When observed at a single site and at daily resolution, rainfall is featured by a large number of zeros (i.e. dry days) and a distribution of non-zero rainfall intensities that strongly differs from the normal distribution . In addition, when observed simultaneously at neighboring locations, rainfall is structured spatially with inter-site dependencies decreasing with the separation distance and vanishing at distances of dozens to hundreds of kilometers depending on the regional climate . In this study we adopt a trans-Gaussian approach to model the above features of rainfall . Trans-Gaussian geostatistics assume that a rainfall observation R(s) performed at location s∈D originates from a latent and standardized (i.e., with zero mean and unit variance) Gaussian random variable Y(s) through a transform function Ψ which combines a truncation at threshold a0 and a monotonic transformation ψ of the non-censored part of the latent variable. The spatial dependencies observed between a set of N rain observations performed at different locations R(si),i=1,…,NG are modeled by assuming that the associated latent variables Y(si),i=1,…,NG form a random vector Y=[Y(s1),Y(s2),…,Y(sNG)] that follows a standardized multivariate Gaussian distribution with covariance function CY.

In the following of Sect.  and in Sect.  we develop a Trans-Gaussian rainfall model tailored to the example dataset introduced in Sect. , which entails choosing parameterizations of ψ and CY that best fit the statistical signature of rainfall in the Island of Hawai`i. A different dataset, in particular from a different climate, could require different parameterizations of these functions without calling into question the non-stationary Trans-Gaussian framework deployed in this study.

Here we assume a Gamma distribution for single-site rain intensity because this distribution has easy-to-interpolate parameters (i.e., with high auto-correlation and low cross-correlation) and because when combined with rain types the Gamma distribution has been shown to accurately model daily rainfall intensity . In addition we assume a Matérn covariance function for the latent Gaussian random field because this model of covariance is very flexible and has shown good skills in modeling the spatial dependencies of most atmospheric variables, including rainfall, at daily resolution . Finally, a geometric anisotropy is added to the model of covariance to allow spatial dependencies to vary with the direction of interest . Overall, the trans-Gaussian model for daily rainfall is defined as: 1R(s)=Ψ(Y(s))=0ifY(s)≤a0Gamma-1(Φ(Y(s));k,θ)ifY(s)>a0 With Y∼N0,CY and CY(h;ν,γa,1,γa,2,ρb)=1Γ(ν)2ν-1QνKνQ and with Q=hT×Σ×h; Σ=V×Λ×VT; V=γa,1γa,12+γa,22-γa,2γa,12+γa,22γa,2γa,12+γa,22γa,1γa,12+γa,22; Λ=γa,12+γa,2200ρb2 where Gamma(⋅;k,θ) is the cumulative distribution function (cdf) of the Gamma distribution with parameters k>0 (shape) and θ>0 (scale); Φ is the cdf of the standardized normal distribution; h is the separation vector between two locations si and sj; Γ is the usual gamma function; Kν is the modified Bessel function of the second kind with parameter ν>0, and ν is the shape parameter of the Matérn covariance function that controls the regularity of the latent process Y at short separation lags; γa,1∈R and γa,2∈R are the two components of the vector defining the major anisotropy axis (whose direction corresponds to the direction of maximum spatial correlation of the latent process Y and whose norm ρa=γa,12+γa,12 controls the variability of Y in that direction); and ρb>0 is the norm of the minor anisotropy axis vector (which is orthogonal to the major anisotropy axis vector).

The trans-Gaussian model of Eq. () is made fully non-stationary in space by allowing the parameters of the marginal distribution and those of the dependence structure to vary through space. To this end, the parameters of the transform function Ψ (i.e., a0, k and θ) are first defined for each observation location separately, and then interpolated at ungauged locations . Making the parameters of the covariance function CY (i.e., ν, γa,1, γa,2, and ρb) non-stationary is more challenging because the resulting covariance function must be a valid model of covariance over the whole domain of interest . Different approaches have been proposed to obtain valid non-stationary covariance models, for instance the convolution of locally stationary covariance kernels , the space deformation of the modeling domain , and the use of a locally varying diffusion operator in stochastic partial differential equations . Here we follow the convolution approach because it provides a direct extension of the Matérn covariance function defined in Eq. () to the non-stationary case, and because the use of locally stationary covariance kernels simplifies the calibration of the model based on sparse rain gauge observations (cf. Sect. ). We adopt the parametrization of to derive the non-stationary and anisotropic covariance function of Matérn type between two observation locations si and sj: 2CY(si,sj;νi,νj,γa,1i,γa,1j,γa,2i,γa,2j,ρbi,ρbj)=21-νijΓ(νi)Γ(νj)Σi14Σj14Σi+Σj2-12QijνijKνijQij with Qij=si-sjTΣi+Σj2-1si-sj, with νij=νi+νj2 and with Σi and Σj defined as in Eq. ().

