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Given the complementarity of gauge observations (high accuracy at point-scale, poor spatial coverage) and SPP estimates (lower accuracy, but better spatial coverage), it is widely accepted that final estimates can be improved by merging the two data sources (Nerini et al., 2015). A number of SPPs already include an initial gauge correction as a final processing step. In TMPA a mean field bias correction is performed at a monthly time-scale (Huffman et al., 1997), while in CHIRPS a weighted average based on inverse distance and decorrelation slope is used to combine a local IR estimate with the five closest gauges (Funk et al., 2015). However, a number of studies have demonstrated that SPPs benefit further from local gauge correction (Vila et al., 2009; Arias-Hidalgo et al., 2013; Ochoa et al., 2014; Nerini et al., 2015).

The most simple and, arguably most important, satellite-gauge (S-G) merging step is removal of mean bias in the satellite rainfall field. However, satellite-gauge errors are dependent on rain- fall intensity and vary spatially and temporally. While such an approach may still be sufficient for correcting mean monthly climatologies due to their gradual spatial variation (Almazroui, 2011), it is agnostic to the complex spatio-temporal structure of rainfall fields and their errors, resulting in large local errors over shorter time-scales (Todini, 2001; Nerini et al., 2015).

Therefore, different approaches have been developed to establish a spatially variable bias and correct SPPs accordingly. Dinku et al. (2014) calculated residuals between satellite estimates and gauge observations at the respective gauge locations and interpolated these residuals with inverse-distance weighting (RIDW) before adding the interpolated surface back onto the orig- inal satellite field. Applied to 10-day accumulations and using a high-density gauge dataset in Ethiopia, the resulting merged fields were found to be comparable or better in terms of correlation, bias and random errors than those from more complex geostatistical methods. In- stead of residuals (additive bias), Arias-Hidalgo et al. (2013) calculated satellite-gauge ratios (multiplicative bias) for monthly TMPA and gauge data in Ecuador and interpolated these between gauge locations using IDW before applying them as multiplication factors to the full

satellite grid, finding that the corrected TMPA data was comparable to gauge calibrations in terms of the performance of hydrological simulations based on the data. Vila et al. (2009); Salio et al. (2015) combined additive and multiplicative biases in a weighted average to correct daily real-time TMPA across South America. The notion here is that a multiplicative bias is unsuitable when the satellite estimate is zero (missed rainfall) and insensitive to magnitude when the gauge rainfall is zero (false alarm). On the other hand, an additive bias may increase the rainfall frequency if zero satellite rainfall estimates are assigned positive intensities (Vila et al., 2009). Condom et al. (2011) applied a similar combination of additive and multiplicative bias correction to mean monthly TMPA estimates over the Peruvian Andes. In their applica- tion, satellite and gauge data was log-transformed prior to calculating bias in order to account for the non-Gaussian distribution of rainfall and obtain a more gradually varying estimate of the spatial bias. An alternative to residual interpolation by IDW is presented by Li and Shao (2010) who define a method based on Gaussian kernel smoothing, which showed lower bias in the merged estimate of TMPA and gauge data in Australia than complex geostatistical meth- ods. Applied to TMPA data in the Peruvian Andes, the method accounts well for the spatial heterogeneity of rainfall and achieves good local bias correction by weighting gauges based on their distance (Nerini et al., 2015). Other methods do not explicitly calculate errors at gauge locations but use other forms of spatially variable averaging to directly combine satellite and gauge measurements. For example, Rozante et al. (2010) used the Barnes objective analysis method in combination with bilinear interpolation to combine gauge measurements and TMPA estimates in a two-step process, while Heidinger et al. (2012) used wavelet analysis to combine the high-frequency signal of gauges (noise) with the more low-frequency signal of TMPA (base) over the Andean plateau.

However, simple interpolations of residual error, do not account for potential variations in rainfall at ungauged locations and are, thus, highly dependent on gauge density. Furthermore, they are deterministic methods and provide no indication of the error associated with the merged estimate in ungauged locations. This, in part, explains the popularity of geostatistical (kriging) methods for S-G merging. Scheel et al. (2011) used ordinary co-kriging to combine daily TMPA and gauge observations over the central Andean plateau. Here, the error covariance of the satellite field is used to approximate the error covariance structure between TMPA and

true precipitation, while the error covariance of the ordinary kriging field (gauge interpolation) represents the error covariance structure between the gauge and true precipitation (Krajewski, 1987). An improvement to this approach is colocated co-kriging (CCK), that avoids instability of redundant secondary variable (satellite data), i.e. higher spatial correlations for the secondary variable that is far more densely sampled (i.e. full raster grid) than the more sparsely sampled primary variable ( ´Alvarez-Villa et al., 2011). However, co-kriging requires estimation of the entire cross-variograms, which is not required for kriging with a trend model or “external drift” (KED), where satellite data is used as a auxiliary variable to gauge data. ´Alvarez-Villa et al. (2011) compared both methods when merging long-term mean annual rainfall fields of TPR and

gauges. KED outperformed CCK in terms of regional cross-validation due to the high linear correlation between the TPR and the gauges ( ´Alvarez-Villa et al., 2011).

