The reference observations used as the predictand for calibration were obtained from the Pacific Rainfall Database PACRAIN, . The PACRAIN dataset comprises daily and monthly rainfall records from a comprehensive collection of rain-gauge stations situated across atolls and islands in the South Pacific region. These records are sourced from various institutions, including the National Institute of Water and Atmospheric Research of New Zealand (NIWA, https://www.niwa.cri.nz, last access: June 2024), the US National Centers for Environmental Information (NCEI, https://www.ncei.noaa.gov/, last access: June 2024), the French Polynesian Meteorological Service (https://meteo.pf, last access: June 2024), the Schools of the Pacific Rainfall Climate Experiment (SPaRCE, https://sparce.ou.edu, last access: June 2024), and the Atlas of Pacific Rainfall . Despite the seemingly ample raw samples within the database, an examination of missing data reveals a significantly reduced number of suitable data points. Two critical considerations arise in this context:

(i)

Bias correction requirements. Achieving robust fits for the various statistical methods employed in bias correction demands a relatively large sample size. This is especially true for effectively characterizing extreme events, a main point in our study due to their paramount importance in numerous hydrological applications.

As a result, we used the final subset of suitable rain-gauge stations presented in Table . The calibrated dataset in this study is the Tropical Rainfall Measuring Mission 3B42_7B Daily product TRMM TMPA Precipitation L3 1 d 0.25° × 0.25° V7,

https://disc.gsfc.nasa.gov/datasets/TRMM_3B42_Daily_7/summary, last access: June 2024

We used principal component analysis (PCA) to obtain representative precipitation patterns and then performed a clustering approach. We chose K-means clustering, a traditional method that divides the feature space into a fixed (K) number of clusters by iteratively finding group centroids that maximize cluster distances and minimize within-cluster dispersion e.g.,.

The variables used for clustering were precipitation, sea-level pressure, day-to-day sea-level pressure difference, and eastward and northward wind components, all extracted from the ERA5 reanalysis . Sea-level pressure and its difference were chosen in order to provide suitable descriptors of the South Pacific Convergence Zone state and the occurrence of tropical cyclones, respectively, that have a major effect on precipitation patterns within this region . Since the geostrophic approximation is not valid near the Equator, we also include the wind components in order to characterize circulation. In order to eliminate redundance and linear dependence among variables to be clustered, we performed a joint principal component analysis (PCA) of all the variables, retaining all PCs explaining up to 80 % of total variance (summing up to 45 PCs) prior to clustering. We chose k=5 different weather types (WTs) as a trade-off between the representativity of each group (at least 2 years of data in each group) and the minimization of the total within-cluster variance. Additional groups did not lead to significant reductions of within-group variance and resulted in less representative WTs (less than 600 d for the 21-year period from 1998–2019), thus potentially affecting the robustness of the conditioned statistical calibration. Furthermore, we assessed the robustness of the obtained classification following a 4-fold cross-validation approach by partitioning the data in four temporal blocks (1979–1988, 1989–1998, 1999–2008, and 2009–2019). PCA was iteratively trained on each fold, and the resulting empirical orthogonal function (EOF) was projected onto the remaining folds. At each iteration, the resulting weather typing yielded consistent results in terms of climatologies and precipitation seasonality. The identification of five distinct daily WTs and their relatively balanced sample sizes ensures a robust conditioning of the calibration process, thereby enhancing the reliability and stability of the calibration results. We provide a summary of the resulting weather-type classification in Appendix . The methodology and main features of the classification are further explained in .

The calibration techniques used for the adaptive methodology include scaling, empirical quantile mapping (eQM), parametric quantile mapping (pQM), and generalized Pareto distribution quantile mapping (gpQM). A more detailed description of the methods is provided in Appendix .

The scaling technique is applied to the raw TRMM data by multiplying it with a correction factor. This factor is computed as the ratio between the mean of the predictand (PACRAIN rain gauge measurements) and the mean of the raw TRMM measurements during the training period.

The eQM method is an adaptation of the approach presented in , which utilizes empirical cumulative distribution functions (eCDFs) for calibration. In its parametric version (pQM), the QM method relies on the theoretical distribution rather than the empirical one, whose parameters are estimated based on the observed and TRMM data. In particular, here it is assumed that both the observed and the simulated intensity distributions can be well approximated by the biparametric gamma distribution , and therefore both shape and scaling parameters need to be estimated for transfer function fitting.

The gpQM approach also utilizes quantile mapping but incorporates the generalized Pareto distribution (GPD) above a certain threshold . The threshold, denoted as u, represents the percentile above which the GPD is used to adjust the wet-day distribution. Below the threshold, the distribution is adjusted to a gamma distribution following the pQM method. This method aims to improve the performance of pQM in the upper tail of the distribution, specifically for extreme events. In this work, two different thresholds are selected: the 95th and the 75th percentiles, resulting in the methods named gpQM-95 and gpQM-75, respectively.

