We introduce topological restricted maximum likelihood (TopREML) as a method to predict runoff signatures in ungauged basins. The approach is based on the use of linear mixed models with spatially correlated random effects. The nested nature of streamflow networks is taken into account by using water balance considerations to constrain the covariance structure of runoff and to account for the stronger spatial correlation between flow-connected basins. The restricted maximum likelihood (REML) framework generates the best linear unbiased predictor (BLUP) of both the predicted variable and the associated prediction uncertainty, even when incorporating observable covariates into the model. The method was successfully tested in cross-validation analyses on mean streamflow and runoff frequency in Nepal (sparsely gauged) and Austria (densely gauged), where it matched the performance of comparable methods in the prediction of the considered runoff signature, while significantly outperforming them in the prediction of the associated modeling uncertainty. The ability of TopREML to combine deterministic and stochastic information to generate BLUPs of the prediction variable and its uncertainty makes it a particularly versatile method that can readily be applied in both densely gauged basins, where it takes advantage of spatial covariance information, and data-scarce regions, where it can rely on covariates, which are increasingly observable via remote-sensing technology.

Regionalizing runoff and streamflow for the purposes of
making predictions in ungauged basins (PUB) continues to be one of the
major contemporary challenges in hydrology. At global, regional and local
scales only a small fraction of catchments are monitored for streamflow

There are a number of approaches to predicting runoff in ungauged catchments,
including process-based modeling

Spatial correlation is generally problematic for linear model predictions,
including the multiple regression approaches commonly used for
regionalization. For example if these models predict a hydrologic outcome,

There are several techniques available to address spatially correlated data.
Within PUB, kriging-based geostatistical
methods

The first suite of methods posits the existence of an underlying point-scale
process, which is assumed to have a spatial auto-correlation structure that
allows kriging to be applied. Because the runoff point-scale process is only
observed as a spatially integrated measure made at specific gauged locations
along an organized network of streams, the spatial auto-correlation structure
of the point-scale process cannot itself be observed. Block-kriging
approaches

Geomorphological considerations of the topology of a river network generally
focus on the channels, and lead to an intuitive conceptualization that
topological interpolation should focus on runoff correlations along flow
paths. The second type of approach embraces this topological structure. It
does not consider a point-scale runoff generation process, but instead models
the hillslope-scale runoff delivery process to the channel network as a
uni-dimensional directed tree

Conceptual flow propagation model. (i) Runoff is generated continuously
by a spatially distributed point process and drained to the stream
network. (ii) When monitored by stream gauges, runoff is spatially integrated
over the corresponding catchment and temporally averaged at the chosen observation
frequency (e.g., daily streamflow). (iii) The model conceptualizes the catchments
as isolated drainage areas (IDA) (A

Inspired by both types of approaches, here we present a method based on the
use of linear mixed models to generate a BLUP for hydrological variables on a
flow network. Rather than using a kriging estimator, we adopt a
REML framework

We first derive the TopREML estimator and its variance for mass conserving
(i.e., linearly aggregated) variables, with extensions to some
non-conservative variables (Sect.

Linear models can be used to make predictions about hydrological
variables along a network, provided that the models explicitly address the
effects of network structure. A mixed linear model approach provides a
suitable framework for this accounting. In this framework, the effects of
observable features on the hydrological outcome are assumed to be independent
of the network, and retain their influence independently, as so-called “fixed
effects”. The role of spatial structure is assumed to lead to correlation
specifically in the residuals

In the linear mixed model framework, the propagation of
hydrological variables through the flow network introduces topological
effects into the covariance structure of that variable. Firstly, linearly
propagated variables, such as annual specific runoff, are discussed.
Nonlinearly propagating variables can in some cases be transformed to allow
the linear solutions to be used (as outlined in Sect.

With these assumptions, observations of

Here we assume that the area-averaged process

With this background, the covariance matrix of

The restricted maximum likelihood approach partitions the likelihood of

Once the variance components

The empirical best linear unbiased prediction of

The variance of the TopREML prediction error can be expressed as

Unlike mean specific runoff, numerous streamflow signatures
(e.g., runoff frequency or descriptors of the recession behavior) are
non-conservative and cannot be expressed as linear combinations of their
values in upstream sub-catchments. In such conditions the derivations in
Sect.

