The goal in hydrology is to learn about processes related to hydrological variables like daily rainfall, annual runoff or the 5th percentile flow. To gain knowledge about the different hydrological processes, relevant data are collected. There are always uncertainties related to the data that must be accounted for in an analysis, and which make a statistical analysis appropriate.

Assuming x=(x1,…xn) is a vector consisting of hydrological variables of interest, e.g. the annual runoff at several locations for a specific year, the observation likelihood π(y|x) expresses how the data y=(y1,…,ym) are connected to the truth x. In the classical frequentist statistical approach, the variables in x are considered unknown but fixed. In the Bayesian approach, however, x is considered to be a quantity whose variation can be described by a probability distribution (see e.g. ). Prior to the analysis, this probability distribution is expressed through what is called a prior distribution π(x). This is constructed based on expert knowledge about the variable(s) of interest. The goal of the Bayesian analysis is to update the prior distribution by using data. Through Bayes' formula, the so-called posterior distribution of x is obtained: 1π(x|y)=π(x)π(y|x)π(y)∝π(x)π(y|x).

Next, the marginal distribution π(xi|y) for xi∈x can be integrated out, and a prediction of xi can be summarized through e.g. the mean, median or the mode of the posterior distribution π(xi|y).

If a complex process is under study, it is sometimes easier to model it by thinking of its mechanisms in a hierarchy of underlying processes or distributions . The annual runoff x can e.g. be thought of as a process that depends on some parameters θ that express the spatial correlation between locations. Here, both x and θ are stochastic variables with prior (and posterior) distributions. A Bayesian model of this type is typically expressed as a three-stage hierarchical model where the first stage consists of the observation likelihood π(y|x,θ), the second stage is the prior distribution π(x|θ), often referred to as the latent model or process model, while the third stage is the prior distribution of the model parameters π(θ). As before, Bayes' formula can be used to make inference about the variables of interest x but also about the model parameters θ given the set of observations y. In this study we use a three-stage hierarchical Bayesian model to model annual runoff.

GRFs are commonly used to model environmental variables like precipitation, runoff and temperature or other phenomena that are continuous in space and/or time. In this analysis, the second stage of the Bayesian hierarchical model consists of GRFs that model the spatial dependency of runoff between catchments. A continuous field {x(u);u∈D} defined on a spatial domain D∈R2 is a GRF if for any collection of locations u1,…,un∈D the vector (x(u1),…,x(un))T follows a multivariate normal distribution , i.e. (x(u1),…,x(un))T∼N(μ,Σ) where μ is a vector of expected values and Σ is the covariance matrix. The covariance matrix Σ defines the dependency structure in the spatial domain, and element (i,j) is typically constructed from a covariance function C(ui,uj). The dependency structure for a spatial process is often characterized by two parameters: the marginal variance σ2 and the range ρ. The marginal variance provides information about the spatial variability of the process of interest, while the range gives information about how the covariance between the process at two locations decays with distance. The range is defined as the distance at which the correlation between two locations in space has dropped to almost 0. If the range and the marginal variance are constant over the spatial domain, we have a stationary GRF.

In this study, the involved GRFs have their dependency structure specified by a stationary Matérn covariance function that is given by 2C(ui,uj)=σ22ν-1Γ(ν)(κ||uj-ui||)νKν(κ||uj-ui||). Here, ||uj-ui|| is the Euclidean distance between two locations ui,uj∈Rd, Kν is the modified Bessel function of the second kind and order ν>0, Γ(⋅) is the gamma function and σ2 is the marginal variance that controls the spatial variability . The parameter κ is a scale parameter, and it can be shown empirically that the spatial range can be expressed as ρ=8ν/κ, where ρ is defined as the distance at which the correlation between two locations has dropped to 0.1. Using a Matérn GRF is convenient for computational reasons because it makes it possible to use the stochastic partial differential equation (SPDE) approach to spatial modelling from which is briefly described in Sect. .