Model calibration starts with the estimation of model parameters at the local scale assuming local stationarity, and continues with the interpolation of these parameters to create a non-stationary model. The data used for calibration are daily rainfall measurements performed by a rain gauge network encompassing a set Network of Ng observation locations sg positioned within the study domain D: sg∈Network⊂D, g=1,…,Ng.

Synthetic rainfall fields reproducing the statistics of the training dataset are generated by stochastic simulation performed at a set of NT target locations ui often located on a regular lattice grid to comply with the requirement of impact models: ui∈Grid⊂D,i=1,…,NT. Note that target locations are denoted by ui to distinguish them from rain gauge locations sg, and we recall here that model parameters at target locations derive from the spatial interpolation of the parameter values estimated at gauge locations.

To simulate synthetic rainfall fields, a realization of the latent field is first generated by geostatistical simulation. A simple way to draw this realization is by Cholesky decomposition of the covariance matrix CY : 5Yus=L×Viid where Yus is the vector containing the realization of the latent field Yus at target locations ui, L is the lower triangular matrix resulting from the Choleski decomposition of the covariance matrix CY evaluated at the target locations (i.e., CY=L×LT) and Viid is an independent and identically distributed standardized normal vector of size NT.

Standard algorithms for the Cholesky decomposition of the covariance matrix have a complexity of O(NT3) point operations and a memory footprint of O(NT2) , which can be prohibitive when more than a few thousands of simulation points are intended. When more target points are requested the simulation based on the Cholesky decomposition of the covariance matrix can be replaced by an approximate but more scalable simulation method, for instance based on a spectral approach that scales linearly with the number of simulation points .

A synthetic rainfall field is finally obtained by transformation of the latent field: 6Rus(ui)=Ψui(Yus(ui)) where Yus(ui) is the realization of latent field at target location ui and Ψui is the local transform function at the same location. Note that an ensemble of Nsim realizations (i.e., Rusk(ui), i=1,…,NT, k=1,…,Nsim) can be obtained by iterating Nsim times the Eqs. () and () with different realizations of the vector Viid.

Rainfall maps and associated uncertainty estimations can be obtained by the stochastic interpolation of the rain gauge observations. To this end, the ensemble of synthetic rainfall fields Rusk described above is modified by a process called conditioning that forces them to reproduce rain gauge observations. An efficient way to perform conditioning in the framework of Geostatistics is conditioning by Kriging . Since conditioning by Kriging requires a target variable with a multivariate Gaussian distribution, it should be performed on the latent field Yusk. Hence, conditioning a synthetic rainfall field requires the computation of the latent counterparts of rainfall observations Y(s)=Ψs-1(R(s)), which are referred to as pseudo-observations. For the observation locations sw where some rainfall is recorded (i.e., R(sw)>0), the pseudo-observation is obtained by simply inverting the parametric transform function ψ. In contrast, at observation locations sd where no rainfall is recorded (i.e., R(sd)=0), the pseudo-observation Y(sd) is not explicitly defined because the latent field Y is censored. To circumvent this problem we propose to use a Gibbs sampler to simulate the censored latent values Y(sd) conditional to the uncensored latent values Y(sw) and under the constraint that Y(sd)<a0(sd) . The Gibbs sampler algorithm adapted to the simulation of the censored pseudo-observations of rainfall is detailed in Appendix . Once the pseudo-observations Y(s) are known at all observation locations, conditioning by Kriging is performed by adding to the original simulations Yusk the result of the simple Kriging of the residuals between the simulations of the latent field Yusk(s) and the pseudo-observations Y(s) : 7Ycsk(u)=Yusk(u)+∑i=1NGλiSK(u)Y(si)-Yusk(si) where λiSK(u) is the solution of the simple Kriging system: ∑j=1NGλjSK(u)CY(si,sj)=CY(u,si),∀i=1…NG. Finally the realizations of the interpolated rainfall field are retrieved by applying the local transform function at each target location: 8Rcsk(u)=Ψu(Ycsk(u)) and the uncertainty of the stochastic interpolation is assessed by evaluating the dispersion of the ensemble of Nsim realizations.