For instance, runoff frequency

A similar reasoning can be applied to predict recession parameters. For
example, the exponential function

The computational implementation of TopREML in R

Taxonomy of the compared regionalization approaches.

Empirical correlograms of the mean specific summer flow recorded at the 57 gauges of the Austrian data set. Distance has a different effect on the correlation between flow-connected (black circles) and flow-unconnected (white triangles) gauges. Both correlograms are well fitted by an exponential function but the spatial correlation range doubles when gauges are flow connected. Both empirical correlograms are constructed using 5 km bins.

Observed streamflow data are used to evaluate the ability of
TopREML to predict streamflow signatures in ungauged basins. The assessment
is based on leave-one-out cross-validations, where the tested model is
applied to predict runoff at one basin based on observations from all the
other basins. After predicting runoff at all available basins in that manner,
the model is evaluated based on its mean absolute prediction error.
Streamflow variables from 57 catchments in Upper Austria

Catchment characteristics of the case studies.

The predictive ability of TopREML was evaluated on (a) specific annual runoff
in Nepal, (b) wet season runoff frequency in Nepal and (c) average summer
streamflow in Austria. The performance of TopREML (TR) was compared to five
other widely used regionalization methods: sample mean (LM

Location of the gauges and related catchments included in the cross-validation analyses in Upper Austria

Conventional geostatistical methods predict runoff by weighing observations
from surrounding basins based on their geographic distance. TopREML also
incorporates the topology of the stream network by including or excluding
basins based on their flow-connectedness. This adds topological information
to the determination of the covariance structure of runoff, at the expense of
discarding information that could be derived from correlations between
spatially proximate regions that are not connected to the gauge of interest
by a flow path. Assessing the net benefits of accounting for network effects
requires being able to control the topology of the network, and thus requires
numerical simulations. A series of Monte Carlo experiments as described in
Fig.

Monte Carlo generation procedure: (i) a spatially correlated Gaussian
field with an exponential covariance function (mean = 30, partial sill = 8,
nugget = 2, range = 3) is generated along a 7

A key advantage of the REML framework is its ability to
avoid the downward bias in the covariance function that affects kriging-based
methods (including Top-kriging) when external trend coefficients are
simultaneously estimated. This bias particularly affects the prediction of
the variance. Again, empirical cross-validation analysis does not allow an
assessment of this bias, because the observation data sets used contained only
one observation per location. Numerical simulations, however, allow many
realizations of the underlying stochastic process to be made at each
location, and thus allow the prediction variance to be compared with the
numerical variance. We evaluate TopREML's ability to predict variances (and
therefore evaluate prediction uncertainties) at ungauged locations using the
Monte Carlo procedure on the synthetic catchments described in Fig.

Results of the comparative cross-validation analyses of

Basin-level predictions of the considered signatures are
presented in Fig.

Figure

In contrast, the analysis revealed significant spatial effects for both
runoff frequency in Nepal, which has a much larger spatial correlation range
than annual streamflow (426 km – presumably set by meteorology and the
correlation range of storm events;

Results of the Monte Carlo experiments.

Results from the Monte Carlo analysis are presented in Fig.

Figure

In Fig.

Cross-validation outcomes suggest that TopREML is an attractive operational
tool for predicting streamflow in ungauged basins. The method performs as well
as the best alternative approach in the prediction of the considered runoff
signatures in Nepal and Austria, and significantly outperforms Top-kriging in
the prediction of modeling uncertainties in the numerical analysis.
Two distinguishing features of TopREML are responsible for these encouraging
results. First, TopREML incorporates the topology of the stream network by
restricting correlations to runoff observed at flow-connected catchments.
This allows TopREML to explicitly model the higher correlation in streamflow
anticipated along channels, but comes at the expense of discarding
correlations with neighboring, but not flow-connected catchments. Such
correlations can, for instance, be driven by large-scale weather patterns.
This tradeoff was investigated in a Monte Carlo analysis showing that
modeling performance increases more rapidly when including distant
flow-connected basins (slope in Fig.

