.-. THE EFFECT OF THE EARTH'S OBLATE SPHEROID SHAPE ON THE ACCURACY OF A TIME-OF-ARRIVAL. LIGHTNING GROUND STRIKE?LOCATING SYSTEM

Paul W.Casper, PE and Rodney B. Bent, Ph.D.

Atmospheric Research Systems, Inc. 2350 Commerce Park Drive, NE,Suite 3 Palm Bay, Florida 32905 U.S.A.

ABSTRACT

The algorithm used in previous technology time-d-arrival lightning mapping systems was based on the assumption that the earth is a perfect spheroid. 'I'hese systems yield highly-accurate lightning locations, which is their major strength. However, extensive analysis of tower strike data has revealed occasionally significant (one to two kilometers) systematic offset errors which are not explained by the usual error sources. It has been determined that these systematic errors reduce dramatically (in some cases) when the oblate shape of the earth is accounted for. The oblate spheroid correction algorithm and a case example is presented in this paper.

INTRODUCTION it becomes essential to base all mathematics on an extremely accurate earth model. Lightning ground strike tracking systems based on the time-of-arrival (TOA) technique, in Tracking systems using terrestrial timing syn- combination with wideband waveform detection chronization signals (e.g., LORAN) are doubly have been in operation for almost a decade. The affected by the earth's oblate shape. The follow- accuracy of these systems has steadily improved ing section identifies the affected parameters and as more has been learned about the fine char- discusses the methods for improvement. Subse- acte ristics of timing synchronization signals, quent sections contain illustrations of the magni- propagation effects on lightning waveforms and tude of each effect. software processing methods for accuracy en- hancement. One of the more recent improve- MATHEMATICAL PROCEDURE ments has been accomplished by refinement of the mathematics to more accurately accommo- The oblate shape of the earth spheroid directly date the oblate shape of the earth spheroid. affects TOA system accuracy in two ways:

Earlier versions of TOA lightning tracking sys- a. Calculation of time clock offsets due to timing tems used mathematics based on a spherical reference signal differential propagation delays approximation of the earth. This is usually not seriously in error, especially if the accurate b. Calculation of lightning stroke coordinates earth's radius for the region of interest is used in given a set of accurate receiver time differences the approximation. However, when accuracies otherwise are approaching a few hundred meters, Different mathematical processes are involved in each of these steps. Derivation of the equations

81-1 is beyond the scope of this paper (consult refer- figure provided by the U.S. Coast Guard for the ences [l,2, 3]), but the methodology for each is LORAN navigational system. This 100 KHz described in the following two subsections. velocity figure is determined as follows:

CALCULATION OF TIME CLOCK OFFSETS V' = Vln (1)

Receiver sites and a terrestrial timing reference where: V' = 100 KHz ground wave velocity signal transmitter are at fixed locations. There- fore, the clock offsets due to timing signal dif- n = index of refraction at the earth's ferential propagation delays need only be calcu- surface for 100 KHz (1.000338) lated one time and stored in the system software. In order to determine these differential offsets, V = free space velocity (299,792,458 very accurate geodesic distances between the me terslsecond) timing transmitter and each LPATS receiver must be first computed. A geodesic is defined as Accurate time clock offsets are then easily calcu- the curve of minimum length between two points lated: on the surface of a spheroid [l]. T12 = V' (D1 - D2) (2) Geodesic distances can be estimated fairly accu- rately by using a perfect sphere model of the where: D1 = geodesic distance from receiver 1 earth with a radius equivalent to the earth radius to the timing transmitter at the mean latitude of the network. However, data to be presented in later sections shows that D2 = geodesic distance from receiver 2 to achieve systematic location errors of less than to the timing transmitter several hundred meters, the actual shape of the earth must be properly accounted for. Equation (2) produces the time offset between clocks in receivers 1 and 2. This figure for each The non-iterative solution by Sodano [l] pro- clock pair in the system is stored by the central vides a highly-accurate means of calculating software and used to correct the time differences geodesic distances. Sodano developed a system actually reported by the receivers. of equations that are easily programmed and solved in double precision using a personal com- CALCULATION OF STROKE puter equipped with a mathematic co-processor. COORDINATES FROM TIME DIFFER- The input to this set of equations is only the lati- ENCES tude and longitude (accurate to the fourth deci- mal place) of the two points of interest. Naviga- The second part of the problem is computation tion receivers using the satellite-based Global of the stroke coordinates given accurate input Positioning System are most convenient for time differences. Razin [2] describes the math- determining the coordinates of receivers to the ematics for accomplishing this computation for required accuracy. both the spherical earth approximation and for the more exact oblate spheroid case. Once accurate geodesic distances are available, Razin does not present the equations for the they must be converted to an equivalent propaga- oblate case. Fell [3] fills in this void. The full set tion time. For this, we need an accurate ground of equations is quite complex and extensive, and wave propagation velocity figure, applicable for the reader is referred to both Razin and Fell for the high-energy frequencies radiated during the the details. The computational procedure will be first ten microseconds of a lightning ground described here to provide general understanding stroke. A figure for frequencies in the 50 to 300 of the process. KHz range is appropriate, such as the 100 KHz