To obtain a comprehensive evaluation of the method's overall performance, we initially focus on an unconditioned intercomparison, referred to as standard calibration hereafter. This evaluation involves assessing the method across the entire time series without considering different weather types. The preliminary findings indicate the presence of low to moderate (negative) biases in the TRMM product. As an illustrative example, we present the results obtained at the Kolopelu station in Fig.  (the upper triangle represents the standard calibration results), which are representative of the overall outcomes observed at other locations (the corresponding plots for the remaining rain gauge locations are included in Appendix ). While certain TRMM indices show negligible biases compared to the rain-gauge stations (such as skewness and SDII), others exhibit significant relative biases, particularly for representing high-precipitation events (such as R10p, R20p, or P98_WetAmount). These findings highlight the necessity of applying some form of calibration to enhance the accuracy of TRMM for impact studies. In the same vein, we include the ERA5 precipitation series to provide a benchmark to contextualize the biases of satellite data. This comparison helps to highlight the strengths of the satellite products while also underscoring the importance of calibration against a reliable reference as satellite product biases remain an issue.

Under the assumption that the satellite and radar error may be proportional to the magnitude of the rain rate , scaling has been previously used for TRMM correction see, e.g.,. However, it was found to be ineffective in mitigating biases in most specific indices related to intense precipitation events and some others related to mean precipitation, such as SDII. Instead of yielding and improvement, scaling had an overall deleterious effect, underscoring the critical importance of considering alternative calibration techniques better suited to TRMM adjustment. As shown in Fig. , the scaling step introduces additional bias to the raw TRMM data, specially for SDII, R10p, R20p, and P98Wet. The scaling step also changes the sign of the bias in these cases but consistently increases its absolute value in both the standard (not conditioned) and the WT-conditioned approaches.

As Fig.  shows, overall the quantile-mapping-based techniques attain similar performance, although this may vary by index or technique. The relative bias of the indices is also similar for both standard and conditioned calibration. This supports 's finding that EQM outperforms scaling, with minor differences between unconditioned and WT-conditioned calibration in bias. This also applies to the parametric quantile mapping variants (pQM and gpQM) for different exceedance thresholds, which are additional methods introduced in this study.

The final column in Fig.  compares the best-performing conditioned technique in each case with the adaptive methodology, combining the optimal calibration technique for each weather type individually. In general, the adaptive methodology surpasses the results achieved by the best-conditioned calibration. The most notable reduction in relative bias is observed in indices that measure high rainfall amounts, such as R10p, R20p, or P98WetAmount. This improvement is significant because conditioned calibration alone did not exhibit substantial enhancements, except for specific cases involving techniques like gpQM. However, only three indices (mean, SDII, and P98Wet) did not show improvement with the adaptive approach and remained nearly unchanged. In summary, the overall results indicate that the adaptive calibration method offers improved adjustment in the upper tail of the distribution, which is where TRMM exhibits the most significant biases. This calibration methodology facilitates enhanced adjustment for extreme precipitation events, with a specific focus on high precipitation indices. Next, in Sect. , we present a more in-depth analysis of the detailed results obtained from the adaptive calibration approach.

We introduce the RF score as a comprehensive measure that accounts for the various indices described earlier (Table ) to evaluate the performance of the adaptive approach in comparison to the standard WT-conditioned method. This measure allows for an easier ranking of methods. Figure  presents the summarized results, considering both unweighted and weighted RF values, where “unweighted” here refers to a version in which all evaluation metrics are equally weighted, and thus no specific weights are assigned by the user. The latter emphasizes the significance of extreme indicators in the evaluation process, as discussed in Sect. .

First of all, we undertake a method intercomparison using the unweighted RF score as a ranking measure (Fig. a). In most stations the adaptive calibration outperforms the standard and WT-conditioned techniques, with some exceptions (Port Vila and Nu'uuli), in which the adaptive approach has a similar performance than the simpler ones. On the contrary, the adaptive method outperforms the standard and WT-conditioned calibration at other locations, such as Kolopelu, Alofi, or Raoul Island. Here, the adaptive calibration yields much higher RF scores than the best scores achieved by the other methods. While the score for the best standard technique at these stations exceeds 0.60, the adaptive calibration achieves values between 0.80 and 0.90, representing an improvement of 33 %–50 %. This significant enhancement in calibration demonstrates an overall improvement that justifies the application of the adaptive method. Furthermore, it is noteworthy that in the worst-case scenario, the adaptive approach will nearly match (and never significantly impair) the performance of the calibration. Furthermore, the limited spatial variability across locations observed in the adaptive method (Fig. , box plots) underpins its potential for a robust application across different locations. This characteristic holds importance in hydrological studies, where spatial consistency between locations at the basin level is typically desired.

