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

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 (

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

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

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 ET

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 >

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 (

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

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

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

The accurate description of the

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

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.

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

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

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:

The water balance equation, Eq. (4), for the catchment is generally solved
by the following:

For recharge

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

The mixed term in Eq. (12) and thus the total mass are commutative in (

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
(

The observables related to measurements by GRACE and discharge from gauging
stations are the total mass anomaly d

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

Average river runoff:

– Average total mass:

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

Time series of recharge

In order to describe the general behaviour of the mass and runoff time
series and their dependence on

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:

For

The relative signal amplitudes (standard deviations normalized with those of
the respective input) of both the catchment mass

Signal amplitudes of total mass normalized by recharge:

The calculated long-term averages of the runoff contributions

The relative signal amplitudes of both catchment and river runoff
(normalized with the respective input

Signal amplitudes in standard deviations for river runoff normalized by recharge:

Both observables, total mass and river runoff, show a non unique behaviour
with respect to combinations in (

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

For the synthetic example with a sinusoidal recharge time series

Phasing of river network mass with respect to recharge time series
displayed versus

The functional form of the phasing

As the catchment runoff

As total mass

Phasing of total mass versus total time constant

It can be approximated by the fitting function

The phase shift between GRACE total mass and river runoff is thus given by the following:

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 (

As the total mass

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 (

Optimization versus runoff:

Optimization versus mass anomalies:

Relative error

For the synthetic case relative errors

For catchments showing a phase shift between total mass and runoff the
description of the system by a single storage approach (

It can be summarized that in contrast to the single storage approach the
cascaded storage approach permits the determination of both time constants
(

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 (

These can be determined by the following calculations.

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

Use of the phase shift

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
(

The

Generally recharge from different approaches and products can serve as input
to the system, such as the following:

from the hydrometeorological products precipitation

from the terrestrial water balance with monthly temporal derivatives of
GRACE measurements and measured river runoff

Here recharge (mm month

Time series of river runoff

Time series of total mass anomalies d

Time series of river network storage

The calculated river runoff

The cascaded storage approach reproduces the phase shift between measured
runoff

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

In order to illustrate the benefits of the cascaded versus a single storage
approach even in the fitting quality, results for a fixed

The statistical characteristics are listed for calculated river
runoff

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 (NS

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, NS

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

GRACE mass, calculated river network mass d

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

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

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 (

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 (

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.

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:

The author declares that there is no conflict of interest.

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.

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.

This paper was edited by Stan Schymanski and reviewed by Andreas Güntner and three anonymous referees.