There are a number of approaches to predicting runoff in ungauged catchments, including process-based modeling e.g.,, graphical methods based on the construction of isolines e.g.,, and statistical approaches. Statistical approaches are often implemented via linear regression, wherein the runoff signature of interest is considered to be an unobservable random variable correlated with observable features of both gauged and ungauged basins (e.g., rainfall, topography). Such linear models are well understood and widely implemented, not only for PUB see review in p.83 but also across a wide variety of fields in the physical and social sciences e.g.,.

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, y, using a matrix X of observed features then the linear model has the form: y=Xτ+η. Here τ is an a priori unknown set of weights that represent the influence of each external trend on the hydrological outcome being modeled. The residuals, η, are the observed variation of y that cannot be explained by a linear relation with X. If the residuals are independent and identically distributed (iid), the best linear unbiased predictions (BLUP) of both y and its uncertainty (i.e., Var(y)) can readily be obtained using ordinary least squares (OLS) regression. Unfortunately, residuals are rarely iid in hydrological applications due to the spatial organization of hydrological processes around the topology of river channel networks. This organization has the potential to introduce non-random spatial correlations with a structure imposed by the river network. To recover a suitable model in which residuals remain independent requires that the model structure be altered to explicitly account for the spatial and topological correlation in the residuals.

There are several techniques available to address spatially correlated data. Within PUB, kriging-based geostatistical methods have been widely used e.g.,. In a geostatistical framework, a parametric function is used to model the relationship between distance and covariance in observations. The ensuing semi-variogram is assumed to be homogenous in space, and predictions at a point are computed as a weighted sum of the available observations. The weights are chosen to minimize the variance while meeting a given constraint on the expected value of the prediction. In ordinary kriging for PUB applications, this given constraint is simply the average of the streamflow signature as observed in gauged catchments. Ordinary kriging can also be extended as “universal kriging” to include a linear combination of observable features . Kriging approaches are widely used to predict spatially distributed point-scale processes like soil properties e.g., and climatic features e.g.,. Although ordinary kriging has also been used to interpolate runoff e.g.,, the theoretical justification for this approach is less robust than for point-scale processes. Runoff is organized around a topological network of stream channels, and the covariance structure implied for prediction should reflect the higher correlation between streamflow at watersheds that are “flow connected” (i.e., share one or more subcatchments), compared to unconnected but spatially proximate catchments. Currently, two broad classes of geostatistical methods accommodate this network-aligned correlation structure.

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 infer the semi-variogram of the (unobserved) point scale so as to best reproduce the observed spatial correlation of the area-integrated runoff at the gauges – a procedure known as regularization. The topology of the network is implicitly accounted for by the fact that nested catchments have overlapping areas, which affect the relation between observed (area integrated) and modeled (point-scale) covariances. Yet, complex catchment shapes complicate the regularization of semi-variograms, meaning that the estimation of the point-scale process becomes analytically intractable and requires a trial-and-error approach in most practical applications (e.g., Top-kriging; ). Top-kriging is an extension of the block-kriging approach that accommodates non-stationary variables and short observation records. Top-kriging provides an improved prediction method for hydrological variables when compared to ordinary kriging or linear regression techniques and was recently extended to account for deterministic trends . Top-kriging represents an important advance for PUB, but it does have a few drawbacks: (i) the regularization process is unintuitive, and requires extensive trial-and-error to determine both the form of a suitable point-scale variogram, and its parameters; (ii) this trial-and-error process is likely to be computationally expensive; (iii) like all kriging techniques, the estimation of the variogram is challenging when accounting for observable features: the presence of an unknown trend coefficient and variogram leads to an underdetermined problem, making consistent estimates for both challenging. p. 166 showed that the presence of a trend tends to impose a spatially inhomogeneous, negative bias on the estimated semi-variogram. The bias increases quadratically with distance, meaning that estimates of the long-range variance (the sill) are strongly impacted by the presence of the trend, leading to an underestimation of the prediction uncertainty. This bias, however, only marginally affects the prediction itself.