Within the geostatistical framework, kriging approaches have shown promising results for interpolation of hydrological variables (see e.g. , or ). In kriging methods, the target variable is represented as a random field, typically a Gaussian random field x(u) defined through a covariance structure and some unknown parameters. The process of interest is observed at n locations u1,…,un, and any unknown parameters can be estimated based on e.g. maximum likelihood procedures. Furthermore, to estimate the value of the variable x^(u0) at an unobserved location u0 a weighted average of the observations is used, i.e. 3x^(u0)=∑i=1nλix(ui), where λi are interpolation weights and x(ui) for i=1,…,n are observed values. The interpolation weights are computed by assuming that x^(u0) is the best linear unbiased estimator (BLUE) of x(u0). That is, we determine x^(u0) by finding the weights that both minimize the mean squared error and that give zero mean expected error . Note that the consequence of the latter is that the kriging weights are restricted to sum to 1, i.e. ∑i=1nλi=1 if we assume that the process is homogeneous in space.

Further, to minimize the mean squared error of the kriging predictor in Eq. (), the covariance function (or variogram) must be estimated and evaluated. The covariance function typically depends on the distance between the observations and the target locations, such that observations measured close to the target location u0 are weighted more than observations further away. In many hydrological applications, the centroids of the catchments are used to compute the catchment distances , but as mentioned in the introduction this can lead to a violation of basic mass conservation laws. The reason is that streamflow variables are connected to (catchment) areas, not single point locations. Catchments are also organized into subcatchments, and this should be considered when computing the kriging weights.

The top-kriging approach suggested by is an example of a kriging method that takes the nested structure of catchments into account. In this method, the streamflow observations are interpreted as areal referenced, and the covariance is computed based on the pairwise distances between all grid nodes in a discretization of the involved catchments. This way, observations from a subcatchment can be weighted more than observations from nearby, non-overlapping catchments. Top-kriging is currently one of the leading methods for interpolation of hydrological variables and is therefore chosen as a benchmark when we evaluate our new interpolation approach.

The framework we suggest is both a framework for spatial interpolation and a framework for record augmentation. There exist several approaches for record augmentation for which many of them are based on developing a linear relationship between the target catchment and one or several catchments with longer time series of runoff . One class of approaches is the maintenance of variance extension (MOVE) methods. MOVE methods are based on developing a linear relationship between the target catchment and the donor(s) catchment(s) by assuming that the sample mean and sample variance of runoff are maintained over time for the target catchment . There are different ways the sample mean and sample variance can be estimated, giving different estimators for the predicted runoff. Another way to develop a linear relationship between a donor and a target catchment is to use simple linear regression . In this article, we use simple linear regression as a benchmark method, in addition to top-kriging.

Assume annual runoff is observed for year 1,…,n in the target catchment and that there exist annual runoff data from some other catchments for year 1,…,n+m. Simple linear regression is performed by first finding a so-called donor catchment for the catchment of interest. This can be e.g. the closest catchment in space or a catchment with similar catchment characteristics (elevation, annual precipitation, vegetation). Next, it is assumed that there is a linear relationship between the annual runoff in the target catchment and the donor catchment, yi=β0+β1xi+ϵi for i=1…,n, where yi is the annual runoff in the target catchment, xi is the annual runoff in the donor catchment, ϵi is normal distributed measurement error N(0,σ2) with fixed (but typically unknown) variance σ2, and β0 and β1 are coefficients that must be estimated. The linear relationship between the two catchments is developed by estimating β0 and β1 by minimizing the sum of least squares, ∑i=1n(yi-(β0+βxi))2. Next, the linear relationship can be used to estimate the runoff in the target catchment yn+1,…,yn+m based on xn+1,…,xn+m with corresponding uncertainty estimates.

In this section we highlight and describe two of the model properties that make the suggested framework different from top-kriging and other geostatistical interpolation methods that are used for hydrological applications.

