Knowledge of
spatio-temporal rainfall patterns is required as input for distributed
hydrologic models used for tasks such as flood runoff estimation and
modelling. Normally, these patterns are generated from point observations on
the ground using spatial interpolation methods. However, such methods fail in
reproducing the true spatio-temporal rainfall pattern, especially in data-scarce regions with poorly gauged catchments, or for highly dynamic,
small-scale rainstorms which are not well recorded by existing monitoring
networks. Consequently, uncertainties arise in distributed rainfall–runoff
modelling if poorly identified spatio-temporal rainfall patterns are used,
since the amount of rainfall received by a catchment as well as the dynamics
of the runoff generation of flood waves is underestimated. To address this
problem we propose an inverse hydrologic modelling approach for stochastic
reconstruction of spatio-temporal rainfall patterns. The methodology combines
the stochastic random field simulator Random Mixing and a distributed
rainfall–runoff model in a Monte Carlo framework. The simulated
spatio-temporal rainfall patterns are conditioned on point rainfall data from
ground-based monitoring networks and the observed hydrograph at the catchment
outlet and aim to explain measured data at best. Since we infer a three-dimensional input variable from an integral catchment response, several
candidates for spatio-temporal rainfall patterns are feasible and allow for an
analysis of their uncertainty. The methodology is tested on a synthetic
rainfall–runoff event on sub-daily time steps and spatial resolution of
1 km

The importance of spatio-temporal rainfall patterns for rainfall–runoff (RR)
estimation and modelling is well known in hydrology and has been addressed
by several simulation studies, especially since distributed hydrologic
models have become available. Many of those studies demonstrated the
effect of resulting runoff responses for different spatial rainfall
patterns

In general, rainfall monitoring networks based on point observations on the ground (station data) require interpolation methods to obtain spatio-temporal rainfall fields usable for distributed hydrologic modelling. Traditional interpolation methods fail in reproducing the true spatio-temporal rainfall pattern, especially for (i) data-scarce regions with poorly gauged catchments and low network density; (ii) highly dynamic, small-scale rainstorms which are not well recorded by existing monitoring networks; and (iii) catchments which are partly covered by rainfall. Consequently, uncertainties are associated with poorly identified spatio-temporal rainfall patterns in distributed rainfall–runoff-modelling since the amount of rainfall received by a catchment as well as the dynamics of runoff generation processes are typically underestimated by current methods.

The effects of poorly estimated spatio-temporal rainfall fields are
visible in particular for semiarid and arid regions, where rainstorms
show a great variability in space and time and the density of ground-based
monitoring networks is sparse compared to other regions

To address the inherent uncertainties described above, stochastic
rainfall generators are used intensively to create spatio-temporal
rainfall inputs for distributed hydrologic models to transform rainfall
into runoff. A large amount of literature exists describing different
approaches for space–time simulation of rainfall fields, including
multi-site temporal simulation frameworks

Applications of spatio-temporal rainfall simulations together with
hydrologic models are straightforward Monte Carlo types, where
a large number of potential rainfall fields are generated driven by
stochastic properties of observed rainstorms or longer time series.
These fields are used as inputs for distributed hydrologic model simulations
to investigate the impact of certain aspects of rainfall like uncertainty
in measured rain depth, spatial variability, etc., on simulated catchment
responses. Rainfall simulation applications are performed in unconditional
mode (reproducing rain field statistics only) or conditional mode,
where observations (e.g. from rain gauges) are reproduced too. The
latter are commonly used for investigating the effect of spatial variability
using fixed total precipitation and variations in spatial patterns

On the other hand, inverse hydrologic modelling approaches have been
developed to estimate rainfall time series based on observed streamflow data. Those approaches require either an inversion of the underlying
mathematical equations for the non-linear transfer function

The goal here is an event-based reconstruction of spatio-temporal
rainfall patterns which best explains measured point rainfall data and catchment
runoff response. For that we looked for potential candidates
for rainfall fields for sub-daily time steps and spatial resolution
of 1 km