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 . Runoff correlation is expected to decrease with the distance along the stream following a known parametric function. However, unlike Euclidian distances, the streamwise distance does not have the necessary properties to provide a solvable kriging system. This issue is addressed in and , where streamflow is modeled as a random process represented by a Brownian motion that starts at the trunk of the tree (i.e., the river mouth) moves upstream, bifurcates and evolves independently on each branch. The resulting model only allows spatial dependence with points that are upstream on the river network and provides a positive definite covariance matrix that is estimated through restricted maximum likelihood (REML). Models of this nature have been successfully tested on stream chemistry data and further developed to also allow spatial auto-correlation among random variables on stream segments that do not share flow, with potential applications to the modeling of the concentration of upstream moving species (e.g., fishes or insects) . While these methods do not account for the streamflow generation process, they avoid the conceptual and prediction uncertainty challenges confronted by kriging techniques.

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 to estimate variance parameters. This reduces the bias on the semi-variogram by allowing the variance to be estimated independently from the trend coefficients . This use of a REML framework to estimate a linear mixed effect model on a topological support is termed topological restricted maximum likelihood (TopREML). The approach is based on the following conceptual assumptions:

Flow generation and propagation: similar to Top-kriging, runoff is assumed to be generated at a point scale on the landscape, from where it is routed to a channel and measured at a gauge (Fig. i). Runoff observations made at any individual gauge (Fig. ii) can be broken up into a local contribution, derived from a never previously gauged catchment area, and an upstream contribution that was previously observed at upstream gauge(s) along the channel (Fig. iii). TopREML disaggregates all flow contributions into a cascade of local components, as observed at each successive gauge, and uses these characteristics to constrain the covariance structure of runoff and to account for the stronger spatial correlations between flow-connected basins.

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. ). We then apply the approach in two case studies to evaluate its ability to predict mean runoff and runoff frequency by comparison to other available interpolation techniques: Sects.  and present leave-one-out cross-validations in Nepal (sparse gauges, significant trends) and Austria (dense gauge network, no observed trends). In both cases, TopREML performed similarly to the best alternative geostatistical method. We then use numerical simulations to illustrate the effect of the two distinguishing features of TopREML: its ability to properly predict runoff using highly nested networks of stream gauges and its ability to properly estimate the prediction variance when accounting for observable features (Sects.  and ). Finally, we discuss the limits and delineate the context in which TopREML – and geostatistical methods in general – can successfully be applied to predict streamflow signatures in ungauged basins (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 η. The residuals are split into two parts: (i) one containing “random effects”, u, that exhibit spatial correlation along the flow network and (ii) a remaining, spatially independent, white noise term, ϵ, which does not have any spatial structure. With these assumptions, the mixed linear model is written as: y=X︸Trends:Explanatoryvariables(N×k)τ︸Coefficients(k×1)+IN︸IdentityMatrix(N×N)u︸Correlatedrandomeffects(N×1)+ϵ︸Residuals,uncorrelatederrors(N×1). To proceed, we assume that u and ϵ (and therefore y) are normally distributed with zero mean and are independent from each other. The variance associated with ϵ is denoted σ2, the variance of u is assumed to be proportional to σ2 according to some ratio, ξ, and finally, u is assumed to have a spatial dependence captured by a correlation structure G, which is related to the spatial layout of gauges along the river network and a distance parameter ϕ (the correlation range). Thus, the random effects can be specified as uϵ∼N00,σ2ξG(ϕ)00IN. To solve this mixed model, five unknowns must be found: σ2, ξ, ϕ, the fixed (τ) and random (u) effects. Once τ and u are known, the empirical best linear unbiased prediction (E-BLUP) of y can be made at ungauged locations . The solution strategy adopted here is to prescribe a parametric form for G(ϕ), allowing the covariance structure along the network to be specified, and the likelihood function for the model to be written in terms of all five unknowns. Identifying the parameter values that optimize this model thus simultaneously solves for the correlation structure, covariance parameters, fixed and random effects. To proceed with the specification of G(ϕ), however, the form of the covariance structure that arises along the network needs to be addressed.