In order to make the framework described in Sect.  computationally feasible, some simplifications of the suggested models are necessary. In general, statistical inference on models including GRFs is slow when the number of target locations is large because matrix operations on dense covariance matrices are required. The computational complexity is particularly large for the areal model, because each grid node in the discretization of the catchments can be regarded as a new target location, and because it includes soft constraints. To solve the computational issues for the centroid and areal models, we utilize the fact that a GRF with a Matérn covariance function can be expressed as the solution of a specific SPDE . This SPDE can be solved by using the finite-element method (see e.g. ), and the result is a Gaussian Markov random field (GMRF). Working with GMRFs is convenient because GMRFs have precision matrices (inverse covariance matrices) that typically are sparse with many zero elements, and efficient algorithms are available for sparse matrix operations (see e.g ). In this work, both GRFs xj(u) and c(u) are approximated by GMRFs.

Another challenge with the suggested models is that we suggest Bayesian models that include a large number of parameters for which the marginal distributions must be estimated. Traditionally, Bayesian inference is done by using Markov chain Monte Carlo (MCMC) methods, but inference can be slow when the dimension of the problem is large . These challenges are met by modelling runoff as a latent Gaussian model (LGM). That is, the latent part x of the hierarchical model in Sect.  consists of only Gaussian distributions. More specifically, the prior distributions for c(u) and xj(u) are modelled as GRFs, and the prior distributions for βj and βc are Gaussian given the model parameters (see the equations in Eq. ). This is convenient, because it allows us to use integrated nested Laplace approximations (INLAs) to make inference and predictions. INLA is a tool for making Bayesian inference for LGMs and represents a fast and approximate alternative to MCMC algorithms. The INLA approach is based on approximating the Bayesian marginal distributions by using Laplace or other analytic approximations, and on numerical integration schemes. The main computational tool is the sparse matrix calculations described in , such that in order to work quickly, the latent field of the LGM should be a GMRF with a sparse precision matrix. This requirement is fulfilled through the SPDE approach as already outlined.

INLA in general provides approximations of very high accuracy for most models but has faced problems for some (more extreme) models with binomial or Poisson data . For Gaussian likelihoods, however, INLA is exact up to numerical integration error. As we use Gaussian likelihoods in this work, we can thus expect INLA to give reliable approximations. The SPDE approach also provides accurate approximations , but it is important that the mesh involved in the finite-element method computations is sufficiently dense relative to the spatial variability and range in the study area.

Because of the high computational speed and accuracy, the INLA and SPDE framework has become quite common to use within different fields of science. See for example , , , , , and . We refer to the R package r-inla for a user-friendly interface for applying INLA and the SPDE approach to spatial modelling. In particular, is recommended for a description of how a model with (catchment) areal data can be implemented in r-inla. Furthermore, we have made code for the centroid model available on http://www.github.com/tjroksva/runoffinterpolation (last access: 29 January 2020; 10.5281/zenodo.3630348) with example data from the catchments in Fig. a.

To assess the framework's ability to fill in missing values of annual runoff, we do interpolation of runoff for the 10 hydrological years 1996–2005 for the 180 fully gauged catchments shown in Fig. a. This is done both for series of annual runoff, and for the annual series of monthly runoff for January, April and June described in Sect. .

The annual time series of monthly runoff are included in the analysis in order to demonstrate the framework’s properties for hydrological variables or areas that are driven by more unstable hydrological processes. For the annual series of monthly runoff, the models from Sect.  are specified as before: considering predictions for January, Qj(Ai) in Eq. () represents the true runoff in January for catchment Ai, year j, such that the GRF c(u) represents the long-term spatial variability in January. Likewise, the GRF xj(u) represents the annual discrepancy from the climate in January, and yij is the observed runoff in January for catchment Ai year j. The models for June and April are specified similarly, and for simplicity we use the same prior distributions for all experiments.

In our assessment of the framework's predictive performance for infill of missing annual observations, the three following methods are compared.