After this introduction the methods are described in Sect. 2. It gives an overview of the methodology and further details for the applied rainfall–runoff model, the Random Mixing and its application for rainfall fields. Section 3 aims to test the methodology. A synthetic test site is introduced which is used to demonstrate and discuss (i) the limits of common hydrologic modelling approaches (using rainfall interpolation) and (ii) the shortcomings of rainfall simulations which are not conditioned on the observed runoff. In contrast, the functionality of the inverse hydrologic modelling approach is illustrated and discussed. In Sect. 4, the inverse hydrologic modelling approach is applied for real-world data by an example of an arid mountainous catchment in Oman. The test site is introduced and results are shown and discussed. Finally, summary and conclusions are given in Sect. 5.

The methodology described here can be characterized as an inverse
hydrologic modelling approach. It aims to infer potential candidates
for the unknown spatio-temporal rainfall patterns from runoff observations
at the catchment outlet, known parameterisation of the rainfall–runoff
model, and rain gauge observations. The approach combines a grid-based
spatially distributed rainfall–runoff model and a conditional random
field simulation technique called Random Mixing

A simple spatially distributed rainfall–runoff model is used
as transfer function to portray the non-linear transformation of spatially
distributed rainfall into runoff at catchment outlets. The model is
dedicated to describe rainfall–runoff processes in arid mountainous
regions, which are mostly based on infiltration excess and Hortonian
overland flow. The model is working on regular grid cells in event-based modes. It is parsimonious in the number of
parameters, considers transmission losses but has no base flow component.
Pre-state information
at the beginning of an event is neglected since runoff processes mostly start
under dry conditions

More specifically, only simple approaches known from hydrologic textbooks
for the simulation of single rainfall–runoff events (no long-term
water balance) are used

Random Mixing is a geostatistical simulation approach. It uses copulas
as spatial random functions

The goal of the inverse hydrologic modelling approach presented herein
is to find a conditional precipitation field

Note that

In order to find such a precipitation field

Flowchart of the Random Mixing algorithm for inverse hydrologic modelling.

Using the given observations

As a next step we assume that the field

As a next step, unconditional standard normal random fields

Once a solution with an acceptable

The next step is to simulate fields

It ensures that

To also honour the observed runoff defined in Eq. (

To test the ability of the methodology a synthetic example was designed.
The example consists of a synthetic catchment partly covered by rainfall.
The synthetic catchment has a size of 211 km

Furthermore, 10 different cells were selected from the spatio-temporal
rainfall patterns to represent virtual monitoring stations of rainfall.
They were chosen in a way that the centre of the event is not recorded.
They are designated as the known “observed” rainfall

Topography, watershed, and observation network of the synthetic catchment.

Rainfall amounts of the synthetic rainfall event. Virtual monitoring stations are marked by crosses.

Time series of rainfall intensities at virtual monitoring stations.

Interpolated rainfall amounts per event by using data of virtual monitoring stations.

At first, hourly rainfall data from virtual monitoring stations were
used to interpolate the spatio-temporal rainfall patterns on a regular
grid of 1 km by 1 km cell size by using the inverse distance method,
which is quite common in hydrologic modelling. Afterwards, the response
of the synthetic catchment was calculated by the RR model. Figure

The Random Mixing approach was used to simulate 200 different spatio-temporal
rainfall patterns conditioned on the virtual rainfall monitoring stations
only. Resulting runoff simulations are displayed in Fig.

Runoff simulations based on simulated spatio-temporal rainfall patterns conditioned at rainfall point observations only (grey graphs) compared to its mean (red graph), runoff observation (blue graph), and simulation based on interpolated rainfall patterns (green graph).

