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NOAA KLM User's Guide

Section 2

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Section 2.4: Interpolating the Level 1b Earth Location Data

The geographic location of AVHRR LAC/HRPT viewed areas presents a problem, since one scan line contains 2048 viewed spots, but only 51 of these are located in the Level 1b data; point 25 being the first located point, and then every fortieth point. The scan is from right to left, when facing in the direction of satellite motion. GAC data are less of a problem, since every eighth point is located. It should be noted, however, that the Earth locations given for GAC areas are not in the center of the area; see Section 2.4.4. All other instruments currently on NOAA satellites have all observed areas located. Since investigators often require the location of all observed points, the locations for points intermediate between those given in the Level 1b data must be obtained by interpolation. The common method is to interpolate linearly separately in latitude and longitude between given points. This study was undertaken to determine the accuracy of such linear interpolation, and if this method should prove insufficiently accurate, to determine an interpolation method which would yield acceptable accuracy.

2.4.1 Method

In order to ascertain the accuracy of an interpolation method, it is necessary to know the true location of every point. In order to simplify the computations, a spherical earth was assumed, permitting the use of spherical trigonometry. The radius was taken to be 6371 km, the approximate mean radius of the actual Earth.

The satellite altitude was assumed to be 850 km with an inclination, β, of 99 degrees, and assumed to be in the first quarter orbit (northbound, in the Northern Hemisphere), and only the right (first) half of the scan was considered. These assumptions will have no effect on the results, since the configuration is completely symmetrical. The trace of the scan line on the earth may be assumed to be a great circle, especially since only the portion of a scan which includes at most five Level 1b located points will be examined at one time. The time for this to occur is about 5 milliseconds, during which the sub-satellite point will have moved about 35 meters. The satellite position was therefore taken as fixed, and the 1024 "true" scan locations calculated, using a scan step of 0.0541 degrees. It should be noted that this scan step, combined with the field of view of the sensor, results in an oversampling of 1.362 samples per field of view; in other words, the LAC data overlap (see Figures 2.4.4-1 and 2.4.4-2). No attempt was made to do any calculations which involved using points on both sides of the nadir point. This is of little consequence, since interpolated positions are most accurate near nadir.

Since all calculations assume instantaneous conditions, it is not necessary to consider Earth rotation. φ0 is the satellite latitude and zero longitude was taken to coincide with the sub-satellite point. From Figure 2.4.1-1, the angle at the sub-point from north to the sub-track is:

alpha sub T= arcsin{ (-cos beta / cos phi sub 0)}
and the angle to the scan line is:
alpha sub L = alpha sub T - 90 degrees

Figure showing relationship of sub-satellite point to Earth

For a viewing nadir angle (scan angle) σ, the corresponding geocentric arc distance is given by:
theta=arcsin{({{R+H}/R} sin sigma)}-sigma


where R is the Earth radius and H is the satellite height (see Figure 2.4.1-2). From the spherical triangle shown in Figure 2.4.1-3, the latitude of the viewed point, φ, may be found to be:
sin phi=sin phi sub 0 cos theta + cos phi sub 0 sin theta cos alpha sub L

and the longitude relative to the sub point by:
cos Delta lambda= {cos phi sub 0 cos theta - sin phi sub 0 sin theta cos alpha sub L}/cos phi

Performing these computations for each scan step, starting at an angle from nadir of half a scan step, defines the "true" Earth locations to which the interpolated locations will be compared.

Figure showing angular relationships between satellite, surface and Earth center

Figure showing relationship between viewed point and latitude and longitude

2.4.2 Results

The results for linear interpolation are shown in Table 2.4.2-1. The point numbers are those of located points, and the range of scan angles are shown for each group. In each group of 39 interpolated points, the mean error and the maximum error are shown, converted to linear distance in kilometers on the surface of the Earth. For all interpolation methods, the variation with sub-point latitude is small, as would be expected as a result of the spherical assumption. For an ellipsoidal Earth, the variation may be somewhat larger. Near nadir the errors are small, but become completely unacceptable by the time the most limbward pair of points are reached.

