2.3.1. Pixel-Based Method

This approach is analogous to at-site frequency analysis of extreme precipitation from rain gauge stations, which was originally applied by the NWS to establish the rainfall frequency isohyetal maps for the US [31]. Treating each 4-km × 4-km HRAP radar pixel as a single station, this is equivalent to considering the domain of study as a dense network of stations that are located 4 km apart from each other. At each pixel, the AMS sample of hourly precipitation, constructed from the 11-year radar dataset, is fitted to the GEV model and the parameters of the distribution are estimated at each pixel using the L-moment method. The quantiles corresponding to different return periods (e.g., 5, 10, 25, 50, 100) can then be estimated at each pixel using the GEV parameters. Confidence intervals for the pixel-based parameter and quantile estimates are constructed using the classical scalar bootstrap procedure suggested by Efron [32]. The bootstrap procedure is used to generate a large number of samples (500 in our case) for each individual pixel.

### 2.3.2. Regional Spatial Bootstrap Method

This is a probability weighted regional method that was originally proposed by Uboldi et al. [17] as a resampling approach for estimation of parameters of the AMS distribution. This technique is based on the generation of a regional sample at any desired location by taking into account data observed at surrounding stations but with decreasing importance when distance increases. Thus, the probability of contribution of a certain station decreases as it goes far away from the desired location. The probability of sampling also takes into consideration the length of the time series at each station, and as such, the possibility of oversampling can be avoided, and the use of short time series is enabled. This method is basically a spatial bootstrap technique in which a regional sample is generated repeatedly from the surrounding locations (pixels in the case of radar data) based on the randomness produced from the probability of data extracting. The procedure of this approach involves formation of a homogenous region, construction of a regional sample, estimation of statistical distribution parameters, repeating the regional sampling and parameter estimation several times as in any bootstrap technique, and finally obtaining a distribution of estimates for each parameter.

The regional sample of size (N) is constructed by extracting (N) observations randomly from all of the available data (M) in a homogenous region. The probability of extraction of each observation is assumed to be proportional to a prescribed Gaussian function (γm) of the distance between the station at the desired location (X) and any other station (Km). Using spatially continuous radar observations (pixel resolution of ~4 km in the current study), the spatial bootstrap methodology is implemented as follows. For each pixel at a desired location X, and by prescribing distance-dependent extraction probabilities, observations from nearby pixels are selected more often than observations from pixels located far away. The probability of extraction of the mth observation located at a pixel (Km) is given by the following relation:

$$\gamma\_{\rm m} = \exp\left\{ -\frac{1}{2} \left[ \frac{\mathbf{d}\_{\rm h}(\mathbf{X}, \mathbf{k\_{m}})}{\mathbf{D}\_{\rm h}} \right]^2 \right\}.\exp\left\{ -\frac{1}{2} \left[ \frac{\mathbf{d}\_{\rm v}(\mathbf{X}, \mathbf{k\_{m}})}{\mathbf{D}\_{\rm v}} \right]^2 \right\}\tag{7}$$

where dh(X, Km) and dv(X, Km) are the horizontal and vertical distance between pixel Km and the pixel at the desired location (X). The Dh and Dv are scale parameters that are selected to impose some degree of smoothing and were chosen in this study to be equal to the standard deviation of the available distances between (X) and (Km).

Normalized by the sum of probabilities of all the observations (M), the probability of extraction of each observation from N set of available observations can be obtained as follows:

$$\overline{\gamma}\_{\rm m} = \frac{\gamma\_{\rm m}}{\Gamma}, \Gamma = \sum\_{\rm m}^{\rm M} \gamma\_{\rm m} \tag{8}$$

By sorting the (M) observations in a descending order according to their probability of extraction (γ m) and assigning each observation a number (m) from 1 to M, a series of sequential ordered dataset is obtained. The cumulative normalized probability of extraction (γ m) of each observation ranges between (0, 1) and the probability of extraction of this cumulative probability is assumed to be uniformly distributed, i.e., (γ m)~U (0,1). A continuous random variable (ρ) is then used to implement a random number generator for a discrete random variable (m) with any prescribed (non-uniform) probability distribution on positive integers up to a generic M. By generating a random number (ρ), the corresponding cumulative probability (γ m) is equal to the generated random number (ρ) and realization number (m) is equal to the first observation that has cumulative probability greater than or equal to the generated probability (ρ).