The suitability of kriging methods, more specifically KED, as a S-G merging method; how- ever, depends on a number of factors. Firstly, it assumes gauge observations to be accurate local quantifications of true precipitation, effectively using the satellite to extrapolate to un- gauged locations. Yet previous discussion of gauge errors has revelaed that depending on the spatial scale and temporal aggregation, the point-area difference can be significant. The use of block kriging may be used to account for this (Todini, 2001; Grimes and Pardo-Ig´uzquiza, 2010), but the ability to obtain accurate estimates of the areal mean depends on the gauge net- work density. If all inter-gauge distances exceed the spatial resolution, the small-scale spatial correlation structure remains uncertain and, hence, block kriging will not necessarily supply an accurate estimate of the areal mean. This introduces a second issue with kriging methods in that they depend heavily on the spatial distribution of the gauge network. A sparse or unevenly distributed gauge network can deteriorate the correlation between the variables and skew the interpolation. This becomes increasingly relevant for shorter time-scales. Scheel et al. (2011) have shown that the correlation between TMPA and gauges is best for large rainfall events, but decreases with shorter time-scales (daily totals or less). This can be linked to rainfall being increasingly spatially intermittent over short time-scales (Villarini et al., 2008; Scheel et al., 2011) and the rainfall intensity distribution increasingly skewed (i.e. non-Gaussian), while a fundamental assumption of kriging is that the variable of interest is normally distributed.

Erdin et al. (2012) compared Box-Cox transformations with different parameter settings as a means ensure normality of precipitation data prior to kriging. While concluding that this is, in general, a suitable method to reduce interpolatioon uncertainty, the authors state that the dif- ference in the final estimate between transformed and untransformed KED-interpolated rainfall fields was not substantial. Furthermore, excessive transformation may introduce positive bias into the kriging and care needs to be taken when a substantial fraction of gauges are dry (Erdin et al., 2012). Given the deviation from normality of Box-Cox transformed data, especially in the tails of the distribution, a normal-score transformation, which always yields an exact trans- formation to a Gaussian distribution, is preferable (Goovaerts, 1997). However, neither of these approaches can address the issue of zero rain, often yielding unrealistically large areas of low intensity precipitation in ungauged locations. A possible solution is to split the rainfall variable into a binary indicator (rainfall occurrence) and a positive rainfall intensity. In “double kriging” the rainfall occurrence indicator is interpolated to delineate the rainfall field, the intensities of which are subsequently assigned using kriging (e.g. KED) of the positive rainfall intensities (Grimes and Pardo-Ig´uzquiza, 2010). However, double kriging is based on the assumption that rainfall occurrence and intensity are entirely independent variables. Furthermore, quantifying an uncertainty estimate for the final estimate is problematic given that the kriging error and the rainfall probability need to be accounted for (Grimes and Pardo-Ig´uzquiza, 2010).

tially smoothed precipitation fields (Grimes and Pardo-Ig´uzquiza, 2010). This can be addressed by conditional simulation, whereby different possible realisations of a rainfall field are generated that conserve the measured gauge values, while also being based on the same model of the spatial correlation structure as optimal interpolation (Grimes and Pardo-Ig´uzquiza, 2010). An ensem- ble of rainfall field realizations will return the mean and variance of the kriging estimates at any point-scale. Conditional rainfall field simulations have been showing to be suitable for merging infrared satellite retrievals (cold cloud duration) with gauge observations of rainfall intensity (Teo and Grimes, 2007; Greatrex et al., 2014). Furthermore, the statistical distribution across an ensemble can characterise the non-Gaussian shape of a probabilistic estimate at a specific point and is thereby more informative than the kriging estimation variance if the uncertainty of variable of interest is non-Gaussian. The individual rainfall field realisations can also easily be back-transformed from a normal state into its original non-Gaussian state. This is advantageous as the ensemble variance (i.e. uncertainty in estimate) can be obtained, while circumventing the challenge of back-transforming the kriging estimation variance. The ensemble variance can also be used further in S-G merging using an inverse-error-weighted averaging approach (e.g. Grimes et al., 1999). Given a corresponding estimate of the satellite rainfall error variance, satellite and gauge estimates can be merged, weighting the local estimate in accordance with the respective uncertainties.

A more recent development is the integration of geostatistics into conditional and Bayesian frameworks to improve S-G merging. Todini (2001) calculated the error between a block-kriged gauge field and a rainfall radar. Subsequently, the spatial covariance structure of the error field is modelled and the final merged best estimate is obtained by minimization of the variance in a Bayesian framework using a Kalman filter (Todini, 2001). This approach has been adopted by Nerini et al. (2015) for combining daily TMPA estimates with a rain gauge network on the northern Peruvian Andes. Sinclair and Pegram (2005) used conditional merging to separately interpolate gauge observations and the colocated radar measurements. The (conditional) spa- tially distributed error field is then added back to the original radar field. Verdin et al. (2015) proposed a Bayesian kriging approach wherein gauge observations are modeled as a linear func- tion of satellite-derived estimates. The resulting residuals are interpolated by ordinary kriging. For the Bayesian framework prior distributions are defined on historical data and posterior dis- tributions from Markov Chain Monte Carlo sampling. Hereby the method generates a complete representation of the variability and a quantification of the uncertainty of the parameters. In estimating the merged precipitation, the posterior distributions also produce probability density functions of precipitation estimates. The method yielded good results when combining pentadal infared-based rainfall estimates from CHIRP with gauge measurements in central America and Colombia. The approach was particularly efficient at removing bias even for relatively low gauge densities; however, fundamental kriging limitations (assumption of Gaussian-distributed primary variable and assumption of spatial stationarity) remain (Verdin et al., 2015). Another source of remaining error is the is introduced by the spatial scale misfit between point-based

gauges and, comparatively high-resolution, 5 km satellite estimates (Verdin et al., 2015). While this issue can be addressed using block kriging, as discussed earlier, it also raises the question as to whether SPPs, particularly those, like TMPA, that represent areal averages over scales of 25 km or larger, should be spatially disaggregated (i.e. downscaled) prior to comparison and possible S-G merging.

2.5. Spatial Disaggregation of Satellite Precipitation Products