Top-kriging. Spatial interpolation with top-kriging. For top-kriging each year (1996–2005) is interpolated independently of other years. Short records on an annual (or monthly) scale do not have an impact on years without data. The default covariance function (or variogram) in the R package rtop was fitted as this gave the most accurate results. This is a multiplication of a modified exponential and fractal variogram model, the same model as used in .

The predictive performance of the three methods is evaluated by cross-validation: the 180 catchments in Norway were divided into 20 groups or folds, each containing nine catchments. In turn each group was left out, and annual or monthly runoff predictions were performed for these so-called target catchments by using observations from the catchments in the other groups. That is, we predict runoff for 1996–2005 for 9 target catchments at once by using data from the remaining 171 fully gauged catchments and repeat the process for all 20 cross-validation folds. To evaluate and compare the three methods described above, we do the following two tests.

UG (ungauged). Assess the methods' ability to fill in missing values for ungauged catchments (denoted UG). That is, the target catchments are treated as totally ungauged, and all their observations are left out of the dataset when the predictions for 1996–2005 are performed.

To make the results comparable, we use the same cross-validation groups for both experiments (UG and PG) and methods (top-kriging, areal model and centroid model) and remove the same set of annual observations for PG across methods. For the PG case, we also compare our models to a method for exploiting short records from the target catchment. The method we choose for comparison is simple linear regression, and we perform linear regression for the PG case as follows.

Linear regression. The closest catchment in terms of catchment centroid is used as a donor catchment, and only catchments outside the target catchment's cross-validation group can be considered. Two annual observations between 1996 and 2005 are randomly drawn from the target catchment, and data from the donor catchment and target catchment are used to fit a linear regression model on the form yi=β1xi+ϵi. Next, the model is fitted as described in Sect.  and used to predict runoff for the target catchment for 1996–2005 (where two of the years are observed). The reason for using a short record of length 2 instead of 1 is that at least two observations are required to fit a linear regression model with uncertainty. Also note that we have omitted the intercept β0 in the regression model, such that we only have two unknown variables (β1 and σ2). We emphasize that estimating the variables statistically based on only two pairs of observations might not give the most reliable estimates, particularly when considering σ2. However, the linear regression results can provide an intuition about the target variable and how correlated the observation locations are over time. A good performance for linear regression in this study suggests that the spatial pattern for the target variable is very stable and that a fair prediction can be obtained by simply using the ratio of runoff between a target catchment and a donor catchment for any chosen year to develop a linear relationship between the two catchments. Hence, a short record can be very valuable. Motivated by this, linear regression is treated as an indicator for when our geostatistical method can be expected to perform particularly well rather than as a recommended method for record augmentation when having only two observations.

To assess the framework's ability to estimate mean annual runoff, we use annual data from 1981 to 2010 from the 260 catchments in Fig. . Recall that these catchments have at least one observation of mean annual runoff between 1981 and 2010, but only 83 of them are fully gauged. This was different from the experiments described in Sect. , where all the test catchments were fully gauged before the cross-validation was performed.

For this experiment, we again compare the performances of top-kriging and the areal and centroid models. The areal model and the centroid model are fitted for several years of annual runoff simultaneously, as before. As a predictor for the mean annual runoff, we use the posterior distribution of the climatic part of the model. This is given by c(uA)+βc for the centroid model, where uA is the centroid of the catchment A of interest. For the areal model it is given by the average c(ui)+βc over the grid nodes ui in the discretization of the target catchment. Note that the climatic part of the model must be re-estimated for each experiment or cross-validation fold.

In order to interpolate mean annual runoff by using top-kriging, we have to compute the mean annual runoff based on the annual observations for all catchments before running the analysis. For catchments with less than 30 annual observations we use the average of the 1–29 available observations as an approximation for the mean annual runoff for 1981–2010. Next, the mean annual runoff is interpolated by using top-kriging where the uncertainty of the observations is specified as a function of record length. This is the suggested approach from for including short records in the top-kriging framework. We set the observation variance for a catchment with record length m to σ^2/m, where σ^ is the average empirical standard deviation for the observed annual runoff taken over the 83 fully gauged catchments in our dataset, in this case σ^=336 mm yr-1. For the top-kriging experiments, we fit the same covariance model as in Sect. .