TopREML also has considerably lower computational requirements than
Top-kriging, both in terms of input data and optimization complexity. Unlike
Top-kriging, where watershed polygons are necessary inputs for the
regularization procedure, vectors are not fundamentally indispensable for
TopREML. Indeed, TopREML does not rely on a distributed point process but
assumes homogenous IDAs. It follows that its only fundamental data
requirement is a table (i.e., a data.frame) of IDAs displaying the observed
regionalization variable and the area, centroid coordinates and network
position (i.e., own ID and downstream ID) of the IDA. When considering
runtime, both methods rely on numerical optimization, but Top-kriging uses it
to back calculate the point semi-variogram in its regularization procedure.
This may substantially increase the dimensionality of the optimization task,
depending on the grid resolution chosen for the discretization of the
catchment areas, which in turn has a highly significant effect on prediction
performances

Despite these encouraging results, TopREML is subject to stringent linearity
assumptions on the nature of the regionalized runoff signature. The predicted
variable should aggregate linearly both on hillslope surfaces and at channel
junctions that are subject to mass conservation. This limitation also affects
block-kriging approaches, as pointed out by

Global distribution of factors affecting model selection.

The regionalization methods assessed in the cross-validation analysis range
from simple linear regressions with strong independence assumptions, to
complex geostatistical methods that allow for both spatial and topological
correlations. Results indicate that while complex methods perform best in
general, there seems to be a threshold beyond which increasing the
complexity of the statistical method does not significantly improve the
prediction performance: while a linear model is better than a simple average
for the prediction of annual streamflow in Nepal (Fig.

Under these conditions, the selection of the optimal method is driven by the interplay between the layout of the gauges and the spatial correlation range of the considered runoff signature. A dense network of flow gauges is necessary for geostatistical methods to properly estimate the semi-variogram and improve on predictions from linear regressions – the case studies suggest that the mean distance between the gauges must be on the order of half the spatial correlation range of the runoff signature. Sparser gauge densities do not allow geostatistical methods to capture spatial correlations and their prediction is effectively driven by the deterministic components of the model, i.e., the intercept and (when available) observable features.

The probability density functions assumed in Eqs. (

An interesting tradeoff arises if observable features are themselves spatially correlated and explain a significant part of the spatial correlation of the predicted variable. Including these observable features in the model reduces the correlation scale of the residuals, possibly crossing the threshold below which geostatistics are not the most parsimonious approach. In Nepal, controlling for rainfall reduced the spatial correlation range of annual streamflow from 21.6 to 7 km – well below the mean distance between the gauges (13.9 km). In that case there is a tradeoff between relying on observable features or variance information to make a prediction, and parsimony and stationarity considerations come into play when selecting the regionalization model. For instance, while parsimony generally prescribes the use of observable features, a climate may be less stationary – and therefore a less reliable external trend – than embedded geology or geomorphology.

Algorithmic chart of the provided TopREML implementation. Dashed frames
and arrows represent vector data and operations and the bold arrow represents the
step requiring numerical optimization. The complexity of the computational tasks
represented by the remaining plain arrows is driven by matrix inversion, which is
of polynomial complexity. In the figure,

Leave-one-out cross-validation results for Austrian summer flow when resampling a subset of the training gauges. Computational performances are represented as the ratio of runtimes for TopREML against Top-kriging. Prediction performances are represented as the ratio of relative errors. TopREML performances when using gradient-based and stochastic optimization algorithms are represented as circles and triangles, respectively. Points represent the median value and error bars represent 90 % confidence intervals over 200 repetitions.

In general, geostatistical approaches improve on the prediction of ungauged
basins if the distance between the stream gauges is significantly smaller
than the spatial correlation scale of runoff. Favorable areas are
characterized by high drainage densities or localized rainfall, in addition
to a high density of streamflow gauges. All three variables are highly
heterogeneously distributed at a global scale, as seen on Fig.

