**4. Discussion**

Accurate information on PFEs are critically needed at high temporal and spatial resolutions to serve in various water resources planning and design purposes. The estimation of PFEs becomes challenging when dealing with short data records to derive precipitation frequencies for large return periods [30]. Using a short sample size to fit the extreme value distributions results in large uncertainties when estimating distribution parameters and quantiles, especially for short durations, e.g., hourly PFEs. Therefore, implementing a regional frequency analysis is an effective means for trading space with time. However, when using precipitation estimates from remote sensing data, e.g., radar or satellite products, applying a robust regional frequency analysis is driven by: (1) accurate estimation of extreme values; and (2) definition of a homogenous region. This study investigated the utility of the spatial bootstrap technique as a potential regional approach to derive precipitation frequencies using radar-based precipitation datasets that typically have short observational records. The spatial bootstrap approach has the advantage over pixel-based estimation to augment the sample size by sampling from a homogenous region surrounding the pixel of interest. Our results indicated that the spatial bootstrap technique can provide spatially smoother distribution parameters and associated quantiles compared to the pixel-based approach, which reduces the unrealistically high variations between neighboring pixels over the fine-resolution radar grid (4-km × 4-km in the case of Stage IV).

Defining the spatial extent of a homogenous region is an important factor to consider when using a spatial bootstrap technique. The selection of the region size is a trade-off problem, in which larger regions will increase the number of pixels and the overall sample size, but at the expense of the homogeneity of the pixels included in the analysis. A larger region will also result in a reduction of the uncertainty of the PFEs (Figure 8). As recommended by [33], it is strongly prefered to base the formation of homogenous regions on site characteristics, using for example geographical delineation, cluster analysis, and principle components analysis. For example, a square region of pixels, as used in our study, might not be appropriate in case of complex terrain. Therefore, in such cases, a careful selection of a homogenous region should include different attributes of the study region such as physiographic catchment characteristics, geographical location attributes, and meteorological factors [41]. At-site or pixel-based statistics can be then used in subsequent testing of the homogeneity of the proposed set of regions.

Our tests on the effect of the regional sample size showed that a longer sample size can significantly reduce the uncertainty associated with large return periods, e.g., 10-year PFEs (Figure 10). When using radar data for PFE analysis, the regional sample size could be increased beyond the actual record length, but without significantly impacting the estimation of the mean PFEs. It is noted that the desired increase of the regional sample size might lead to over-sampling by including observations of similar events in the same synthetic sample; however, the spatial bootstrap method avoids such problems by assigning distance-dependent probabilities to individual observations, rather than to specific pixel sites. While our study focused on the frequency analysis of precipitation at hourly scale, the same regional approach can be implemented at longer durations, e.g., 6-h and 24-h, to derive DDF or IDF curves required for design purposes.

The results of the radar-based PFE were assessed versus those from the NOAA Atlas 14 that were developed using a gauge-based regional frequency analysis. The comparison indicated that pixel-based approach was highly sensitive to observational and sampling variability, and as such can yield much higher or lower PFE estimates compared to the gauge-based PFE. On the other hand, region-based spatial bootstrap approach was less sensitive to sampling effects and short records of radar data, thanks to its regional sampling mechanism. The spatial bootstrap technique provides more realistic representation of the PFE confidence intervals and thus can be considered more reliable when assessed against the reference NOAA Atlas 14 frequency estimates. Since spatial bootstrap technique is less sensitive to outliers, it can be more robust when applied using data that typically contain outliers in extreme precipitation, such as the case of most real-time radar products, including the Stage IV product [25]. The spatial bootstrap approaches are still prone to the systematic biases that are inherent to most radar-rainfall products. Conditional biases, which impact the extreme rainfall values and propagate into the PFE estimation process, need to be adjusted at the radar-rainfall estimation phase before being used for PFE applications. Isolating the effect of the inaccurate estimation of the extreme values by the radar product from other factors, e.g., selection of homogenous region or sample size, is beyond the scope of our study. A future work, e.g., through some simulation-based approach, can quantify how the systematic biases in extreme value estimation can mix with other factors and how they individually (and combined) affect the overall PFE results.