81-2 The first step in the process is to map the re- actual tower strikes. Although the cluster pattern ceiver coordinates from the spheroid (earth) is identical to the tower pattern, there is an onto an osculating sphere, internal to the apparent systematic offset to the southwest. Af- spheroid, which is tangent to the spheroid at a ter investigation of all the usual sources of point Po. This tangent point is selected near the systematic error (receiver coordinates, computa- center of the receiver network. Figure 1 illus- tional error, etc.), it was determined that the er- trates this arrangement in two dimensions for ror was substantially due to time clock propaga- clarity. Once all the receiver coordinates (PR) tion offsets and the spherical approximation are mapped onto the osculating sphere (PB), the stroke coordinate solution mathematics. Here solution process proceeds as if the earth is a per- we will examine the error due to time clock off- fect sphere, using the Pk set of receiver coordi- sets and address mathematical error in the next nates. When the stroke coordinates on the section. osculating sphere are computed (PIs), they are mapped back to the sphtvoid (Ps) using a very I:igure 4 is a blow-up of the area around towers simple equation pair. #1 and #2. A 200m x 200m grid has been su- perimposed to aid in scaling distance. The cen- Note that the osculating sphere receiver coordi- troid of the cluster near tower #1 is seen to be nates need only be computed one time. They can approximately 590 meters southwest of the then be stored as fixed constants in the LPATS tower. The same is true of the #2 tower and central computer software and used with the re- stroke cluster. The circles around the tower indi- ported time differences to calculate each stroke's cate the approximate attractive radius of each coordinates. The burden on the real time central tower due to its height. It was determined from computer software is substantially unchanged the data base that receiver triad 1-4-5 was used to from the all-spherical case. locate virtually all the strokes in the Bithlo tower clusters, which is in fact the triad the system ERROR DUE TO TIME CLOCK OFFSETS should have used since it is the optimum combination (least affected by timing errors for Offsets in the receiver time clocks, because of the Bithlo area). We will concentrate on this re- spherical approximation error in the offset cor- ceiver triad to examine the spherical approxima- rection constants, can be significant. Here we tion time clock offset error. will examine an actual case to illustrate the point. The centroid of the actual tower #1 cluster is The five Florida LPATS network receiver loca- shown in Figure 5 (point A). Point B is a theo- tions are shown in Figure 2, plus the Jupiter, retical point, computed as follows: Florida, LORAN transmitter that is used for synchronization. This network has been in op- a. A stroke location at the exact coordinates of eration for several years, and a substantial the tower was assumed. archive of data has accumulated. In an effort to objectively assess the accuraj of this network, b. The exact propagation times to receivers 1,4 the data base was searched over a complete light- and 5 were computed using the Sodano ning season in the vicinity of three very tall, method. attractive objects ("Bithlo towers") to determine if an expected pattern of strikes existed. Each of c. nerelative stroke time differences seen by these towers are over 1,000 feet in height. The each receiver pair were then computed using location of the three Bithlo towers relative to the the propagation times. These time differ- network is shown on Figure 2. The pattern of ences should be exactly what the system strikes in the vicinity of these towers is shown in actually saw, to the extent that it is possible Figure 3. Note that there is an unmistakable to predict them. stroke cluster near each tower, which can only be

8 1-3 d. The 100 KHz timing signal propagation computation given receiver time differences. times from Jupiter to receivers 1, 4 and 5 The magnitude of this error contribution is il- were computed in two ways: 1) with lustrated here. spherical approximation mathematics, and 2) wiih Sodano's oblate spheroid mathematics. Refer again to Figure 5. With the time clock Secondary correction factors (due to earth offset errors removed, the solution moves from B conductivity effects) were added to all times. to C. Point C was calculated using the same solution mathematics as used by the system at the e. Time clock offset errors were computed by time, which was based on a spherical earth taking the differences between the spherical approximation. If we instead use the oblate and oblate times by receiver. spheroid mathematics as described previously, point C moves to D, which is only 205 meters Point B in Figure 5 is what results when actual from the tower. The net result of removing the time differences from sic P "c" are corrupted with time clock offset and improving the the time clock offset error3 picdicted in step "e" mathematical accuracy is a 50 percent reduction This very closely approximates the operatinl; of error from 405 meters (point B) to 205 meters condition of the system at the time, which was (point D) . using time clock offset correction constants computed with spherical approximation The residual error of 205 meters is still under mathematics. Point B is 405 meters southwest of investigation but is believed to be due to ap- the tower, short of the actual 590 meters, but proximations in the derivation of the oblate certainly explaining the majority of it. Point B mathematics. Further findings will be presented was computed with spherical approximation at the conference. coordinate solution mathematics, exactly the same as the system was using. DISCUSSION