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. ). Consider a set of streamflow gauges monitoring a watershed as illustrated in Fig. ii. Because of the nested nature of the river network, the catchment area related to any upstream gauge is entirely included within the area drained by all downstream gauges. To account for the network structure, the catchment at any location along a stream can be subdivided into the IDA that are monitored for the first time by an upstream gauge. This is illustrated in Fig. iii, and leads to a subdivision into non-overlapping areas, each associated with the most upstream gauge that monitors them. In making this subdivision, it is implicitly assumed that the timescales at which a hydrological variable is propagated in the channel are negligible compared with the timescales on which hillslope effects operate (a generally valid assumption for small to moderately sized watersheds; see ). IDAs can be associated with both gauged locations and ungauged locations. In what follows, indices i, j, k and m are used to refer to gauged sites, while index n refers to ungauged sites where a prediction is to be made.

With these assumptions, observations of yi made at gauge i can be expressed as a linear combination of contributions from the upstream IDAs: yi=∑k=ik∈UPiakyk′, where yk′ is the contribution of the IDA related to gauge k (that is, yi is equivalent to yi′ only if there are no gauges upstream of gauge i); UP is the set of IDA monitored by gauges that are located upstream of i; ak=Ak/∑m=iUPAm≤1 is the surface area of the drainage area k normalized by the total watershed area upstream of gauge i. The covariance between observations of y made at different gauges can then be expressed as Covyi,yj=Eyiyj-EyiEyj=∑k=ik∈UPi∑m=jm∈UPjakamEyk′ym′-∑k=ik∈UPiakEyk′∑m=jm∈UPjamEym′ With Eyk′ym′=Covyk′ym′+Eyk′Eym′, we have Covyi,yj=∑k=ik∈UPi∑m=jm∈UPjakamCovyk′,ym′, where Covyk′,ym′ is the covariance between the contributions of sub-catchments k and m. By summing over UP in Eq. () (rather then the complete set of available gauges), the model assumes no correlation between runoff observed at flow-unconnected gauges.

Here we assume that the area-averaged process y′ is drawn from a second-order stationary random process, and that the covariance between yk′ and ym′ will depend only on the relative position of sub-catchments m and k, given some specified correlation function ρ(⋅) of the distance ckm between the centroids of the two sub-catchments . We assume that this function is well approximated by an exponential function ρ(ckm,ϕ)=exp(-c/ϕ). A justification for this assumption, which reproduces the streamflow variances observed in our case studies well (Fig. ), is derived for strongly idealized conditions in Appendix . Finally, because the observations made at the gauges represent an area-averaged process, the averaging generates a nugget variance σ2 that is homogenous across observations. The nugget consists of the variance of processes that are spatially correlated over scales smaller than the sub-catchments (see Appendix ) and of measurement errors at the gauges.

With this background, the covariance matrix of y can be expressed as Covyi,yj=ξσ2∑k=ik∈UPi∑m=jm∈UPjakamρckm,ϕ+σ2=σ2⋅ξU[A⋄R]UT+IN, where σ2=Var(yk′,yk′), Ui,j=1{j∈UPi}, A=aaT, and where Ri,j=ρ(ci,j,ϕ). [⋅⋄⋅] denotes the element-by-element matrix multiplication. The matrix G describing the correlation between the random effects in Eq. () is finally G(ϕ)=U[A⋄R(ϕ)]UT. The topology of the network is described by the matrix U, which ensures that only those catchments that are on the same sub-network (upstream or downstream) of the considered gauge are utilized in the determination of the covariance of y. This spatial constraint comes at the expense of neglecting potential correlations with neighboring catchments that are not flow-connected, and the effects of this tradeoff are investigated in the Monte Carlo experiment described in Sect. . The effect of spatial proximity is addressed by use of the Euclidian distance between catchment centroids (matrix R), and the effect of scale is accounted for by weighting by the catchment area of the IDAs (matrix A).