Lastly, the decreasing returns to improvements in the complexity of the model
also suggest that the performance of statistical methods for PUB is
ultimately bounded by the spatial heterogeneity of runoff generating
processes. Statistical methods resolve parts of that heterogeneity using the
spatial distribution of observable features (linear regressions) and/or based
on the analysis of the variance of a sample of the predicted variable
(geostatistics). Yet very important parts of the hydrological activity
related to storage and flow path characteristics take place underground: they
cannot be observed and included in the statistical models

We introduced TopREML as a method to predict runoff signatures in ungauged basins. The approach takes into account the spatially correlated nature of runoff and the nested character of streamflow networks. Unlike kriging approaches, the restricted maximum likelihood (REML) estimators provide the best linear unbiased predictor (BLUP) of both the predicted variable and the associated prediction uncertainty, even when incorporating observable features in the model.

The method was successfully tested in cross-validation analyses on mass conserving (mean streamflow) and non-conservative (runoff frequency) runoff signatures in Nepal (sparsely gauged) and Austria (densely gauged), where it matched the performance of the best alternative method: Top-kriging in Austria and linear regression in Nepal. TopREML outperformed Top-kriging in the prediction of uncertainty in Monte Carlo simulations and its performance is robust to the inclusion of observable features.

The ability of TopREML to combine deterministic (observable features) and stochastic (covariance) information to generate a BLUP makes it a particularly versatile method that can readily be applied in densely gauged basins, where it takes advantage of spatial covariance information, as well as data-scarce regions, where it can rely on covariates with spatial distributions that are increasingly observable thanks to remote-sensing technology. This flexibility, along with its ability to provide a reliable estimate of the prediction uncertainty, offer considerable scope to use this computationally inexpensive method for practical PUB applications.

The aim of this analysis is to explore the likely forms of a correlation structure between spatially aggregated processes, given that the underlying point-scale processes are also spatially correlated. In order to maintain tractability, the analysis will consider a strongly idealized case. While we anticipate deviations from the results in non-ideal situations, we nonetheless interpret this idealized analysis as offering insight that constrains the choice of correlation function in the TopREML analysis.

Assuming that the underlying point-scale process

To proceed, we make the assumption that the area of the drainage areas

Following

We also assume that the underlying point-scale process is second-order
stationary and follows an exponential correlation function:

Inserting Eqs. (

We note that within this analysis, the spatial aggregation of the point-scale
process creates a nugget variance arising from spatial correlation scales
smaller than the subcatchments. The nugget variance can be derived (for this
idealized case) by considering the average covariance of points within the
catchments:

We describe runoff occurrence as a binary random variable taking
the value of 1 if an increase in daily streamflow occurs and 0 otherwise. If
runoff events are uncorrelated in time, the random variable follows a
Bernouilli distribution with frequency

In a simple situation with two upstream sub-basins described by the random
variables

Extending the above derivation to multiple sub-basins and neglecting the
covariance term leads to a linear relation between runoff frequencies at
gauge

An algorithmic chart of TopREML, as implemented in the provided
script, is presented in Fig.

A resampling analysis was performed on Austrian data set to evaluate the
runtime and predictive performance of each method as a function of the
topological complexity of the considered region (as proxied by the size of
the considered sample of gauges) and the considered semi-variogram model. We
randomly selected one validation gauge, and resampled the remaining gauges
randomly (no repetition) to obtain the chosen sample size. The resampled
gauges were used to estimate summer flow at the validation gauges using
TopREML and Top-kriging, and successively assuming an exponential
(differentiable) and a spherical (non-differentiable) variogram. In each
case, relative error and runtime were recorded. This process was repeated 200
times for each sample size. Results (shown in Fig.

The authors would like to thank Michèle Müller, Morgan Levy, David
Dralle, Gabrielle Boisramé and two anonymous reviewers for their helpful
review and comments. Data have been graciously provided by the Department of
Hydrology and Meteorology of Nepal, the HKH-FRIEND project and the