*3.2. Radar-Based PFE using Regional Sptail Bootstrap*

Before presenting the results of PFEs using the spatial bootstrap technique, we first examine the traditional at-site method (pixel-based in case of using radar datasets). The pixel-based estimation procedure described in Section 2.3.1 was applied to the radar dataset to estimate the GEV distributional parameters and the corresponding PFE for different return periods ranging from 2 to 100 years. Confidence intervals for the estimated parameters and PFEs were also derived using classic scalar bootstrap sampling at each pixel. The parameters and PFEs for each pixel are represented using the mean of the 500 runs of bootstrap. The difference between the lower 5% and upper 95% quantiles of the bootstrap samples are used to quantify the uncertainty in the estimates. The confidence intervals are calculated using a non-parametric method, in which a probability is initially assigned to the sorted

values of the sample ((0.5/n), (1.5/n), ([n—0.5]/n)), where n is the sample size (n = 500 bootstrap runs). The quantiles are then computed by setting the probability to be equal to the confidence limit required, e.g., 0.95, 0.90, 0.05, or 0.1. The first and last value in the bootstrap sample are assigned to the quantiles for probabilities less than (0.5/n) and greater than ([n—0.5]/n), respectively. Figures 2 and 3 show the GEV parameters estimated at each pixel using the pixel-based and regional spatial bootstrap methods, respectively, over the domain of study covering Louisiana.

**Figure 2.** The GEV distribution parameters (shape, scale, and location parameters) from the pixel-based approach. Left Panels: Mean of 500 bootstrap runs. Right Panels: The confidence width (95–5% percentiles).

**Figure 3.** The GEV distribution parameters (shape, scale, and location parameters) from the spatial bootstrap (region-based) approach. Left Panels: Mean of 500 bootstrap runs. Right Panels: the confidence width (95–5% percentiles).

The shape parameter, estimated from the average of 500 bootstrap runs, varies between positive and negative values mostly between [−0.5, 0.5]. The 5% and 95% confidence of the shape parameter have values below −0.5 and above 1 due to the sampling variability. The scale parameter, in most pixels, falls in the range between 5 and 20, with some subtle spatial patterns. The location parameter has noticeable spatial gradients similar to those of the MAM (Figure 1a) where the location parameter increases from north to the south and as we get closer to the Gulf boundary. The sampling effect on both of the scale and location parameters is evident in the 5% and 95% confidence limits. The corresponding PFEs are displayed for two representative return periods of 2 and 10 years (Figures 4 and 5). The PFE results show significant variability in space with clear gradients from north to south. The uncertainty associated with these estimates is fairly large, especially for large return periods, e.g., 50 and 100 years (figures not shown). The spatial maps also show clear signs of irregularities and inconsistency in the spatial variability of the estimated quantiles, which are mostly noticed for large return periods.

**Figure 4.** The rainfall depth (in mm) and the confidence width (95–5% percentiles) corresponding to 2-year return period from the pixel-based (upper panels) and spatial bootstrap approaches (region-based) (lower panels).

**Figure 5.** The rainfall depth (in mm) and the confidence width (95–5% percentiles) corresponding to 10-year return period from the pixel-based (upper panels) and spatial bootstrap approaches (region-based) (lower panels).

The confidence limits are estimated using the spatial bootstrap technique for 500 runs using a moving window of 11 × 11 pixels (R = 5). Compared to the pixel-based approach, the results suggest that the spatial bootstrap approach reduced the estimated parameters and resulted in narrower confidence intervals. For instance, the mean shape parameters, in most of the pixels, went down to the range [−0.2, 0.2] with a noticeable reduction in the width of the uncertainty bounds. The reduction in the dispersion of the estimated parameters is attributed to the gain from the repeated sampling from the surrounding pixels, which is the main advantage of a regional estimation as opposed to using information available at each pixel only. Sampling from a homogenous region resulted in smoother fields of the GEV parameters with less sampling variability. Because of the short record available in each pixel, only 11 years, the pixel-based estimation varies considerably from one pixel to another, which was circumvented when using the regional spatial bootstrap estimation with the moving window at each pixel. Increasing the size of the moving window to (21 × 21) or R = 10 pixels resulted in lower variability and more smoothness for the estimates transition between the pixels (figure not shown), but possibly at the expense of losing details in the spatial patterns.

Figures 4 and 5 display the PFEs using the GEV distribution parameters for return periods of 2 and 10 years. Improvements in the smoothness of the different PFEs can obviously be seen when using the spatial bootstrap approach over the pixel-based approach. The smoothness in the PFEs patterns by the spatial bootstrap resembles to a great extent the smoothing algorithm performed by Durrans et al. [4] who used simple distance-weighted averaging procedures to spatially smooth the estimates of sample L-moments. Their smoothing algorithm reduced the effects of sampling variations caused by the short time series used, only eight years in their study.