The areal and centroid models and top-kriging are again evaluated by cross-validation. The 83 fully gauged catchments from Fig.  were divided into four folds containing 20, 20, 20 and 23 catchments respectively, and in turn observations from each fold were removed and predicted. This was done for varying record lengths for the target catchments, more specifically when 0, 1, 3, 5 or 10 randomly drawn annual observations from the target catchments were included in the likelihood. We denote these settings UG, PG1, PG3, PG5 and PG10. Note that while we only are able to assess the predictive performance for the 83 fully gauged catchments in Fig. , data from the remaining 177 partially gauged catchments in Fig.  are used in the observation sample. This is in addition to the data from the fully gauged catchments from the other folds.

To evaluate the predictions we use the root mean squared error (RMSE) and the continuous rank probability score (CRPS). Having m pairs of observations and predictions, the RMSE is computed as RMSE=1m∑j=1m(yj*-y^j*)2, where yj* is the observed value and y^j* is the corresponding predicted value. In our analysis, the posterior mean is used as the predicted value for the areal and centroid models.

The CRPS is defined as CRPS(F,y*)=∫-∞∞(F(s)-1{y*≤s})2ds, where F() is the predictive cumulative distribution and y* is the actual observation . For the methods we test (areal, centroid, top-kriging and linear regression), F() is a Gaussian distribution with mean equal to the predicted value and standard deviation equal to the standard deviation of the prediction.

For the experiments related to infill of individual years, the CRPS and RMSE are first computed for each of the 180 catchments in the dataset based on 10 pairs of predictions and observations. The average RMSE and CRPS over all catchments are used as a summary scores. For the experiments related to predictions of mean annual runoff, there is only one (mean annual) prediction for each catchment, and the RMSE and CRPS over all catchments are reported. Both the CRPS and the RMSE are negatively oriented such that low scores mean better predictions.

To be able to compare the RMSE and CRPS across methods, we use a paired Wilcoxon signed-rank test . This is a non-parametric test that does not require normal distributed data. The null hypothesis of the test is that the median difference between pairs of data (in this case pairs of RMSE or CRPS values) follows a symmetric distribution around zero. The alternative hypothesis is that the difference between the data pairs does not follow a symmetric distribution around zero. If the null hypothesis is rejected, it indicates that one of the methods gives a significantly smaller RMSE or CRPS than another method.

In addition to the RMSE and the CRPS, we report the 95 % coverage of the experiments. The 95 % coverage is computed by calculating the amount of the actually observed runoff values that are within the corresponding 95 % posterior prediction intervals. Here, we make posterior prediction intervals for top-kriging and linear regression by assuming that the predictions are Gaussian. A 95 % coverage close to 0.95 is optimal and indicates that the model provides an accurate representation of the underlying uncertainty.

We also want to compare our mean annual runoff results with other studies of mean annual runoff, more specifically the studies collected in . In , the absolute normalized error (ANE) and the squared correlation coefficient (r2) are used as evaluation scores. The ANE is computed as 14ANE=|y^*-y*|y*, where y* and y^* are the observed and predicted value as before. The ANE divides the absolute difference between the actual observation y* and corresponding prediction y^* by the observed runoff and is therefore scale independent. An ANE close to zero corresponds to an accurate prediction.

Finally, the squared correlation coefficient between m pairs of observations and predictions is computed as 15r2=(Cor{(y1,…,ym),(y^1*,…,y^m*)})2, where Cor(⋅,⋅) denotes the Pearson correlation. An r2 close to 1 indicates a high correlation between the predicted and observed values.

We now present the results related to the framework's ability to predict runoff for individual, missing years for the annual time series of annual and monthly runoff for a 10-year period (1996–2005). First, we present the results for the ungauged catchments (UG), before we proceed to the partially gauged catchments (PG) that have short records of length 1.

