupled (liquid) and uncoupled
(solid) storage compartments. Remote sensing of the catchment snow coverage
by MODIS facilitates the separation of the coupled liquid storage
(proportional to river runoff) on the uncovered areas and the uncoupled
frozen part on the snow-covered areas. The coupled liquid storage determined
in this way actually constitutes a LTI system; i.e. the hysteresis can be
fully explained by a phase shift. This fulfils the prerequisites for the
cascaded storage approach and thus permits an application to boreal
catchments as well. As a consequence, the principle of the cascaded storage
approach is not limited to fully humid climatic conditions. It permits an
application to other climatic regions as well, provided that the coupled and
uncoupled storage compartments can be separated.
The description of monsoonal regions for example, which play an important
role in the global water budget, is a considerable challenge. For these
regions with seasonally dry periods, high-precipitation events during the wet
season lead to distinct runoff in parallel from overland and groundwater,
with different time constants τSurface and τGW, and to
time-dependent uncoupled storage compartments like soil or isolated open
water bodies, which do not contribute to discharge. All these storages have
to be addressed for an adequate description of the drainable storage
volumes. The uncoupled storage compartments have to be quantified by remote
sensing (soil moisture and open water body altimetry from satellites) and
subtracted from the total catchment mass measured by GRACE, which of course
is a major task.
Conclusions
The test of the cascaded storage approach with synthetic recharge data has
shown that the parameter optimization versus either mass anomalies or runoff
reproduces the time constants (τC, τR) for
both the catchment and the river network in a unique way with high
accuracy, yet with an ambiguity for τR<τC or τR>τC, and thus in
the related storage volumes. This problem can only be solved by reasonable
assumptions or, preferably, by additional information on the volume of river
network or flood areas, which can be taken from ground-based observations or
remote sensing like GIEMS flood areas and water levels from altimetry.
Numerical tests have also shown that the description of a system (showing a
phase shift) by a single storage approach can only address the total
drainable storage and thus leads to phasing differences between the
calculated and measured runoff or storage and to considerable errors in the
time constant of the total system.
The application to the full Amazon basin shows that the system behaviour
including the time lag can be described by a simple conceptual model with a
catchment and a river network storage in sequence and an adjustment of only
two parameters, the time constants. The storage amplitudes for the total
drainable water storage and the time lag to runoff are described with high
precision. Calculated river network volume and the observed flood area
correspond to GIEMS observations and the newest model results (including river
routing; Getirana et al., 2017a) and are in phase with river discharge. This
independent quantification of the river network volume permits an
investigation of the relationship between flood areas, flood volumes, river
runoff and calculated river network with its additional information and
might provide insights into river hydraulics, i.e. routing times and the
mass–area–level relationships of flooded areas.
As the optimization performance is comparable for either reference, the
observed river runoff or GRACE anomalies, a calculation with given recharge
and an optimization versus measured GRACE data can be used to determine
both the river discharge and the drainable storage volumes, even for
ungauged basins. For these cases the availability of accurate recharge data
determines the accuracy of runoff and storage calculations at present.
However, for ungauged basins the use of moisture flux divergence still
provides quite acceptable results based on remote sensing and atmospheric
data exclusively.
For the case that river discharge is available for a sufficient period of
time in order to adapt the two time constants (τC, τR) sufficiently, the cascaded storage approach facilitates a simple
“fully data-driven” determination of the drainable water storage volumes
MT(t), MC(t) and MR(t) at other times directly from
observations of GRACE or river discharge without the necessity of new model
runs. This permits data gaps to be closed or even forecasts within the
period of the time lag to be made.
As the spatial resolution of GRACE and the accuracy of moisture flux
divergence is limiting the applicability of the cascaded storage approach to
global-scale catchments at the moment, any improvement in the spatial or temporal resolution and accuracy of GRACE and hydrometeorological data
products will tremendously increase the number of catchments which can be
described by this approach in future.
Abbreviations in the mathematical descriptions
AbbreviationDescriptionUnits: general or for applicationNRecharge = (precipitation - actual evapotranspiration)Volume area-1 time-1 (mm month-1)MCStorage mass catchmentMass density in equivalent water height (mm)τCTime constant catchmentTime unit (month)RCRunoff catchmentVolume area-1 time-1 (mm month-1)ωMCPhasing catchment massTime unit (month)MRStorage mass river networkMass density in equivalent water height (mm)τRTime constant river networkTime unit (month)RRRunoff river networkVolume area-1 time-1 (mm month-1)ωRRPhasing river runoffTime unit (month)MTStorage mass total systemMass density in equivalent water height (mm)τTTime constant total systemTime unit (month)ωMTPhasing mass total systemTime unit (month)RoObserved river runoffVolume area-1 time-1 (mm month-1)GRACEGRACE mass anomalyMass density in equivalent water height (mm)GIEMSFlood areaArea (km2)Prefix “d”Indicates signal anomalies from long-term mean (anomalies)Suffix “m”Indicates mean values on the intervals
Data availability
In the Supplement calculations and data are provided in an Excel workbook for the synthetic case and for the Amazon catchment.
The supplement related to this article is available online at: https://doi.org/10.5194/hess-24-1447-2020-supplement.
Competing interests
The author declares that there is no conflict of interest.
Special issue statement
This article is part of the special issue “Integration of Earth observations and models for global water resource assessment”. It is not associated with a conference.
Acknowledgements
The author would like to thank Nico Sneeuw and Mohammad Tourian from the
Institute of Geodesy, Stuttgart, for the handling of GRACE data; Harald Kunstmann and Christof Lorenz from the Institute of Meteorology and Climate
Research, Garmisch, for the provision of moisture flux data; and Catherine
Prigent, Observatoire de Paris, for the provision of flood areas from the
GIEMS project.
Review statement
This paper was edited by Stan Schymanski and reviewed by Andreas Güntner and three anonymous referees.
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