<p> 1 Online appendix B: Application of Savitzky-Golay smoothing 2 to POP12 and QPF06 3 4 to 5 6 “The US National Blend of Models Statistical Post-Processing of 7 Probability of Precipitation and Deterministic Precipitation Amount” 8 9 by Thomas M. Hamill et al. 10 11 See online appendix A of Hamill et al. (2015) for more background on Savitzky-Golay (S-</p><p>12 G) smoothing and the general rationale for its use. The main difference in this algorithm versus </p><p>13 the one documented previously is: (a) the use of higher-resolution terrain data sets for </p><p>14 determining the relative terrain roughness, which is subsequently used to specify how much to </p><p>15 weight the smoothed field relative to the unsmoothed, and (b) the use of different constants in </p><p>16 the combination in accordance with the finer-resolution terrain. As before, a window size of 9 </p><p>17 grid points and using a third-order polynomial is used in the S-G smoothing. </p><p>18 The smoothing algorithm follows several steps: </p><p>19(a) Copy the input forecast data we seek to smooth into a work array, be it POP12 or QPF06. </p><p>20 Adjust the work array values at grid points outside the CONUS borders to be a weighted linear </p><p>21 combination of data from nearby points inside the CONUS, for reasons explained later. Apply </p><p>22 S-G smoothing to the resulting work array.</p><p>23(b) Adjust the final values for points outside the CONUS to taper from the smoothed values </p><p>24 produced in (a) above to raw input value as one considers grid points further and further away </p><p>25 from the CONUS.</p><p>26(c) Produce the final smoothed field inside the CONUS from a linear combination of the S-G </p><p>27 smoothed (the work array) and the raw input data, with more weight to the S-G smoothed value </p><p>28 in flat terrain and more weight to the raw input data in rugged terrain. </p><p>29 30 We now describe each of these steps in more detail. Before S-G smoothing, we first do </p><p>31 a preliminary smoothing to points outside the CONUS, step (a) above. Why a preliminary </p><p>32 smoothing? The S-G algorithm uses a 9 x 9 stencil of points as inputs for smoothing. Suppose </p><p>33 there were radical differences in the data values the CONUS border because quantile mapping </p><p>34 was applied previously inside the CONUS, and no quantile mapping was applied outside the </p><p>35 CONUS. Then when applying S-G smoothing along the border in step (b) below, the points just</p><p>36 inside the CONUS border could be deleteriously affected by points with uncorrected data </p><p>37 outside the border. Adjusting the work-array values outside the CONUS minimizes such </p><p>38 problems.</p><p>39 The approach for pre-smoothing was inspired by a simple objective analysis procedure </p><p>40 akin to a single-pass Barnes analysis (Barnes 1964). The work array output value assigned to a</p><p>41 given point outside the CONUS consisted of a weighted combination of the work array input </p><p>42 values from nearby grid points inside the CONUS. Define a length scale L=3 grid points for the </p><p>43 spatial weighting function. Define a cutoff radius R (= 10 grid points). Then for a given location </p><p>44 (i, j) outside the CONUS, define SL as the set of indices of all M CONUS grid points within a </p><p>45 cutoff radius distance from (i, j), SL = [(i, j)1, …, (i,j)M]. Assume there was also a set of </p><p>46 associated distances DL = [D1, …, DM] that provided the distance of each in grid points from (i, j).</p><p>47 For the kth of M grid point locations, the distance-dependent weighting function wk is calculated </p><p>48 according </p><p>49 , (B1)</p><p>50 which is Gaussian in shape. Then the final smoothed work array output value assigned to this </p><p>51 location is</p><p>52 (B2) 53 where p(i,j)k is the input work array value at the kth of the M associated land locations within the </p><p>54 cutoff radius. For points inside the CONUS, is simply set to the input work array value p(i,j). In</p><p>55 the exceptional case where M=0, then is set to the average over the whole CONUS.</p><p>56 We now apply the S-G smoothing filter to the resulting work array values, producing a gridded </p><p>57 output. Again, see online appendix A of Hamill et al. (2015) for more details on the S-G </p><p>58 smoothing.</p><p>59 Now, the second step, (b) above. This algorithm has now produced smoothed values </p><p>60 inside the CONUS domain. We return in step (c) to consider how to use these smoothed </p><p>61 values. But first we describe the procedure for setting the final value at points outside the </p><p>62 CONUS? In these locations we were not provided with training data for post-processing, though</p><p>63 we now have two potential sources of information: (1) the smoothed field produced through eqs.</p><p>64 (B1) and (B2) above, reflecting a weighted combination of values from inside the CONUS, and </p><p>65 (2) the raw model guidance. Were we to use the raw values, be they POP12 or QPF06, we </p><p>66 would sometimes notice a discontinuity along the CONUS boundaries as a consequence of </p><p>67 applying quantile mapping inside the CONUS and no quantile mapping outside the CONUS. </p><p>68 The resulting field may be visually unacceptable to forecasters. Accordingly, we decided to </p><p>69 taper between the two, using mostly the S-G smoothed field for grid points outside but adjacent </p><p>70 to the CONUS, the raw model guidance for grid points some grid points distant from the </p><p>71 CONUS border, and a weight of the two in between. For a grid point outside the CONUS that </p><p>72 is d grid points away from the nearest point inside the CONUS, we produce a weight for the </p><p>73 smoothed ws according to </p><p>74 if d < 16 (B3)</p><p>75 = 0 if d ≥ 16.</p><p>76 The resulting output field for points outside the CONUS is then ws ✕ the S-G smoothed field + (1</p><p>77 - ws ) ✕ the raw input data. 78 The final step in the smoothing algorithm, part (c) above, is to linearly combine the S-G </p><p>79 smoothed field with the unsmoothed field inside the CONUS, weighting the unsmoothed more in</p><p>80 regions of great terrain roughness. The difference here relative to the algorithm described in </p><p>81 Appendix A of Hamill et al. (2015) is a change to using a higher-resolution terrain height data </p><p>82 set when defining the terrain roughness. The ⅛-degree terrain elevations are shown in Fig. B1 </p><p>83 below. Let fu(i,j) be the unsmoothed forecast datum at location (i,j) , be it POP12 probabilities or</p><p>84 QPF06, and let fs(i,j) be the S-G smoothed forecast datum. The final forecast product f(i,j) is </p><p>85 then</p><p>86 , (B4)</p><p>87 where the weight is determined from the local standard deviation of the terrain height in a 3 x 3</p><p>88 box centered on the grid point (i,j) of interest. Define a pre-weight e as</p><p>89 -13). (B5)</p><p>90 The weight then is</p><p>91 w = e if 0.0 < e < 0.6</p><p>92 = 0 if e ≤ 0.0</p><p>93 = 0.6 if e ≥ 0.6 (B6)</p><p>94 A map of this weight is shown in Fig. B2.</p><p>95</p><p>96 References:</p><p>97 Barnes, S. L., 1964: A technique for maximizing details in numerical weather map analysis. J. </p><p>98 Appl. Meteor., 3, 396 409.</p><p>99 Hamill, T. M., M. Scheuerer, and G. T. Bates, 2015: Analog probabilistic precipitation forecasts </p><p>100 using GEFS Reforecasts and Climatology-Calibrated Precipitation Analyses. Mon. Wea.</p><p>101 Rev., 143, 3300-3309. Also: online appendix A and appendix B.</p><p>102 103 104 Figure B1: Terrain elevations at ⅛-degree grid spacing.</p><p>105 106 Figure B2: Weight applied to the unsmoothed input fields.</p>