The uncertainty in daily rainfall maps derived from the stochastic interpolation of rain gauge observations increases with the distance to the rain gauges, and the variance of the interpolation uncertainty tends to the variance of the rainfall signal itself at grid points far from any rain gauge. This leads to a high uncertainty of rainfall interpolation in sparsely gauged areas and at the edges of the interpolation domain. To reduce the uncertainty of daily rainfall maps in data-poor areas, it is possible to generate daily rainfall maps conditioned not only to daily rain gauge observations, but also to monthly rainfall totals derived for instance from monthly rainfall maps incorporating additional observations recorded by rain gauges operating at the monthly resolution, as well as information about long-term rainfall patterns .

In the present framework, conditioning daily rainfall maps to monthly totals is made complicated by the fact that daily rainfall can originate from different rain types within a given month. This implies resorting to different statistical models for the spatial modeling of the daily rainfall fields, and makes it difficult to use direct conditioning approaches. We therefore use a Markov Chain Monte Carlo (MCMC) simulation approach to deal with the complex statistical setup imposed by the coexistence of multiple rain types . Among MCMC approaches we select the Metropolis within Gibbs algorithm to condition the daily rainfall simulations to monthly totals, in which a Gibbs sampler is used for simulating spatial patterns of daily rainfall conditional to rain gauge observations and a Metropolis acceptance rule is leveraged for conditioning the local sum of daily rainfall to a monthly total.

The main elements of this algorithm are introduced hereafter and a full description is provided in Appendix . Let m be the tag of a target month encompassing ND days dt,t=1,…,ND, and consider a simulation grid encompassing NT target locations ui,i=1,…,NT. Assume that the monthly rainfall amount R(ui,m) is known at each target location, and that daily rainfall observations R(sj,dt) are available at a set of NG rain gauge locations sj,j=1,…,NG (distinct from the target locations ui). Then the main steps of the Metropolis within Gibbs algorithm are:

Initialization. a.

At each rain gauge location and for each target day, compute pseudo-observations Y(sj,dt) by inverting Eq. () (rainy observations) or by Gibbs sampling following Appendix  (dry observations).

The main limitation of this algorithm is its low computational efficiency, which is typical of MCMC techniques . In consequence, the Metropolis within Gibbs algorithm is only applied to a restricted set of locations, which are selected in such a way that the density of conditioning locations becomes relatively homogeneous in space.

To acknowledge the temporal variability of the rainfall patterns on the Island of Hawai`i the 20-year dataset is split into rain types within which the statistics of rainfall are as homogeneous as possible. The main patterns of daily precipitation are delineated by unsupervised clustering of days based on rain gauge observations. We follow the approach of and use the following indicators to characterize daily rainfall fields at the scale of the island: (1) the proportion of rain gauges recording a wet day, (2–3) the shape and skewness parameters of the distribution of daily rainfall intensity recorded across the island, and (4–6) the first three components of the Karhunen–Loève expansion of the (preliminarily standardized) rain gauge observations. Based on these indicators, the clustering is performed using a Gaussian Mixture Model (GMM; see e.g., ), which approximates the joint pdf of the indicators as a weighted sum of multivariate normal distributions. In the present case the indicators characterizing rainfall intensity (1–3) and rainfall spatial distribution (4–6) are assumed to be only slightly correlated to each other and the covariance matrices used in the GMM are therefore assumed to be diagonal. The number of clusters, hereafter referred to as rain types, is set to six as a compromise between taking into account the diversity of rainfall patterns and keeping the analysis parsimonious.

The rainfall data from the six rain types are processed separately, and one rainfall model is set up for each cluster. Hence, the observation dataset is split into rain types prior to model calibration, and one set of model parameters is estimated for each rain type separately in order to account for the temporal variability of daily rainfall patterns. Figure  displays for each rain type the seasonality of occurrence, the spatial patterns of mean daily rainfall observed by the rain gauge network, as well as the estimated model parameters. Rain typing results show strong contrasts in seasonality and spatial distribution of rainfall between rain types, which translates into contrasting maps of model parameters. It is interesting to notice that the stronger spatial dependencies (denoted by bigger ellipses of anisotropy in Fig. , right column) occur in the North-East of the island, in the wet windward areas. In these areas the spatial dependencies tend to be significantly stronger in the direction parallel to the topography than across it, which is in line with strong rainfall gradients being correlated with gradients of topography. One can also notice the prevalence of rain type 5 during summer months, which brings us to link this rain type with trade wind conditions. This view is reinforced by the observed pattern of daily rainfall, which combines all distinctive marks of trade winds induced orographic rainfall, namely moderately wet windward (East facing) slopes, very dry inland areas because of the presence of the TWI, and some thermally driven rain showers on the Kona coast on the west side of the island.

