Online Appendix B: Application of Savitzky-Golay Smoothing

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-

12 G) smoothing and the general rationale for its use. The main difference in this algorithm versus

13 the one documented previously is: (a) the use of higher-resolution terrain data sets for

14 determining the relative terrain roughness, which is subsequently used to specify how much to

15 weight the smoothed field relative to the unsmoothed, and (b) the use of different constants in

16 the combination in accordance with the finer-resolution terrain. As before, a window size of 9

17 grid points and using a third-order polynomial is used in the S-G smoothing.

18 The smoothing algorithm follows several steps:

19(a) Copy the input forecast data we seek to smooth into a work array, be it POP12 or QPF06.

20 Adjust the work array values at grid points outside the CONUS borders to be a weighted linear

21 combination of data from nearby points inside the CONUS, for reasons explained later. Apply

22 S-G smoothing to the resulting work array.

23(b) Adjust the final values for points outside the CONUS to taper from the smoothed values

24 produced in (a) above to raw input value as one considers grid points further and further away

25 from the CONUS.

26(c) Produce the final smoothed field inside the CONUS from a linear combination of the S-G

27 smoothed (the work array) and the raw input data, with more weight to the S-G smoothed value

28 in flat terrain and more weight to the raw input data in rugged terrain.

29 30 We now describe each of these steps in more detail. Before S-G smoothing, we first do

31 a preliminary smoothing to points outside the CONUS, step (a) above. Why a preliminary

32 smoothing? The S-G algorithm uses a 9 x 9 stencil of points as inputs for smoothing. Suppose

33 there were radical differences in the data values the CONUS border because quantile mapping

34 was applied previously inside the CONUS, and no quantile mapping was applied outside the

35 CONUS. Then when applying S-G smoothing along the border in step (b) below, the points just

36 inside the CONUS border could be deleteriously affected by points with uncorrected data

37 outside the border. Adjusting the work-array values outside the CONUS minimizes such

38 problems.

39 The approach for pre-smoothing was inspired by a simple objective analysis procedure

40 akin to a single-pass Barnes analysis (Barnes 1964). The work array output value assigned to a

41 given point outside the CONUS consisted of a weighted combination of the work array input

42 values from nearby grid points inside the CONUS. Define a length scale L=3 grid points for the

43 spatial weighting function. Define a cutoff radius R (= 10 grid points). Then for a given location

44 (i, j) outside the CONUS, define SL as the set of indices of all M CONUS grid points within a

45 cutoff radius distance from (i, j), SL = [(i, j)1, …, (i,j)M]. Assume there was also a set of

46 associated distances DL = [D1, …, DM] that provided the distance of each in grid points from (i, j).

47 For the kth of M grid point locations, the distance-dependent weighting function wk is calculated

48 according

49 , (B1)

50 which is Gaussian in shape. Then the final smoothed work array output value assigned to this

51 location is

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

54 cutoff radius. For points inside the CONUS, is simply set to the input work array value p(i,j). In

55 the exceptional case where M=0, then is set to the average over the whole CONUS.

56 We now apply the S-G smoothing filter to the resulting work array values, producing a gridded

57 output. Again, see online appendix A of Hamill et al. (2015) for more details on the S-G

58 smoothing.

59 Now, the second step, (b) above. This algorithm has now produced smoothed values

60 inside the CONUS domain. We return in step (c) to consider how to use these smoothed

61 values. But first we describe the procedure for setting the final value at points outside the

62 CONUS? In these locations we were not provided with training data for post-processing, though

63 we now have two potential sources of information: (1) the smoothed field produced through eqs.

64 (B1) and (B2) above, reflecting a weighted combination of values from inside the CONUS, and

65 (2) the raw model guidance. Were we to use the raw values, be they POP12 or QPF06, we

66 would sometimes notice a discontinuity along the CONUS boundaries as a consequence of

67 applying quantile mapping inside the CONUS and no quantile mapping outside the CONUS.

68 The resulting field may be visually unacceptable to forecasters. Accordingly, we decided to

69 taper between the two, using mostly the S-G smoothed field for grid points outside but adjacent

70 to the CONUS, the raw model guidance for grid points some grid points distant from the

71 CONUS border, and a weight of the two in between. For a grid point outside the CONUS that

72 is d grid points away from the nearest point inside the CONUS, we produce a weight for the

73 smoothed ws according to

74 if d < 16 (B3)

75 = 0 if d ≥ 16.

76 The resulting output field for points outside the CONUS is then ws ✕ the S-G smoothed field + (1

77 - ws ) ✕ the raw input data. 78 The final step in the smoothing algorithm, part (c) above, is to linearly combine the S-G

79 smoothed field with the unsmoothed field inside the CONUS, weighting the unsmoothed more in

80 regions of great terrain roughness. The difference here relative to the algorithm described in

81 Appendix A of Hamill et al. (2015) is a change to using a higher-resolution terrain height data

82 set when defining the terrain roughness. The ⅛-degree terrain elevations are shown in Fig. B1

83 below. Let fu(i,j) be the unsmoothed forecast datum at location (i,j) , be it POP12 probabilities or

84 QPF06, and let fs(i,j) be the S-G smoothed forecast datum. The final forecast product f(i,j) is

85 then

86 , (B4)

87 where the weight is determined from the local standard deviation of the terrain height in a 3 x 3

88 box centered on the grid point (i,j) of interest. Define a pre-weight e as

89 -13). (B5)

90 The weight then is

91 w = e if 0.0 < e < 0.6

92 = 0 if e ≤ 0.0

93 = 0.6 if e ≥ 0.6 (B6)

94 A map of this weight is shown in Fig. B2.

96 References:

97 Barnes, S. L., 1964: A technique for maximizing details in numerical weather map analysis. J.

98 Appl. Meteor., 3, 396 409.

99 Hamill, T. M., M. Scheuerer, and G. T. Bates, 2015: Analog probabilistic precipitation forecasts

100 using GEFS Reforecasts and Climatology-Calibrated Precipitation Analyses. Mon. Wea.

101 Rev., 143, 3300-3309. Also: online appendix A and appendix B.

102 103 104 Figure B1: Terrain elevations at ⅛-degree grid spacing.

105 106 Figure B2: Weight applied to the unsmoothed input fields.