So far, we have presented an evaluation of the framework's ability to fill in missing annual observations of runoff for a 10-year period (1996–2005). We now present the evaluation of the framework's ability to predict mean annual runoff for a 30-year period as a whole (1981–2010), as described in Sect. .

Figure  shows the RMSE and CRPS for the predictions of mean annual runoff for top-kriging and the areal and centroid models as a function of record length (0, 1, 3, 5 and 10). The record length is the number of annual runoff observations available from the target catchment in the cross-validation. We find that top-kriging again performs best for the ungauged case (short record length 0), while the centroid model performs slightly better than the areal model for ungauged target catchments. Furthermore, the RMSE and CRPS decrease with increasing record length for all three methods. However, Fig.  shows that our areal and centroid models outperform top-kriging for record lengths larger than 0: the overall difference between our framework and top-kriging is around 30–60 mm yr-1 in terms of RMSE, which is a considerable difference when the RMSE values are around 100–200 mm yr-1.

Furthermore, we notice the large increase in predictive performance when including a (very) short record of length 1 (PG1 in Fig. ). The reduction in RMSE and CRPS is 45 %–50 % from the UG to the PG1 case for the areal and centroid models. These results are thus comparable to the results we obtained for the experiments related to infill of missing annual values (Sect. ).

To be able to compare our findings with other studies, we also included plots of the ANE and the squared correlation coefficient (r2) for the experiments. These can be found in Figs.  and , and are referred to in the discussion (Sect. ). Also, according to these scale-independent evaluation criteria, the overall results are that for ungauged catchments top-kriging performs best, while when there are short records available, our framework performs better.

In our work, we evaluated two versions of our suggested framework by predicting annual runoff and mean annual runoff for Norway. The results showed that our areal-referenced method and our point- (or centroid-)referenced method gave very similar results in terms of posterior mean (see e.g. Figs.  and ). We did not find a trend describing when one of the methods performed better than the others. In prior to the analysis, we would expect the areal model to perform better than the centroid model for ungauged, nested catchments since the areal model takes the water balance and the nested structure of catchments into account. However, these properties did not have a notable impact on the predicted posterior mean runoff for this particular dataset. This is not an extraordinary result as similar results have been obtained by other studies that have compared top-kriging (areal-referenced approach) to ordinary kriging (point-referenced approach): the point-referenced approaches often perform similarly to the areal-referenced approaches .

A possible explanation for the similar performance of the centroid and areal model in this study is that the proportion of nested catchments in our datasets was relatively low: only 30 % of the catchments in Fig.  were nested, while the percentage of nested catchments was 53 % among the fully gauged catchments in Fig. . Furthermore, most of the nested catchments only have one overlapping catchment. The water balance constraints of the areal model might be more important for datasets where there is a higher percentage of nested catchments in an area with high spatial variability. One example is shown in .

It is also possible that the water balance constraints of the areal model have some drawbacks. One example is if there is poor data quality for a subcatchment in the dataset. Then we impose an inaccurate but relatively strict constraint on the runoff in this catchment's drainage area. This will have an impact on the predictions for all overlapping catchments, and how the predicted runoff is distributed here. In this sense, the areal model is less flexible than the centroid model and requires better data quality.

The water balance constraints of the areal model also makes it computationally more expensive than the two other models. Top-kriging used around 1 min for the interpolation of mean annual runoff presented in Sect.  for one cross-validation fold. The centroid model used around 30–40 min for the interpolation, but provided results for both mean annual runoff and runoff for 30 individual years at the same time. Its run time is thus similar to the run time of top-kriging per year. The areal model on the other hand used around 6–7 h to do the calculations on the same computational server. Hence, from a practical point of view, the centroid model might be the most convenient version of our suggested framework for many applications. However, note that if the posterior uncertainty is important, the areal model gives a more realistic representation of uncertainty than the centroid model (see Fig. ). The centroid model also treats small and large catchments equally, which can be problematic for some applications and study areas.