Building on the ability of the NSTG model to simulate realistic rainfall fields between observation locations, we next assess its ability to predict rainfall at ungauged locations, which corresponds to rainfall mapping. This is evaluated through a cross-validation procedure, in which we iteratively select a target rain gauge, remove it from the training dataset (as well as the data from all gauges belonging to a 5 km-radius circular area centered on this target gauge in order to mitigate the effect of near co-located gauges), re-estimate model parameters for the new training dataset, predict rainfall at the target location, and finally compare the predicted and observed rainfall values. The same procedure is applied to all gauges sequentially, and ultimately the cross-validation implemented here is equivalent to a leave-one-out approach. The quality of the rainfall prediction is evaluated with respect to the ability of the model to estimate rainfall occurrence as well as rainfall intensity during wet days. The prediction of rainfall occurrence is evaluated at each gauge location through the following metrics: the mean bias of the predicted rainfall occurrence (MB-RO), the mean absolute error of rainfall occurrence (MAE-RO), and the Brier score of rainfall occurrence (BS-RO) that measures the accuracy of the probabilistic prediction : 12MB-RO(s)=1NT∑t=1NT(R^(s,t)>0-Robs(s,t)>0)13MAE-RO(s)=1NT∑t=1NT|R^(s,t)>0-Robs(s,t)>0|14BS-RO(s)=1NT∑t=1NT(1Nsim(∑sim=1Nsim[Rsim(s,t)>0])-Robs(s,t)>0)2 where R^(s,t) is the prediction of rainfall at location s and day t (here the median of 100 simulations is selected as rainfall prediction for the NSTG model), and Nsim is the number of conditional simulations (here Nsim=100). To complement the evaluation of rainfall occurrence prediction, we assess the prediction of rainfall intensity conditionally to the occurrence of rainfall (in the following, the occurrence of rainfall is always determined from observations). The prediction of rainfall intensity during wet days is evaluated through the following metrics, evaluated solely on wet days: the mean bias of the predicted intensity (MB-RI), the mean absolute error of rainfall intensity (MAE-RI), and the mean continuous ranked probability score of rainfall intensity (MCRPS-RI, ) that measures the accuracy of the probabilistic prediction and corresponds to the integral of the Brier score associated with all possible rainfall intensity thresholds greater than zero: 15MB-RI(s)=1Nw∑tw=1NwR^(s,tw)-Robs(s,tw)16MAE-RI(s)=1Nw∑tw=1NwR^(s,tw)-Robs(s,tw)17MCRPS-RI(s)=1Nw∑tw=1Nw∫r=0+∞(FRsim(s,tw)(r)-[r≥Robs(s,tw)])2dr where Nw is the number of wet days observed at the target location (i.e., days such that Robs(s,tw)>0), and FRsim(s,tw) is the cdf of the rainfall forecast derived from the 100 simulations Rsim(s,tw).

The performance of the NSTG model is benchmarked against a Climatically Aided Interpolation (CAI) framework , which has been applied to map monthly rainfall in Hawai`i and is customized hereafter to enable the mapping of daily rainfall. CAI is used in place of BSNLG2022 to benchmark rainfall mapping because the latter model only allows for multi-site modeling and therefore lacks the interpolation skills necessary for mapping. CAI involves firstly to derive a mean Climatological Rainfall Pattern (CRP) at each gauge location for the period of interest, secondly to define RainFall Anomalies (RFA) to the CRP, thirdly to interpolate CRP and RFA at ungauged locations, and fourthly to retrieve the interpolated rainfall fields by back-transformation of RFA. To allow a fair comparison with the NSTG model, one CAI model is set up for each rain type separately and the exact same dataset is used for model calibration. In this setting and for a given rain gauge location sg∈Network⊂D we define the CRP as the mean daily rainfall during the rain type of interest (i.e., CRP(sg)=1Nobs∑j=1NobsRj(sg)), and the RFA of a given day j as: RFAj(sg)=log⁡Rj(sg)CRP(sg)+1. The interpolation of the CRP at ungauged locations is performed by Ordinary Kriging using a stationary Matérn covariance function with Nugget, and the interpolation of RFA is performed by Ordinary Kriging using a stationary Matérn covariance function without Nugget. With the above definitions the CAI model can be understood as a deterministic spatial interpolator (i.e., it provides a single estimation of daily rainfall at each target location, which is in contrast with stochastic interpolation that provides an ensemble prediction), which accounts for the non-stationarity of rainfall marginal distribution through the CRP, but assumes a stationary covariance for the spatial interpolation.