Table 2.4.2-1. Errors for Linear Interpolation Between Adjacent Located AVHRR Points for Latitude = 40 degrees.
Point Number Scan Angle Range(degrees) Mean Distance(km) Maximum Distance(km)
From To
1- 2 -54.073 -51.909 2.5082 3.8583
2 - 3 -51.909 -49.745 1.7198 2.6449
3 - 4 -49.745 -47.581 1.2518 1.9248
4 - 5 -47.581 -45.417 0.9497 1.4604
5 - 6 -45.417 -43.253 0.7427 1.1422
6 - 7 -43.253 -41.089 0.5944 0.9142
7 - 8 -41.089 -38.925 0.4844 0.7450
8 - 9 -38.925 -36.761 0.4004 0.6159
9 - 10 -36.761 -34.597 0.3348 0.5150
10 - 11 -34.597 -32.433 0.2825 0.4346
11 - 12 -32.433 -30.269 0.2401 0.3694
12 - 13 -30.269 -28.105 0.2052 0.3157
13 - 14 -28.105 -25.941 0.1760 0.2708
14 - 15 -25.941 -23.777 0.1513 0.2327
15 - 16 -23.777 -21.613 0.1301 0.2002
16 - 17 -21.613 -19.449 0.1118 0.1719
17 - 18 -19.449 -17.285 0.0957 0.1471
18 - 19 -17.285 -15.121 0.0814 0.1252
19 - 20 -15.121 -12.957 0.0685 0.1054
20 - 21 -12.957 -10.793 0.0569 0.0876
21 - 22 -10.793 -8.629 0.0463 0.0712
22 - 23 -8.629 -6.465 0.0365 0.0561
23 - 24 -6.465 -4.301 0.0274 0.0422
24 - 25 -4.301 -2.137 0.0193 0.0297

The accuracy is improved by a three point interpolation in latitude and in longitude, using the Lagrangian interpolation algorithm (see Section 2.4.3 for a discussion of this algorithm). The use of the Lagrangian method does not seem to be critical, and any three point interpolation seems to be equally good. As shown in Table 2.4.2-2, the accuracy near nadir becomes even better, while the limbward interval improves markedly, the maximum error being only 2/3 km.

Table 2.4.2-2. Errors for Lagrangian Interpolation Between Three Adjacent Located AVHRR Points at Latitude = 40 degrees.
Point Number Scan Angle Range (degrees) Mean Distance(km) Maximum Distance km)
From To
1- 2 -54.073 -51.909 0.4251 0.6758
2 - 3 -51.909 -49.745 0.2495 0.3961
3 - 4 -49.745 -47.581 0.1598 0.2534
4 - 5 -47.581 -45.417 0.1088 0.1724
5 - 6 -45.417 -43.253 0.0776 0.1229
6 - 7 -43.253 -41.089 0.0574 0.0908
7 - 8 -41.089 -38.925 0.0436 0.0691
8 - 9 -38.925 -36.761 0.0340 0.0538
9 - 10 -36.761 -34.597 0.0270 0.0428
10 - 11 -34.597 -32.433 0.0219 0.0346
11 - 12 -32.433 -30.269 0.0180 0.0285
12 - 13 -30.269 -28.105 0.0150 0.0237
13 - 14 -28.105 -25.941 0.0127 0.0201
14 - 15 -25.941 -23.777 0.0109 0.0172
15 - 16 -23.777 -21.613 0.0094 0.0149
16 - 17 -21.613 -19.449 0.0082 0.0130
17 - 18 -19.449 -17.285 0.0073 0.0116
18 - 19 -17.285 -15.121 0.0066 0.0104
19 - 20 -15.121 -12.957 0.0060 0.0094
20 - 21 -12.957 -10.793 0.0055 0.0087
21 - 22 -10.793 -8.629 0.0051 0.0081
22 - 23 -8.629 -6.465 0.0048 0.0076
23 - 24 -6.465 -4.301 0.0046 0.0073