When considering predictions for ungauged catchments, the results showed that top-kriging provided better results than our two suggested models. Figure  showed that the three methods failed and succeeded for many of the same catchments, but that our models failed slightly more than top-kriging on average. We also see an indication that our models fail more than top-kriging for ungauged catchments that are located further away from other catchments. See for example the catchment that is located south-east in Fig. . For ungauged catchments located far away from other catchments (relative to the spatial range), the predicted value will go towards the intercept βc for our two Bayesian models. For top-kriging, the predicted value will always be a weighted sum of the observations from the neighbouring catchments. This can explain the difference in performance here. Apart from this, we do not find a pattern for which catchments top-kriging performs better (mean elevation, location and the magnitude of the observed value were investigated).

While top-kriging performed best for ungauged catchments, our framework outperformed top-kriging when there were some available data from the target catchment. This was the case both when predicting mean annual runoff, and runoff for individual years. The results showed that the potential gain of including (very) short records in the modelling in Norway was large. An explanation is that the annual runoff in Norway is mainly controlled by orographic precipitation. Since the orographic precipitation is driven by topography and westerly winds are dominating, the precipitation patterns are repeated each year and we obtain hydrological spatial stability with σc≫σx. The mountains in Norway also lead to rapid weather changes in space, here expressed through a low climatic spatial range ρc. Consequently, the information gain from neighbouring catchments is often low for ungauged catchments, and information from the target catchment can be very valuable. It is also convenient that Norway has a humid climate where only around 10 %–20 % of the annual precipitation evaporates.

The evaluation based on annual time series of monthly runoff gave us an indication of how the framework can be expected to behave for other climates and countries: for areas where the annual runoff is driven by unstable weather patterns and hydrological processes, short records can not be expected to contribute to as large improvements in the predictions as for the Norwegian annual data (see the predictions for April in Fig. b). This might be the case for countries and areas where most of the runoff can be explained by convective precipitation, where the aridity index is large or for variables for which storage effects are significant. However, the monthly predictions for January and April also illustrated that we safely can include (very) short records in the model, even if year-specific effects explain most of the spatial variability of runoff. By this we have demonstrated that our models represent a framework for safe use of short records regardless of record length and climate and with the benefit that we do not need to consider the choice of donor catchment as in other comparable methods.

Norway is a country with a moderate gauging density. The framework has not been tested for a denser gauging density. We suppose that there is less to gain from including short records if the gauging density is large relative to the spatial range: here the information obtained from neighbouring catchments could be sufficient. However, a high density of gauged catchments and a close distance to neighbouring catchments do not always guarantee good predictability at an ungauged catchment . It is for example often difficult to predict runoff in ungauged catchments that are very small and/or located close to weather divides. We believe that for such catchments, our method for including short records can be useful regardless of gauging density (as long as the study area is characterized by repeated runoff patterns over time).

There exist several other studies of mean annual runoff in the literature, and some of them are compared in terms of the ANE in the chapter about annual runoff in . According to Figure 5.27 in , an ANE between 0.05 and 0.5 is a typical result for regions like Norway where the potential evapotranspiration is less than 40 % of the mean annual precipitation. Figure  showed that the median ANE obtained for our suggested models is around 0.12 for ungauged catchments, i.e. in the lower range of ANE values in . When a short record of length 1 or 5 was available (PG1 and PG5), the median ANE was as low as 0.05 and 0.03 for our methods.

In Fig. 5.30 in there is also a subplot showing the ANE for predictions of mean annual runoff for ungauged catchments in Austria. Here, geostatistical models (top-kriging) and process-based models (conceptual hydrological models) provided the best predictions according to the ANE, with a median ANE around 0.1. The results we obtain in Fig.  for the ungauged catchments are thus comparable to the results from Austria. This is reasonable as the Austrian climate is humid, like the Norwegian, and the western part of the country is dominated by mountains (the Alps) and has similar climate characteristics to Norway.