Figure a–d displays the results of the cross-validation focusing on the spatial prediction of rainfall occurrence. It shows that the NSTG model performs very well except for a few stations located in sparsely gauged areas and at the transition between a domain where rainfall is very frequent (North-East coast and mountain slopes) and another domain with very rare precipitation (summits in the center of the island and North-West coast). In that context, the withdrawal of the target rain gauge from the training and conditioning datasets leads to a lack of information about the exact location of the gradient of rainfall occurrence, and in the absence of such information the model generates a smooth transition which misrepresents the actual gradient. This artifact is absent from more densely gauged areas (e.g., the West and North-East coasts), and one can therefore infer that the density of the rain gauge network is crucial to get accurate rain occurrence maps in areas with strong rainfall gradients. When comparing the above results with the CAI, one can notice that the NSTG model outperforms the CAI benchmark model both in terms of bias (Fig. b) and magnitude of the prediction errors (Fig. c). This can be explained by the tendency of the CAI benchmark model to over-predict very light rain in place of dry conditions because it does not explicitly account for the zero-inflated nature of daily rainfall data. In contrast, the explicit modeling of this feature in the trans-Gaussian framework allows the NSTG model to map rainfall occurrence more effectively.

Figure e–h investigates the interpolation of rainfall intensity conditionally to the occurrence of rainfall. One can notice that in contrast with rainfall occurrence, rainfall intensity varies substantially between nearby locations, which is particularly visible in the densely gauged area in the East of the island (Fig e). This feature is not perfectly captured by the NSTG model nor by the CAI benchmark model, and both models smooth-out the local variability of rainfall intensity. This is because all data from the gauges located in a 5km radius around the target gauge are discarded from the cross-validation, which prevents the models from learning the local behavior of rainfall and leads to biases of opposite signs for neighbouring gauges (Fig f). The second artefact in rainfall intensity mapping is the imperfect depiction of rainfall intensity gradients in sparsely gauged areas, which is caused by the same reasons as discussed for rainfall occurrence. When comparing the skills of the two rainfall mapping approaches, one can notice that the CAI benchmark model marginally outperforms the NSTG model when averaging the MB-RI and MAE-RI metrics over all gauges, but that the patterns and overall magnitude of the interpolation errors are very similar for both models. This suggests that using a non-stationary covariance in the NSTG model does not improve rainfall mapping in sparsely gauged areas, even though the non-stationary covariance effectively improves the realism of the stochastic rainfall field simulations (cf. Sect. ) used as a basis for the stochastic interpolation.

On another note, Fig. d and h show that a significant advantage of the NSTG model is its ability to provide probabilistic estimates of rainfall by means of an ensemble of conditional simulations, which CAI cannot do. The better scores obtained for the probabilistic forecasts (BS-RO in Fig. d, and MCRPS-RI in Fig. h) than for their deterministic counterparts (MAE-RO in Fig. c, and MAE-RI in Fig. g) show the additional information brought by the ensemble approach.

Section  showed that, like every interpolation method, the NSTG model has decreasing skills in rainfall mapping when the density of the rain gauge network decreases. This is often unavoidable in mountainous regions because the rough topography makes the setup and maintenance of rain gauge networks challenging. To improve the representation of rainfall gradients in poorly gauged areas, we propose conditioning daily rainfall maps not only to daily rain gauge observations as demonstrated in Sect. , but also to monthly totals derived from monthly rainfall maps that incorporate more information about rainfall patterns than is provided by the daily rain gauge network.

Conditioning daily maps to monthly totals is obtained by adding a Metropolis within Gibbs step in the processing framework (see Sect. ), and hereafter we apply this step at 200 virtual stations spread throughout the Island of Hawai`i (white crosses in Fig. , bottom left) to compensate for the large gaps existing in the rain gauge network. The impact of conditioning daily rainfall maps to monthly totals is illustrated for the months of January, April, July and October 2018, which have been selected to cover a wide range of weather conditions and sample all rain types observed in Hawai`i.