There remain the 24 scan points before the first located point (23 points after the last located point) the locations of which cannot be interpolated, but must be extrapolated. Table 2.4.2-3 presents the values extrapolated from the first three located points. Since the errors change rapidly, values are given for each point. In addition to the total error in kilometers, the individual errors in latitude and longitude are also presented. The error for point 25 is exactly zero, since this is a located point. The location errors grow rapidly, being over 5 km for the limbward point

Table 2.4.2-3. Errors for Lagrangian Interpolation From Three Limbward Located AVHRR Points at Latitude = 40 degrees.
LAC Point Scan Angle (degrees) Latitude Error (degrees) Longitude Error (degrees) Distance Error (km)
1 55.425 0.0056 -.0633 5.3122
2 55.371 0.0052 -.0588 4.9389
3 55.317 0.0048 -.0546 4.5818
4 55.263 0.0045 -.0505 4.2403
5 55.209 0.0041 -.0466 3.9140
6 55.155 0.0038 -.0429 3.6026
7 55.101 0.0035 -.0394 3.3055
8 55.047 0.0032 -.0360 3.0225
9 54.993 0.0029 -.0328 2.7531
10 54.939 0.0026 -.0297 2.4969
11 54.884 0.0024 -.0268 2.2535
12 54.830 0.0021 -.0241 2.0226
13 54.776 0.0019 -.0215 1.8038
14 54.722 0.0017 -.0190 1.5968
15 54.668 0.0015 -.0167 1.4012
16 54.614 0.0013 -.0145 1.2166
17 54.560 0.0011 -.0124 1.0428
18 54.506 0.0009 -.0105 0.8794
19 54.452 0.0008 -.0086 0.7260
20 54.398 0.0006 -.0069 0.5824
21 54.343 0.0005 -.0053 0.4483
22 54.289 0.0003 -.0039 0.3232
23 54.235 0.0002 -.0025 0.2070
24 54.181 0.0001 -.0012 0.0994
25 54.127 0.0000 0.0000 0.0000

Table 2.4.2-4 shows the same information except with the extrapolation done with a five point Lagrangian algorithm. This shows a marked improvement, with the maximum error being only slightly greater than 1 km. This would probably be sufficiently accurate for almost all purposes. It should be noted that while the nominal Earth field of view of AVHRR is 1.1 km, this is true only at nadir. The limbwardmost coverage is 6.5 x 2.3 km. A 1 km location error is thus much less than the size of a field of view.

Table 2.4.2-4. Errors for Lagrangian Interpolation From Five Limbward Located AVHRR Points at Latitude = 40 degrees.
LAC Point Scan Angle (degrees) Latitude Error (degrees) Longitude Error (degrees) Distance Error (km)
1 55.425 0.0014 -.0121 1.0231
2 55.371 0.0013 -.0111 0.9388
3 55.317 0.0012 -.0102 0.8595
4 55.263 0.0011 -.0093 0.7850
5 55.209 0.0010 -.0085 0.7150
6 55.155 0.0009 -.0077 0.6493
7 55.101 0.0008 -.0070 0.5878
8 55.047 0.0007 -.0063 0.5302
9 54.993 0.0007 -.0056 0.4764
10 54.939 0.0006 -.0050 0.4261
11 54.884 0.0005 -.0045 0.3793
12 54.830 0.0005 -.0040 0.3358
13 54.776 0.0004 -.0035 0.2953
14 54.722 0.0004 -.0031 0.2577
15 54.668 0.0003 -.0026 0.2230
16 54.614 0.0003 -.0023 0.1909
17 54.560 0.0002 -.0019 0.1613
18 54.506 0.0002 -.0016 0.1341
19 54.452 0.0002 -.0013 0.1091
20 54.398 0.0001 -.0010 0.0862
21 54.343 0.0001 -.0008 0.0654
22 54.289 0.0001 -.0006 0.0465
23 54.235 0.0000 -.0003 0.0293
24 54.181 0.0000 -.0002 0.0139
25 54.127 0.0000 0.0000 0.0000