The inverse modelling approach was used to simulate 107 different spatio-temporal
rainfall patterns which are conditioned on the virtual rainfall and
runoff monitoring stations, and runoff simulation results better than
NSE values of 0.7. Afterwards a refinement was carried out by
selecting only those simulations with nearly identical runoff simulation
results compared to observations. These simulations are characterized
by NSE values larger than 0.995. Figure

Event-based rainfall patterns conditioned
at rainfall point observations only for the top three runoff simulations
in Fig.

Comparison of hydrographs for the synthetic catchment shown by the observed runoff (blue) and rainfall–runoff simulation results based on interpolated rainfall patterns (green), a simulated ensemble of spatio-temporal rainfall patterns conditioned at rainfall and runoff observations (grey) and their mean value (red), and mean ensemble rainfall patterns (black).

However, the inference of a three-dimensional input variable by using
an integral output response results in a set or ensemble of different
solutions. Rainfall amounts of the selected 20 realisations above
20 mm event

Deriving an average rainfall pattern by calculation of the mean value
per grid cell over all realisations of the ensemble for each time
step, a smoother pattern is obtained, which looks more similar to
the true one but has smaller rainfall intensities. Using this mean
ensemble pattern for calculating the runoff response leads to an underestimation
of the observed hydrograph as shown by the black hydrograph in Fig.

In addition, data of the virtual monitoring stations (the observation) have been always reproduced and are equal for each rainfall simulation. This means that each realisation reproduces the point observation of rainfall without any uncertainty. Only the grid points between the observation differ within the three-dimensional rainfall field and contain the stochasticity given by rainfall simulations conditioned on the observed values. In this context, the ensemble can be used as a partial descriptor of the total uncertainty. It describes the remaining uncertainty of precipitation if all available data are exploited under the assumption of error-free measurements, reliable statistical rainfall models, and known hydrologic model parameters.

Selected realisations of spatial
rainfall amounts per event with similar performance in resulting runoff,
obtained by the inverse modelling approach for simulating spatio-temporal
rainfall pattern:

The real-world example is taken from the upper Wadi Bani Kharus in
the northern part of the Sultanate of Oman. It is the starting point
for the present study and part of our multi-year research on hydrologic
processes in this region. The headwater under consideration is the
catchment of the streamflow gauging station of Al Awabi, with an area
of 257 km

Real-world case study: catchment of gauge Al Awabi and sub-daily monitoring network for runoff and rainfall.

Rainfall amounts and altitudes of rainfall gauging stations from 12 February 1999.

The real-world data example was performed for the runoff event from
12 February 1999 with an effective rainfall duration of 3 h.
The simulated runoff for the interpolated rainfall pattern shows an
underestimation of the peak discharge as well as a time shift of the
peak arrival time compared to the observation (Fig.

Comparison of hydrographs for the real-world catchment shown by the observed runoff (blue) and rainfall–runoff simulation results based on interpolated rainfall patterns (green), a simulated ensemble of spatio-temporal rainfall patterns conditioned at rainfall and runoff observations (grey) and their mean value (red), and mean ensemble rainfall patterns (black).

Differential maps of spatio-temporal rainfall patterns for three consecutive time steps (simulation – interpolation).

An inverse hydrologic modelling approach for simulating spatio-temporal rainfall patterns is presented in this paper. The approach combines the conditional random field simulator Random Mixing and a spatial distributed RR model in a joint Monte Carlo framework. It allows for obtaining reasonable spatio-temporal rainfall patterns conditioned on point rainfall and runoff observations. This has been demonstrated by a synthetic data example as well as a real-world data example for single rainstorms and catchments which are partly covered by rainfall.