The restricted maximum likelihood approach partitions the likelihood of y∼NXτ,σ2(ξG+IN) into two parts, one of which is independent of τ . This allows the determination of fixed effects and the variance parameters of the model (here σ2, ϕ and ξ) to be undertaken separately. The variance parameters are then estimated by maximizing the restricted log likelihood expression λR(σ2,ϕ,ξ)=-12log⁡detXTH-1X+log⁡detH+νlog⁡σ2+1σ2yTPy, where det(⋅) is the matrix determinant operator, ν=N-k, H=IN+ξG, P=IN-WK-1WT, W=[X:IN], R is the correlation matrix in Eq. () and K is the block matrix: K=XTXXTXIN+ξ-1G-1. The REML estimators σ2^ and ϕ^ that maximize λR can be obtained through numerical optimization.

Once the variance components ϕ^ and ξ^ are estimated, the fixed effect coefficients τ^ and the random effects ũ can be obtained by solving the linear system : K(ϕ^,ξ^)τ^ũ=Xyy,

The empirical best linear unbiased prediction of ỹn at an ungauged site n can be computed by summing the fixed and random effect predictions yñ=xnTτ^+uñ=xnτ^+gnTG-1ũ, where xn is the vector of fixed covariates at ungauged site n and gn a correlation vector between site n and each gauge. Knowing ϕ^, gn can be readily obtained from the relative position of site n and the gauges in the river network.

The variance of the TopREML prediction error can be expressed as Varỹn-yn=VarxnTτ^-τ+gnTG-1ũ-u=xnTVarτ^-τxn+gnTG-1Varũ-uG-1gn+2xnTCovũ-u,τ^-τG-1gn. The covariance matrix of the error on τ and u in Eq. () can be expressed as a function of the inverted model matrix K : Covτ^-τũ-u=σ2K-1. This provides Varỹn-yn-=σ2xnTK11-1xn+gnTG-1K22-1G-1gn+2xnTK12-1G-1gn, where K11-1, K22-1 and K12-1 are k×k, N×N and k×N partitions of the inverted K matrix. If ϵ is an error that is truly iid and does not affect the true value of yn (e.g., measurement errors), then Eq. () corresponds to the mean square error of the TopREML prediction of yn. If, by contrast, ϵ represents random variations of the true value of yn that are correlated over short distances (and so do not appear correlated in our data), then ϵ should be included in Eq. () and the prediction variance becomes Varỹn-yn+=Varỹn-yn-+σ2, because ϵ and u are independent. In reality ϵ is likely composed of both spatially correlated and iid error components and the true variance will be somewhere between these two bounds .

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.  cannot be applied and the correlation structure in Eq. () will lead to biased REML predictions. The effect of the network structure on streamflow can nonetheless be accounted if the nonlinearities can be neglected or eliminated through algebraic transformations.

For instance, runoff frequency λ is defined as the probability, on daily timescales, that a gauge will record a positive increment in streamflow . Provided all sub-basins are large enough to significantly contribute to streamflow, a runoff pulse at any of the upstream sub-basins causes a streamflow increase at the gauge. Therefore, runoff frequency does not scale linearly through the river network. It can nonetheless be shown (see Appendix ) that if runoff pulses occur independently for each sub-basin, the logarithm of the complement to runoff probability (i.e., ln(1-λ)) propagates linearly throughout the network, enabling the application of TopREML to predict runoff probability at ungauged catchments.

A similar reasoning can be applied to predict recession parameters. For example, the exponential function Q(t)=Q0exp(-krt) is a widely used approach to model base flow recession, where Q(t) is the discharge at time t, Q0 the peak discharge and kr the recession constant which can be considered to represent the inverse of the average response time in storage . Because expected values scale linearly, the average response time at a gauge can be modeled as a linear combination of the mean response times of the upstream IDAs. Therefore, although recession constants themselves do not propagate linearly, their value in ungauged basins can be estimated by taking the inverse of TopREML predictions of average response times.