2.4.3 Lagrangian Interpolation

Given a set of n coordinates, xi, yi, where i = 1 to n, a general value of y is given by

y = sum from i=1 to n of (y sub i L sub i

where
L sub i =product from j=1(not equal to i) to n of{ (x - x sub j)/(x sub i - x sub j)}

x being the value corresponding to the y which is being sought (the productsymbol indicates a product). It should be noted that the given x points must all be different. However, the points need not be equally spaced, and it is not necessary that they be in order (Meeus, 1991).

A schematic computer program for performing Lagrangian interpolation follows:

n Number of input points
x(1),y(1) x(2),y(2), ... x(n),y(n) Input points
x0 X for which Y is to be interpolated
... ...
y0=0 Interpolated value of Y
for i=1 to n
L=1
for j=1 to n
if j ≠ i then L=L*(x0-x(j))/(x(i)-x(j))
next j
y0=y0+L*y(i)
next i

2.4.4 Location of GAC Spots

GAC values are calculated on board the satellite by the following procedure (paraphrased from "Advanced TIROS-N Program", Programming and Control Handbooks for NOAA-KLM and NOAA-N, Section 5.5.3.2.3):

Processed GAC earth data is derived from the earth view portion of every third AVHRR scan line. The starting AVHRR scan is not specified, but the GAC lines are tagged with the times of the AVHRR scans from which they are derived.

The data for each of the five AVHRR channels is processed in accordance with the following five sample-averaging algorithms:

(1) Select only every third AVHRR scan line for data processing. Start with the first AVHRR data sample in the selected scan line.

(2) From the selected scan line, obtain 5 contiguous AVHRR data samples.

(3) Retain the data from the first four samples, and discard the data from the fifth sample.

(4) Form a sum by adding together the data from samples 1, 2, 3 and 4. Form the sum in 12-bit precision.

(5) Divide the sum by four to obtain an "averaged" GAC data word. Round the quotient to 10 bits.

(6) Repeat steps (2) through (5), starting with the next AVHRR data sample, a total of 409 times to generate the required GAC data words for one scan line. Since each AVHRR line includes 2048 data samples, the final 4 samples of the scan line will be skipped.

(7) Repeat steps (1) through (6) for all the AVHRR scan lines.

Thus, the first AVHRR data sample of each selected scan line is averaged into the firs GAC sample of its line, but the last four AVHRR samples of the line are skipped.

The earth location of the centers of 51 selected AVHRR data samples out of 4028 from each scan line are calculated by the Advanced Earth Location Data System during the preprocessing of the data on the ground, and they are included in the level 1B AVHRR data sets. The 25th data sample of a scan and every 40th thereafter are selected. These same earth locations are included in GAC level1B data sets for certain GAC samples. For an earth located GAC spot, the earth location matches the missing AVHRR sample from the 5 used in generating that GAC spot. The 5th GAC data sample of each GAC line and every 8th sample thereafter are earth located in level 1B.

Figure showing Earth locations of LAC and GAC spots near nadir

As shown in Figure 2.4.4-1, the earth location given for a located GAC spot is NOT that of the center of the spot, the location of the center of the skipped fifth AVHRR spot of the five which constitute a GAC.

Position of two LAC spots that straddle the nadir

Figure 2.4.4-2 shows the two LAC spots which straddle nadir. At this position, the Earth location given is displaced 0.4 km from the center of the corresponding GAC spot. At large nadir angles, the displacement will be larger, but since the Earth field of view will also be stretched in the scan direction (to 6.5 km for the limbwardmost LAC position), the proportional difference will remain the same.

Amended May 1, 2006


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