The proposed framework was compared to the methods of rainfall interpolation and conditional rainfall simulation. Reconstruction of event-based spatio-temporal rainfall patterns has been feasible by the inverse approach, if runoff observation and catchment's spatial drainage characteristic represented by the RR model with spatial distributed travel times of overland flow are considered. As shown by the synthetic example, the rainfall pattern obtained by interpolation did not match the observed rainfall field and runoff. If rain gauge observations do not portray the rain field adequately, a “good” interpolation result in the least-square sense is not a solution of the problem. This is the case in particular for small scale rainstorms with high spatio-temporal rainfall variability and/or rainfall data scarcity due to insufficient monitoring network density. By rainfall simulations conditioned on rain gauge observation only, reasonable spatio-temporal rainfall fields are obtained, but with a wide spread in resulting runoff hydrographs. A large number of simulated rainfall fields is required to find those realisations which match the observed runoff, since the amount of possible conditioned rainfall fields is much higher than the amount of rainfall fields matching point observation and runoff. By applying optimisation, rainfall fields are conditioned on discharge too, and appropriate candidates for spatio-temporal rainfall patterns can be identified more reliably, faster, and with reduced uncertainty.

The inference of a three-dimensional input variable by using an integral output response results in a set of possible solutions in terms of spatio-temporal rainfall patterns. This ensemble is obtained by repetitive execution of the optimisation step within the Monte Carlo loop. It can be considered as a descriptor of the partial uncertainty resulting from spatio-temporal rainfall pattern estimates (under the assumption of error-free measurements, reliable statistical rainfall models, and known hydrologic model parameters). Realisations of the ensemble vary in rainfall amounts, intensities, and spatial extent of the event, but they reproduce the point rainfall observation exactly and yield to similar runoff hydrographs. This allows for deeper insights in hydrologic model and catchment behaviour and gives valuable information for the reanalysis of rainfall–runoff events, since rainstorm configurations leading to similar flood responses become visible. As shown in the example, operating with an ensemble mean is less successful in matching the runoff observation compared to an application of the whole ensemble due to smoothing effects.

The approach is also applicable under data-scarce situations as demonstrated by a real-world data example. Here, the flexibility of the approach becomes visible, since simulated rainfall patterns also allow for overcoming a shift in the timing of runoff. Therefore, the approach can be considered as a reanalysis tool for rainfall–runoff events, especially in regions where runoff generation and formation are based on surface flow processes (Hortonian runoff) and in catchments with wide ranges in arrival times at catchment outlets such as mountainous regions or distinct drainage structures, e.g. urban and peri-urban regions.

Nevertheless, further research and investigations are required. Examples
presented in this paper are based on an hourly time resolution and
1 km

The proposed framework is a first step that only aims at reconstructing spatio-temporal rainfall patterns under the assumption of fixed hydrologic model structure and parameters. Certainly, hydrologic model uncertainty is of importance. But instead of changing the model to fit the observed discharge, we estimate rainfall fields which fit the model and the discharge by doing reverse hydrology. As such plausible rainfall fields can be identified, the corresponding model and the rainfall field is plausible. Thus, the framework can be applied to proof hypothesis about hydrologic model selection or to explain extraordinary rainfall–runoff events by using a well calibrated, spatial distributed hydrologic model for the catchment of interest. In this context, further research is dedicated to providing a common interface within the Monte Carlo framework to exchange the hydrologic model and allow for broader use within the community. Also, further sources of uncertainties (e.g. model parameters, observations) need to be considered to contribute for the solution of the hydrologic modelling uncertainty puzzle.

All data (except for confidential data) can be requested via email (jens.grundmann@tu-dresden.de).

The supplement related to this article is available online at:

JG and AB conceived and designed the study. JG and SH performed the analysis and wrote the paper. AB contributed to the interpretation of the results and commented on the paper.

The authors declare that they have no conflict of interest.

The corresponding author wishes to thank the Ministry of Regional Municipalities and Water Resources of the Sultanate of Oman for providing the data for the real case study and supporting the joint Omani–German IWRM-APPM initiative. Research for this paper was partly supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, no. 403207337, BA 1150/24-1) and partly by the Energi Simulation Program. Furthermore, we acknowledge support by the Open Access Publication Funds of the SLUB/TU Dresden. Edited by: Nadav Peleg Reviewed by: two anonymous referees