Figure  compares the reference HCDP monthly rainfall maps (Fig. , top row) with the monthly accumulations obtained by summing the 30 or 31 daily rainfall maps derived from daily rain gauge observations only (Fig. , middle row), and with the sum of daily maps also conditioned to HCDP monthly totals at 200 virtual stations (Fig. , bottom row). The daily rainfall fields associated to Fig.  middle row and Fig.  bottom row are displayed in Sect. S3–S6. Results show that conditioning daily rainfall maps to monthly totals enables to transpose the features of the reference HCDP maps into the daily rainfall maps derived from rain gauge observations, and in their monthly sum displayed in Fig.  bottom row. In particular, one can notice in Fig.  bottom row the emergence of a monthly spatial maximum at an ungauged location that is located on the East side of Mauna Loa for the four months of interest. In addition, the spatial gradients of rainfall occurring in the North and South of this wet area are sharper when the daily maps are conditioned to monthly totals, leading to more realistic rainfall patterns in the South-East and East feet of Mauna Loa. At higher altitudes, the wet-dry gradient occurring at the East of Mauna Kea and Mauna Loa summits is more sinuous when adding the monthly conditioning, which again improves the realism of the monthly rainfall patterns. It is worth mentioning that the improved gradients and patterns of monthly rainfall obtained by conditioning daily maps to monthly totals propagate to the daily rainfall maps displayed in Sect. S3–S6 in supplementary material, and that this improvement does not introduce any spurious smoothing nor a drizzle effect (i.e., the under-estimation of dry areas coupled with an over-estimation of very low rainfall intensities) in the daily rainfall maps.

To assess how the conditioning to monthly totals impacts daily rainfall maps, we perform a cross-validation study narrowed to three gauges located on the East side of Mauna Kea and on the months of January, April, July and October 2018. This area has been selected because it experiences a strong spatial gradient of rainfall and because it is only sparsely gauged, which leads to a limited performance in rainfall mapping as discussed in Sect. . The cross-validation is performed as in Sect.  and we use the same evaluation metrics (i.e., Mean Bias – MB, Mean Absolute Error - MAE, and Mean Continuous Ranked Probability Score – MCRPS) but for conciseness these metrics are evaluated hereafter on a monthly basis and on the entire time series (i.e., without separation between rainfall occurrence and rainfall intensity). We compare three methods of rainfall mapping: (1) the stochastic interpolation of daily rain gauge observations with the parameters of the NSTG model assumed to be unknown at the target location, which corresponds to the setting tested in Sect. ; (2) the stochastic interpolation of daily rain gauge observations with the parameters of the NSTG model assumed to be known at the target location, which corresponds to the hypothetical case of a rain gauge that has been discontinued but whose past data are used to estimate the parameters of the model at that location; (3) the stochastic interpolation of daily rain gauge observations with the parameters of the NSTG model assumed to be unknown at the target location, and with an additional conditioning to the monthly total derived from the HCDP reference map.

Figure  displays the results of the cross-validation for the three settings described above. Results show that the conditioning to monthly totals (black lines in Fig. ) drastically improves the stochastic interpolation when the parameters of the model are unknown at the target location (dark blue lines in Fig. ). The improvement is major for the mean bias (90 % overall improvement) because the conditioning to monthly totals forces the mean monthly bias to be near zero, and the improvement is also very substantial for the mean absolute error (46 % overall improvement) and the mean CRPS (44 % overall improvement). This proves that conditioning to monthly totals not only improves the spatial patterns of rainfall embedded in monthly and daily maps as shown in Fig.  and in Sect. S3–S6, but also substantially improves point prediction – that is, the actual quality of the daily rainfall maps. It is also worth noticing that in the case of a sparsely gauged area the conditioning to monthly totals leads to better results than knowing the true value of model parameters (light blue lines in Fig. ), with an overall improvement of 83 % for the mean bias, 15 % for the mean absolute error, and 13 % for the mean CRPS. This suggests that knowing the local monthly rainfall accumulation brings more information to the stochastic interpolation of daily observations than knowing the long term rainfall statistics at the target location, and supports the idea that combining different temporal resolutions (here daily and monthly) in rainfall mapping effectively improve the high-resolution maps while ensuring consistency between maps at different resolutions.