As mentioned earlier, it is also possible to assign arbitrary weights to the indices involved in the RF score, giving more importance to specific precipitation characteristics, such as the representation of extremes. In this study, we selected index weights that prioritize extreme precipitation indices (Table ). This weighting aims to guide the calibration towards better adjustment in the upper tail of the distribution, thereby achieving improved correction for extreme precipitation events beyond a certain threshold. In this way, the increased influence of extreme indices favors methods like gpQM, which specialize in adjusting the upper tail using a GPD (generalized Pareto distribution). The findings are illustrated through the box plots presented in Fig. , showing that the scores of gpQM95 and gpQM75 exhibit higher values in the weighted version (Fig. b) compared to the unweighted version (Fig. a). Moreover, the differences in their performance can be visualized in Fig. .

The analysis of the RF weighting configuration demonstrates its dual impact: not only does it affect the overall score, but it also influences the selection of techniques for each weather type (Fig. ). In our weighting scheme, which prioritizes performance in extreme precipitation indices, the gpQM approaches often appear as the best choice, for instance, at the A'oloau site for WTs 1 to 3, Kolopelu (WTs 1 and 2), or Alofi (WT1, Fig. ). This outcome indicates the suitability of the gpQM technique over the conventional eQM and pQM methods when the representation of extreme indices is prioritized. For instance, at A'oloau, the overall adaptive calibration RF score improves from approximately 0.65 to 0.85, representing a 30 % increase, with the best standard calibration method changing from eQM or pQM to gpQM75 in WT1, WT2, and WT3 (Fig. ).

In the same vein, at the Port Vila station in Fig. a, the adaptive calibration score is lower compared to the score of gpQM75. However, when applying weighting with a focus on high rainfall indices (Fig. b), the adaptive calibration undergoes enhancements and achieves a competitive score higher than the unweighted version. It is worth noting that while the inclusion of weights leads to changes and improvements in the adaptive calibration for certain stations, it has no effect on others. For instance, stations like Rarotonga or Nu'uuli exhibit no changes in the composition of the adaptive calibration regardless of the weighting applied. The particular case of the Rarotonga site emphasizes the fact that model assessment is sensitive to user choices, such as the set of indices and measures used for validation and, as in this case, the weights assigned to each of them. As a result, when more weight is assigned to the performance of extreme precipitation indices (weighted RF evaluation), the adaptive calibration approach performs better than the standard calibration with the best-performing method (pQM, RF ∼ 0.75). On the contrary, the standard unconditioned calibration using the PQM method performs (marginally) better under the unweighted RF scheme. In the case of Nu'uuli station, the adaptive calibration did not improve the performance, although it also attained a similar high RF score (∼ 0.85/0.75 unweighted/weighted RF scores; Fig. a and b). Compared to other locations in Figs. – (Appendix ) and Fig.  (Kolopelu), Nu'uuli has moderate raw TRMM biases that are effectively corrected by both empirical and parametric quantile mapping (Fig. ). gpQM performs similarly in general, but fails in reproducing distributional skewness and RV20_max in the case of the adaptive approach.

Therefore, the results show that adaptive method attains an overall better performance in all stations, as highlighted in the box plots in Fig. . Additionally, the adaptive calibration demonstrates a narrower interquartile range (IQR) compared to the other methods in Fig. a, indicating lower variability. Only gpQM95 calibration shows a narrower IQR range but obtains significantly lower scores. We attribute this result to the limited robustness of the fit of the extreme function due to the high-percentile threshold, which greatly reduces the sample size (see Table ). The other methods individually considered exhibit wider RF variability ranges and lower values as compared to the adaptive approach.

Quantile mapping can preserve the relative changes in the simulated data, such as trends and patterns see, e.g.,, while scaling may distort them by applying a constant factor. As a result, scaling shows limitation in representing some of the extreme precipitation indices, such as P98Wet and RV20_max, in most of the locations, as well as the SDII (see Figs. – and Fig. , for Kolopelu). On the other hand, eQM and pQM show an overall good performance in most precipitation indices, and eQM is most often the best method for the “standard” calibration (Fig. ).

The gpQM method has the potential to improve the modeling of local extreme precipitation by better adjusting extreme events above a given percentile (75 and 95 in this study), by fitting a generalized Pareto distribution on the threshold exceedances as the transfer function for these points see, e.g.,. This may happen, for instance, in weather types 1 and 2, which are associated with enhanced tropical cyclone frequency and extreme precipitation events (, and Fig. ), and it is shown at Kolopelu (Fig. a and b for unweighted/weighted RF score) and A'oloau (Fig. b), where gpQM75 is the best-performing calibration method, or WT1 at Alofi (with gpQM95). The limitation, however, arises from the sample size required to perform a robust fit; this is a major challenge in the adaptive approach since only the observations of the given WT are used, and it is exacerbated for higher-percentile thresholds that further limit the number of points for a robust fit. This limitation also exists for the other quantile mapping techniques, although, in this case, the number of observations, even for the less frequent weather type (approximately 300 d for the calibration period in this study; see Sect. ), can ensure a robust fit. Nevertheless, scaling requires less data and is easier to implement than quantile mapping. The assumption of the scaling of a linear relationship between raw satellite and observed precipitation, even though it may not hold true in most occasions, can still be reasonable , yielding a good approximation on some occasions (see, e.g., the overall RF results obtained in Port Vila in Fig. ).