The computational implementation of TopREML in R is described in Appendix  and an operational script is provided as a Supplement to this manuscript. To run the script, two vector data sets (e.g., ESRI Shapefiles) are needed as inputs – one containing the catchments where runoff is available and another containing the basins where predictions are to be made. Catchment polygons and explanatory and predicted variables must be provided as attributes of the vector polygons. The way in which the catchment polygons are nested provides the topology of the stream network. TopREML uses the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm to maximize the restricted log likelihood, though stochastic algorithms are required if a non-differentiable (e.g., spherical) covariance function is selected. The selection of initial values for σ2, ϕ and ξ is a key user input that may affect the performance of optimization algorithms by causing them to converge to a local extrema. We found that initial values of [σ02,ϕ0,ξ0]=[σLM2,Eckm,1] worked well in our case studies, with σLM2 the variance of the OLS residuals of the linear model and Eckm the average distance between IDA centroids.

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 and 52 catchments in Nepal are used in two separate leave-one-out analyses. The location of the gauges is shown in Fig. , and Table  provides a summary of relevant catchment characteristics. Further details on the data sets are provided in for Austria and in Nepal. The two regions differ significantly with respect to gauge density (high in Austria and low in Nepal) and in the nature of the runoff signature and observable features. The Nepalese data set provides specific runoff and wet season runoff frequency as well as gauge elevation and bias-adjusted annual rainfall derived from the Tropical Rainfall Measurement Mission 3B42v7 data set . Gauge elevation and annual rainfall are used as observable features for specific runoff . The Austrian data set was directly taken from the rtop package , where mean summer runoff observations are provided to demonstrate Top-kriging. The Austrian data set did not contain additional observable features and previous studies have found spatial proximity to be a significantly better predictor of runoff than catchment attributes in Austria .

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 (LM0), linear regression (LM), universal kriging (UK) and Top-kriging (TK). As shown in Table , these methods cover a wide spectrum of incrementally specific assumptions and comparing them provides an assessment of the value added by increased model complexity for regionalization of these streamflow parameters. Code to implement all four methods is readily available in R, with dedicated packages available for Top-kriging – rtop – and universal kriging – gstat .

Basin-level predictions of the considered signatures are presented in Fig.  for the three cross-validation analyses described in Sect. . Figure  also provides box plots summarizing the distribution of the ensuing cross-validation errors. In the three analyses, the prediction errors related to TopREML were comparable to the best alternate method: a linear model for annual specific runoff (Nepal) and Top-kriging for runoff frequency (Nepal) and summer runoff (Austria).

Figure a presents results for annual specific runoff in Nepal and shows that observable features play a significant role in the prediction of runoff. The linear model showed a highly significant effect of annual precipitation (τ^yearlyPrecip(LM)=0.99, t-stat: 9.1) a moderately significant effect of altitude (τ^meanElev(LM)=0.39, t-stat: 2.5) and an overall fit of R2=0.63. The positive sign of the altitude coefficient can be attributed to the effects of glacial melt on runoff, which are more significant at higher altitudes, while the average effect of evapotranspiration explains the negative and noisy intercept of -313 mm yr-1. While including rainfall and altitude in the model decreased the median absolute error by 43 % (LM to LM0), further increasing the complexity of the model by allowing for spatial (UK) and topological effects (TK and TR) did not improve the predictive performance: residuals from the linear regression appeared to be correlated at a range shorter than the distance between the gauges in Nepal. Indeed, fitting the empirical semi-variograms with exponential functions revealed spatial correlation ranges that were on the order of the mean distance between IDA centroids for annual streamflow (21.6 km), and significantly below that distance (7.0 km) for the regression residuals. Nonetheless, the lack of parsimony of TopREML did not appear to affect its predictive performance, which almost perfectly reproduced the performance of the linear model – the most parsimonious method.

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; ), and summer average streamflow in Austria, which has a range of 19.1 km but is sampled by a much higher density of streamflow gauges than in Nepal. Allowing for spatial correlation in the residuals (UK) decreased the median absolute error by 11 % compared to the linear model (LM) for runoff frequency in Nepal and 31 % for summer runoff in Austria. Accounting for topological effects further reduced errors by 33 % (runoff frequency) and 40 % (summer runoff) for both TopREML and Top-kriging methods.