We had to include the time clock offset errors We have seen that moderate systematic error in determined in step "e" to simulate actual system the TOA LPATS can arise in two ways from conditions and calculate point B. We can just as using a spherical earth approximation: 1) de- easily remove the errors and observe the change termination of time clock offsets, and 2) calcu- in system accuracy that should result. Point C lation of stroke coordinates from detected time indicates the solution location with zero time differences. In the example presented in Section clock offset error, using the step "ctt time 3, removing time clock offset error resulted in an differences and spherical approximation 820-meter shift in the computed coordinates. In coordinate solution mathematics. This point is this particular example, the radial position error 415 meters northeast of the tower, which is happened to remain about the same, but this was actually a slight degradation in accuracy. only a fortuitous result. With a different triad of However, there remains a second source of error receivers or a stroke in a different location, it is due to inaccurate solution mathematics, which possible that resulting error could be even larger was based on a spherical earth approximation. than the original error as long as spherical This error source is examined in the next section. solution mathematics is still used. Multiple errors can sometimes combine such that ERROR CAUSED BY SOLUTION remaining error is larger when one error source MATHEMATICS is removed. It is important to reiterate that time clock offset error is quite easy to avoid in a As noted at the beginning of Section 2, approx- system with a terrestrial timing reference signal imating the earth as a perfect sphere affects not by using accurate propagation time prediction only the accuracy of time clock offset calcula- software that accommodates the oblate earth tions, but also the accuracy of stroke coordinate shape (i.e., the method of Sodano). In systems

8 1-4 that are synchronized by satellite signals, the earth's oblate shape is handled differently, by including in the line-of-site calculations the elevation of the receiver site above sea level combined with the accurate earth radius for the applicable latitude.

The example illustrated that oblate solution mathematics can also provide a substantial sys- tematic error reduction, in this case 50 percent. Even though the oblate mathematics is more complex than sphcrical mathematics, almost all of the additional complexity is involved in the initial one-time sysbm set-up constant calculations, and there is pi acrizally no additional load on the real time LPATS Central Analyze1 software.

The accuracy of all lightning ground strike tracking technologies in current use is affected by the earth's oblate shape. It has been shown in this paper that errors in a TOA system, due to approximating the earth as a perfect sphere, can be significant enough to warrant attention, especially when inherent system accuracy is good enough to be limited in large part by this error source.

REFERENCES

111 O'Conner, J. J., "TransformationsApplicable to Missile and Satellite Technology", Technical Memorandum ETV-73-27, Air Force Eastern Test Range, Patrick Air Force Base, Florida, 26 April 1973 (specifically, Appendix F Part A entitled "Sodano's Noniterative Solution of the In- verse Geodetic Problem").

[2] Razin, S., "Explicit (Noniterative) LORAN Solution", J. of Inst. of Navigation, Vol. 14, No. 3, Fall 1967, pp. 265-269.

[3] Fell, H., "Comments on LORAN Conversion Algorithm", J. of Inst. of Navigation, Vol. 22, No. 2, Summer 1975, pp. 184-185.

8 1-5 Spheroid Osculating Sphere (Earth) \

Figure 1: Mapping a Point PR on the Spheroid to a Point on an Osculating Sphere.

8 1-6 Bithlo Tower Data UITKUOE 2861

2810 * * * ** c * 28.9 1 * '1 ** * ** * ** ** ** * * ** t 28 56 -31 12 31 11 -a1 io ill 09 dl 08 -31 01 -a C6 -81 :5

LCNOTUDE

Figure 3: Bithlo Tower Lightning Strikes

avg Lat./Lon. = 2a.w6a, -81.096a

NTowers No. 1 and No. 2

Canter of graph: Latitude: 28.6034 N Longitude: -81.0903 W

Grid size: 2kmx2km

Cell size: 200m x 200m Figure 4: Strikes to Bithlo Towers #1 & #2

81-7 Towe

415 m

Centroid of the stroke cluster for tower #1. 0Theoretical solution, with predicted timeclock offsets. 0Theoretical solution, with timeclock offsets removed. Theoretical solution, timeclock offsets removed and math error removed.

Figure 5. Triad 145 Solutions, as Affected by Timeclock Offset

8 1-8