#### *3.3. Comparison Against Gauge-Based PFE*

In this section, the NOAA Atlas 14 gauge-based PFEs [18] are contrasted against the corresponding frequencies estimated using the two approaches presented earlier; pixel-based and spatial bootstrap regional estimation methods. The gauge-based AMS used in the Atlas 14, as well the corresponding PFEs with their 90% confidence intervals, were acquired from the NOAA's Hydrometeorological Design Studies Center (HDSC) web-based data server. We used the gauge-based PFE from the NOAA Atlas as a reference to assess the robustness of the spatial bootstrap method when (a) estimating PFEs with short radar samples, or (b) in cases of having outliers in the radar AMS sample. However, it is important to note that this comparison does not imply that PFEs from gauges are the true estimates, simply because they also have their own uncertainties caused by sampling variability and the estimation process itself [36]. Nevertheless, the comparison will provide some insights into the performance of the regional spatial bootstrap method in deriving PFEs using radar-based estimates.

The NOAA Atlas 14 applied a regional frequency analysis approach that is different from the spatial bootstrap technique used in the current study. The main difference is in how the regional sample is constructed from the homogenous region formed for each station. In the Atlas 14 method, a homogeneous region is defined for each gauge by grouping the closest 10 stations. The 10 stations are then added to or removed from the region based on factors such as distance from a target station, elevation difference, difference in MAMs at various durations, and inspection of locations with respect to mountain ridges. The AMS for a network of 33 hourly gauges in Louisiana is retrieved from the HDSC and used in the current study to identify differences in the AMS constructed from the radar QPE versus those from the gauges (Figure 6).

**Figure 6.** (**a**) the mean annual maxima rainfall depth extracted from NOAA Atlas 14 gauges and the corresponding radar-pixel (each color represents one of the 33 gauges in Louisiana retrieved from NOAA HDSC web-based data server). (**b**) same as (**a**) but reporting the coefficient of variation. (**c**) comparison of AMS from gauge data and radar-based estimates (for common period 2002–2010) at the location of two example NOAA Atlas-14 gauges (indicated in Figure 1a).

Figure 6a shows that the radar-based QPE product has an overall lower value (AMS) than the corresponding gauge-based AMS, with an average underestimation of 9 mm. Such underestimation of radar-based precipitation can be partially attributed to the areal estimation of precipitation in case of radar pixel as opposed to point gauge. However, given the high resolution of radar (4-km × 4-km), the effect of point-area discrepancies is negligible for small areas. For instance, according to the values given in TP-29 [37], the percent of area-to-point precipitation in case of hourly rainfall and for areas of less than 16 km2 is higher than 95%. In terms of the variability of the AMS, 20 gauges experience higher coefficient of variation (average = 0.34) compared to the corresponding radar pixels (average = 0.25). A higher variability in the gauge-based AMS is attributed to longer record available (with an average of 38 years for gauges in Louisiana) compared to only 11-years of radar-based AMS that are used in this study.

Three representative gauges (from the NOAA Atlas 14), are selected for further comparison analysis (Table 1).


**Table 1.** The NOAA Atlas 14 gauges selected in our study to evaluate the spatial bootstrap approach when using radar-based precipitation estimates.

The three gauges are located in the southwest and southeast climate divisions of Louisiana (Figure 1a). Figure 6c shows plots of the AMS extracted from gauge (1) and the coincident pixel for 9 years covering the common period (2002–2010). The gauge-based AMS is available for 49 years from 1962 to 2010 which is a long record compared with the 11-year radar QPE data used in the current study (2002–2012). It is noted that the 2003 annual maximum from the radar QPE is much higher compared to that of the corresponding gauge, which suggests that this particular value might be an outlier. The mean and standard deviation of the AMS for this pixel are 58 mm and 37 mm respectively. When excluding the outlier observation, the standard deviation for the AMS is only 14 mm, which indicates the high variability that might result from this individual value. Moreover, upon applying the Grubbs-Beck (GB) outlier detection test, this observation is considered an outlier at a 5% level of significance.

Figure 7 shows the effect of including annual maxima events from neighboring cells by comparing the range of AMS (minimum and maximum hourly precipitation in AMS sample) at the location of three gauges in Louisiana (indicated in Figure 1). Again, the outlier identified at the pixel coincides gauge (1) is reflected in the range of AMS formed when using regional approach. In addition, it is very evident in gauges (2) and (3) that sampling from neighboring cells, i.e., forming a regional sample, can significantly increase the range of extreme precipitation events that are not captured in the 11-year AMS sample of the pixel. For example, in case of gauge (3), the AMS sample from a region of R = 10 can range between 16.8 mm and 109.5 mm, which covers the range of gauge-based AMS (between 28.4 and 102 mm). When using a pixel-based approach at the same gauge location, sampling from an 11-year AMS can only ranges between 29.9 mm and 63.1 mm.