Results from the Monte Carlo analysis are presented in Fig. , showing the outcomes of the two numerical experiments described in Sect. .

Figure a and b shows the effect of network complexity on the performance of TopREML relative to the baseline performance of universal kriging. This effect is measured as the difference in the relative errors of the two methods as a function of Nouter, the ratio of basins beyond the spatial correlation range of runoff that are flow-connected, and Ninner, the ratio of basins within range that are not flow-connected. The effect is expected to increase with Nouter and decrease with Ninner, reaching zero when 100 % of observed basins lie within the spatial correlation range and 0 % of the basins beyond the range are flow-connected. In that case (not shown in the figure), TopREML and universal kriging perform similarly and the mean difference in the relative error of the two methods is zero. Figure a shows that the relative performance of TopREML improves with the number of flow-connected catchments that are located beyond the spatial correlation range, and which are therefore not properly accounted for by universal kriging. Conversely, Fig. b shows that the relative performance of TopREML decreases with decreasing network effects within the spatial correlation range. A linear regression of the relative performance of TopREML against Nouter and Ninner showed that both trends are significant and in the expected direction. However, the positive coefficient associated with Nouter (9.1, t-stat: 11.9) is larger in absolute value and more statistically significant than the negative coefficient associated with Ninner (-2.6, t-stat: -2.6), which suggests that the benefits of including distant flow-connected basins outweigh the costs of discarding nearby (but unconnected) IDAs.

In Fig. c, the Monte Carlo analysis showed that model uncertainty is well predicted by TopREML and strongly underestimated by Top-kriging, both with and without considering an external trend. Including a trend in the model reduces the prediction variance of TopREML – this effect is expected because the variance explained by the trend is no longer included in the modeling error ϵ. The decrease in the prediction variance is well modeled by TopREML, which predicts the observe model uncertainty almost exactly.

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. a) than it decreases when discarding nearby unconnected basins (slope in Fig. b). Further, empirical correlograms of Austrian summer runoff (Fig. ) reveal significantly lower and shorter-ranged spatial correlations when basins are not flow-connected. Both results suggest that the benefit of accounting for network effects on correlations outweighs the cost of losing some information on the correlation between unconnected basins. Second, the REML framework provides an unbiased estimation of variance parameters, even when accounting for observable features. This allows TopREML to accurately predict modeling uncertainties even for highly trended and auto-correlated runoff signatures, as visible in the Monte Carlo analysis presented on Fig. c. By contrast, the expected downward bias in the kriging estimation of partial sills is clearly visible in the underestimation of prediction uncertainties by the Top-kriging method.

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 . By contrast, the dimensionality of the optimization in TopREML is driven by the number of catchments, not an arbitrary grid. More importantly, TopREML admits a well-defined objective function, the restricted likelihood, that is differentiable if the selected variogram function is differentiable. This allows gradient optimization methods to be used, which are much less computationally intensive than the stochastic algorithm required by Top-kriging. The resampling analysis shown in Appendix  suggests that TopREML reduces the computation runtime by an order of magnitude, relative to the implementation of Top-kriging in the rtop package, for comparable 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 , who suggest that Top-kriging can still be applied, in an approximate way on non-conservative variables. Here we assert that hydrologic arguments can be used to convert some non-conservative variables into linearly aggregating processes using simple algebraic transformations. This theoretically more robust approach was here successfully tested in a cross-validation analysis of runoff frequency in Nepal.

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. a), accounting for spatial (UK) and topological (TR) correlation does not further improve predictions. In that situation, parsimony prescribes selecting the least complex of the best performing methods.

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.

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.

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. . The multiplicity of local settings likely explains the large diversity of existing regionalization methods and suggests that the selection of the optimal regionalization approach has to be made locally.

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 . This residual spatial heterogeneity can ultimately only be resolved through a better understanding of the particular catchment processes governing runoff in the considered region. Approaches coupling statistical regionalization with process-based models that assimilate both a conceptual understanding of catchment-scale processes and the random nature of runoff (e.g., ) are particularly promising.