The spatial bootstrap regional approach requires the formation of a homogenous region surrounding each pixel, from which a regional sample can be constructed. The identification of homogenous regions is a non-trivial step in the regional frequency analysis, and it may require subjective judgement [33]. A homogenous region is the area including a group of sites, or pixels as in the case of radar fields, that share similar physical characteristics. The advantages of working with a homogeneous region is that the historical data available within the region can be pooled to get an efficient estimate of parameters and hence a more robust quantile estimate [34]. Hosking & Wallis [33] strongly preferred to base the formation of homogenous regions on site characteristics (e.g., by using geographical delineation, cluster analysis, or principle components analysis) and to use the at-site statistics only in subsequent testing of the homogeneity of the proposed set of regions. Conventional regionalization techniques identify a fixed set of sites to form a contiguous region, resulting in fixed-boundary regions without smooth transitions.

Burn [10] presented the Region of Influence (ROI) approach for defining homogenous region, in which every site can have a potentially unique set of gauging stations to be used in the estimation of at-site extremes. The ROI technique is recommended as it avoids the transition problems across fixed boundaries by introducing smooth change in the estimates across the boundaries of the regions. The selection of the radius of influence is a trade-off problem, in which a large radius R will increase the number of sites included in each ROI, but at the expense of the homogeneity of the set of sites included. Conversely, a small radius R will ensure the homogeneity of the sites included, but the information transfer will be decreased due to the smaller number of sites. In this study, the ROI approach is applied by using a square window with an area equivalent to (2R + 1)2 bounding the pixel of interest (R is the radius of influence and is used here to refer to the number of pixels considered in the horizontal or vertical directions between the central pixel and the edge of the square window). The window forms a homogenous region and constructs the regional sample for the target pixel (central pixel) using the pixels lying inside this window.

Since the choice of a homogenous region, or the window size, should be based on climatic and physical characteristics, the US Climate Divisions are used in this study to provide an indication for the reasonable range of the radius of influence (R). Louisiana has nine Climate Divisions (Figure 1) and the average area of each climate division is approximately covering a window with side length of about 31 pixels, which corresponds to R = 15 pixels.

**Figure 1.** (**a**) Spatial distribution of the hourly Mean Annual Maxima (MAM; in mm). (**b**) The average month of AMS occurrence in each pixel. (**c**) The average 6-h of AMS occurrence in each pixel. Results are based on the period of study (2002–2012).

Therefore, we chose R = 15 as a threshold for identifying the homogenous regions to estimate PFE. In this study, we tested different square windows ranging from R = 3 pixels to R = 15 pixels (results are only shown for R = 5 and 10) to study the effect of the region size on the uncertainty of the estimates. For example, increasing the window size to 21 × 21, by setting R = 10 pixels, allows for many more pixels (M = 441 × 11; 441 pixels with AMS of 11 observations in each pixel) to be included in the region of each target pixel. The scale parameters in Equation (7), i.e., Dh and Dv, are chosen to be approximately equal to the standard deviation of each radius of influence (for R = 5, Dh = Dv = 1 pixel and for R = 10, Dh = Dv = 3 pixels). The regional sample size is chosen to be the same as the actual number of years available in the radar dataset, i.e., N = 11 (sampled out of the M observations).

In order to reduce the likelihood of extracting annual maxima that might come from the same event, a constraint is added in such a way that the gap in the time stamp of any two annual maxima extracted from two different pixels must be greater than 6 h. This criterion is evaluated using the Julian Date in which the 6 h represent 0.25 day. For instance, if the extracted annual maximum occurs in a certain Julian Date (JD), then any new annual maximum must have a new Julian Date greater than (JD + 0.25) or smaller than (JD − 0.25). This restriction might not be necessary in the case of gauge-based PFE analysis since the gauges are separated with relatively large distances, and therefore it is less probable to have annual series in two gauges that share exactly the same events. On the other hand, the application of this conditioned extracted annual maxima is critical to the radar-based annual maxima, since they are provided on a uniform grid with high spatial resolution (4-km × 4-km in our dataset).