**Figure 7.** The range of AMS from gauge data, radar pixel, and radar-based regional sample considering a radius of 5 and 10 pixels. Each bar ranges between the minimum and maximum value in AMS sample extracted at the location of gauge (see Figure 1a for gauges locations).

To assess the impact of outliers, we estimated the PFE quantiles at the specific location of gauge (1) using both the pixel-based and the spatial bootstrap methods. The unusually high radar AMS value (in 2003) resulted in a rather higher mean PFEs and wider uncertainty bound when using the pixel-based approach (Figure 8), while the spatial bootstrap technique was much less influenced by it.

**Figure 8.** Precipitation Frequency Estimates (PFE) and (95–5%) confidence limits using different estimation approaches at the location of Gauge (1).

For example, for 25-year return period, the pixel-based estimation resulted in a mean PFE of 120 mm compared to 96 mm and 82 mm estimated in Atlas 14 (using gauges) and spatial bootstrap (using AMS from radar pixels with R = 5 pixels), respectively. The (95–5%) percentile difference of 25-year return period dropped down from 115.5 mm when using the pixel-based estimation to 68 mm using spatial bootstrap with a radius of influence R = 5 pixels (compared to 51 mm from the regional approach adopted in Atlas 14). Further reduction in mean PFEs and confidence limits are also noticed in Figure 8 when using spatial bootstrap while augmenting the regional sample using a larger radius of influence (e.g., R = 10). The reduced effect of possible outliers is one of the benefits of the spatial bootstrap technique since the combined use of multiple pixels enables reducing the impact of such very rare events (Uboldi, et al., 2014). Owing to longer records available for precipitation in most of the gauges (49 years for Gauge 1), the gauge-based PFEs from Atlas 14 have narrower uncertainty bounds for larger return periods (>10-year). Overall, the PFE results using the spatial bootstrap method are closer to those of the Atlas 14 than the pixel-based approach. For example, the spatial bootstrap method (with R = 5 pixels) resulted in an overall mean absolute percentage deviation (assuming Atlas 14 as our reference) of 15% compared to 25% when using pixel-based approach. When compared to NOAA Atlas 14 PFEs, the spatial bootstrap method resulted in smaller confidence intervals for shorter return periods (less than 10 year-return period).

Gauge (2) represents an example where AMS sample is constructed for a different period (1948–1981) compared to radar (2002–2012). Unlike gauge (1), the precipitation frequencies estimated by the NOAA Atlas 14 approach are quite larger than those estimated by the radar QPE dataset when using the pixel-based estimation. Figure 8 shows lower mean estimates for the quantiles and very narrow confidence intervals in the pixel-based estimation compared to Atlas 14 regional estimation method. For example, 5-year PFE using pixel-based estimation resulted in a confidence interval of 16 mm compared to 36 mm from gauged-based PFE. The lower quantiles estimates can be attributed to an overall underestimation in radar-based AMS values as compared to those estimated by gauge (average underestimation in mean annual maxima is 14.4 mm), while the less variability is due to the small standard deviation of the AMS. This narrow confidence bounds discloses one of the limitations of using the conventional bootstrap resampling with small sample sizes, since it will never generate an observation either larger or smaller than the maximum or minimum AMS observation [38].

When applying the spatial bootstrap technique over Gauge (2), it resulted in lower PFEs estimates compared to the NOAA Atlas 14 estimates. While the mean PFEs from both the pixel-based estimation and the regional spatial bootstrap techniques are very comparable, the spatial bootstrap resulted in wider confidence intervals. This is attributed to the addition of observations, other than those included in the pixel sample (Figure 7), that introduced more variability to the quantile estimates in a way that makes them closer to those derived by the gauge-based PFE from NOAA Atlas 14. Unlike the expected reduced variability in regional-based approach, the variability increased with opening the moving window to larger size (Figure 9; using R = 10) to take advantage from more pixels surrounding the pixel that coincides with Gauge (2). For example, 50-year return period PFE has a higher (95–5%) confidence width of 67 mm when using the spatial bootstrap (R = 10 pixels) compared to only 24.4 mm from pixel-based estimation. An underestimation in radar-based PFEs can be attributed to the conditional bias that is typically manifested in radar QPE products [16]. The conditional bias characterizes the performance of QPE products at different ranges of the rainfall amount [39]. An increase in the QPE bias at high rainfall rates [40] propagates in the radar-based PFE analysis regardless of the PFE estimation method and results in an overall underestimation of the PFE quantiles as illustrated in Figure 9c.

**Figure 9.** Precipitation Frequency Estimates (PFE) and (95–5%) confidence limits at the location of Gauge (2) based on (**a**) NOAA-Atlas14, (**b**) pixel-based approach, and (**c**,**d**) regional spatial bootstrap with a region of R = 5 pixels (**c**) and R = 10 pixels (**d**).