Furthermore, report an r2 (squared correlation coefficient) between 0.60 and 0.99 for studies done by cross-validation of around 250 catchments, or for studies using models based on spatial proximity like our suggested framework (Figure 5.25 and Figure 5.26 in ). The r2 for our two models was shown in Fig. , and we see that it lies between 0.91 and 0.99. This is in the higher range of values obtained by comparable studies.

In this article, we proposed two models for runoff that are Gaussian. However, runoff is truncated at zero and typically not Gaussian distributed which we also can see from the histograms in Fig. b and . The consequence of the Gaussian assumptions is that there is nothing in the models that prevents them from predicting negative runoff. Negative values appear for both the areal and centroid models due to the uncertainty given by σy, but this is also a problem for the top-kriging technique. Another source of negative values is that the climatic part of the model (c(u)+βc) can be negative in some areas. This is a fully valid result because the other model components could still ensure positive predictions for most catchments and years. However, it can become a problem if we are unlucky and the year-specific GRF does not make up for the negative climatic GRF for one specific year. To avoid negative values, it is possible to log transform the data before performing an analysis. However, this is only valid for the centroid model, as the log transform is not compatible with the linear aggregation performed by the areal model (Eq. ).

In the areal model, negative values also appear as a consequence of requiring preservation of water balance. If there are inconsistent or poor data over nested catchments, negative runoff in parts of a catchment can be the only option to fulfill the water balance requirements. To avoid negative runoff it is important that the discretization of the study area is fine enough to capture rapid changes in runoff over nested catchments. Catchments that are significantly influenced by human activities should also be removed from the analysis as these can influence both the water balance and the significance of the climatic field c(u) relative to the annual field xj(u).

In our study, some negative values were produced for the monthly predictions, as we can see in Fig. b. However, this is not common and happened for only 1.2 % of the predictions of missing monthly data and for a few data points for the missing annual data for the areal and centroid models. For predictions of mean annual runoff, negative values almost never appear as such effects typically are averaged out. Note that unphysical results also appear for top-kriging and other interpolation methods, either in terms of violating the water balance or in terms of negative values. These model weaknesses should be noted such that the modeller is able to choose what is most important in a real modelling setting. In this case it is a choice between (1) avoiding negative values by log transforming the data before using top-kriging or the centroid model or (2) imposing water balance constraints through the areal model.

Finally, we want to highlight what we think are the main areas of use for our suggested framework. First, our results showed that our main benefit compared to top-kriging was connected to exploiting short records from the target catchment. For this reason, we think that our method is suitable as a pre-processing method for making inference about the (mean) annual runoff in partially gauged catchments before doing a further analysis with other statistical tools or process-based models. One possible approach for runoff estimation could for example be a two-step procedure where we (i) use the centroid or areal model as a record augmentation technique to predict runoff for the partially gauged catchments in the dataset and (ii) use top-kriging to predict runoff in ungauged catchments. Here, the results from step (i) can be used as observed values in top-kriging together with data from fully gauged catchments. Differences in observation uncertainty between fully gauged and partially gauged catchments should here be taken into account.

Secondly, we see that the parameter values of the suggested model provides interesting information about the study area. More specifically, if the marginal variance of the climatic GRF σc dominates over the marginal variance of the year-specific GRF σx, it suggests that the spatial variability is stable over time and that short records of runoff can have a large impact on the model, particularly if also ρc is small. This information can be used by decision makers to e.g. motivate the installation of a new (possibly temporary) gauging station as this might improve the long-term estimates only a year after installation for this catchment. Likewise can the model and its parameters be used to assess whether a gauging station is redundant and can be shut down. However, to exactly quantify the importance of a gauging station, all model variances (σx2, σc2, σβ2, σy2) and ranges (ρx, ρc) must be taken into account, as well as the distances between the donor catchments and the target catchment. Computing this gain is outside the scope of this article but an interesting topic for further research that is related to the field of decision theory and the value of information .