J. Lin and R. S. Stein,

Coseismic folding, earthquake recurrence, and the 1987 source mechanism at Whittier Narrows, Los Angeles Basin, California, Journal of Geophysical Research, 94, pp. 9614-9632, 1989.

[Online article] [Table 3A]

APPENDIX: DATA REDUCTION AND ERROR ANALYSIS

In spirit leveling, height differences are measured between adjacent BMs by sighting a horizontal telescope on graduated rods. Sources of leveling error are dominated by elevation-dependent systematic errors. These include improper calibration of the graduated leveling rods and atmospheric refraction of the line of sight between the rods and the horizontal telescope. The NGS survey follows Ist Order Class II (single-run) procedures [Federal Geodetic Control Committee (FGCC), 1984], while the LACO surveys furnish unadjusted double-run field elevations, with no corrections for atmospheric refraction error, rod scale error, or rod thermal expansion. Briefly, we find that the 1987 NGS leveling has little rod scale error, whereas the typical rod scale error in 1986 LACO leveling is large, at -120 ppm times the topographic height difference. If uncorrected, this would result in a 25-mm error over the 200-m height difference of line 1, 8 mm over line 2. The cumulative atmospheric refraction errors for both 1987 NGS and 1986 LACO surveys are 7-8 mm. Since the corrections are similar for both surveys, refraction corrections are unnecessary. The cumulative random error, which accumulates with the square of the leveling distance, is 9.5 mm over line 1, and 8.5 mm over line 2. Uncertainty is assigned to each observation based on the expected random error. Other sources of noise in the data include random and systematic survey errors and non tectonic subsidence caused by fluid and gas withdrawal. .

Surveying Errors

The 1987 and 1986 rod scale error. Rod scale error arises when leveling rods are shorter or longer than indicated by the graduations. Rod scale errors can result in severe elevation-dependent survey errors [Strange, 1980; Jackson et al., 1981; Stein, 1981]. The 1987 NGS leveling was carried out with invar rods which were calibrated at the National Bureau of Standards with a laser interferometer. The rod graduations are accurate to 0.03 mm at the 95% confidence level. This yields a elevation-dependent survey error of only +-10 ppm times the topographic height difference. Since the LACO rods used in the 1975 and 1986 surveys did not meet FGCC [1984] standards for design, construction and maintenance, and since they were not calibrated before or after use, we investigated rod errors.

We examined all surveys for elevation-dependent leveling error by comparing measured elevation changes with topography along the leveling route. Comparison of the 1987-1986 elevation changes with the topography along leveling route (Figures A1a and A1c and A2a and A2c) yields no visually obvious correlations. A more rigorous statistical analysis was carried out following Stein [1981], in which tilt, the difference in elevation change between adjacent BMs, normalized by the distance between the BMs, was regressed against the slope, the topographic height difference between the same pair of BMs. To separate the influence of the earthquake-related deformation on line 1, we divided this N-S trending profile into two segments (north and south), so that each segment has roughly uniform tilt. The results of the statistical analysis are shown in Table Al. For the northern segment of line 1, the inferred mean elevation-dependent error in 1987-1986 elevation changes is -155+-82 ppm. Combining this with the -36+-130 ppm found for the southern segment of line 1 and the -147 +-63 ppm found for line 2, we obtain an average elevation-dependent error of -119+-5 ppm. Since the rod scale error in the calibrated 1987 NGS leveling is small, we attributed all of the elevation-dependent error found in 1987-1986 profiles to the 1986 LACO survey. Note, however, that the correlation shown in Table Al is not significant at 95% confidence level, indicating a large margin of uncertainty.

Table A1: Tilt-Slope Statistics of the 1987-1986 Elevation Change

Line No.	Regression of Tilt on Slope, ppm	Correlation Coefficient, r	Number of Observations, n
Line 1-north	-155 ± 82	-0.206	75
Line 1-south	-36 ± 130	-0.034	59
Line 2-all	-147 ± 63	-0.237	80
Weighted average	-119 ± 55

To test independently the possible scale error of the LACO rods used in the 1986 survey, the 1986 LACO rods were calibrated at the Navy Gage and Standards Center in January 1988, 2 years after the 1986 survey and following considerable additional field use. Based on visual inspection by the LACO field party chief, we are reasonably certain that the rods that we calibrated are the rods used in 1986 survey, although no records nor rod serial numbers were found. At the time of 1988 calibration, the two rods were in poor condition. The frames were bowed from over tensioning of the graduated tape, and the tapes were kinked, bent, and rusted; there was no spring tension suspending the tapes in the rod frames. The calibration records as shown in Figure A3 suggest rod scale errors of -215 to -240 ppm; when the nonlinearity of the error is taken into account, the field errors would correspond to -317 to -421 ppm, 15 times the typical scale error in NGS leveling roods [Stein, 1981]. The appearance of the rods at the time of calibration suggests that they might have been damaged repeatedly in use, although it is not clear how much of the abuse occurred after the 1986 survey. Since no original manufacturer's calibration certificates exist (the rods are apparently 8-10 years old), we cannot assess whether the rods shortened with time. If, during the 1986 survey, the rods were indeed in their 1988 condition, then about 80 mm of elevation-dependent leveling error is expected over the 200 m of differential topography along line 1.

Next we apply the 1988 calibration to the 1986 survey to obtain corrected 1987-1986 elevation changes; these are shown in Figures A4a and A5a. The correction technique is described by Stein [1981]. We then test for residual elevation-dependent error in the corrected data. Positive elevation-dependent correlations with a magnitude of 227 ppm (line 1) and 203 ppm (line 2) now appear in the data. The correlation is significant at the >99% confidence level. This suggests that when the -317 to -421 ppm graduation error is applied to the 1986 survey, the rods are over corrected by about 227 ppm. Thus the actual rod scale error in 1986 survey appears to be -156+-78 ppm.

Table A2: Tilt-Slope Statistics of the 1986-1975 Elevation Change

Line No.	Regression of Tilt on Slope, ppm	Correlation Coefficient, r	Number of Observations, n
Line 1-north	212± 64	0.328	75
Line 1-south	277± 58	0.384	59
Line 2-all	203± 50	0.411	80
Weighted average	226± 78

Based on the two independent analyses, we conclude that the elevation-dependent error in 1986 LACO survey is -120+-55 ppm and that the 1988 calibration is not applicable to the 1986 survey because of 1986-1988 rod deterioration. A -120 ppm correction is thus applied to the 1986 survey (Figures A1b and A2b), and the corrected leveling is used throughout this study.

The 1975 rod scale error. The 1975 LACO rods were, to the best of our knowledge, never calibrated and were destroyed in about 1980. An estimation of the 1975 rod scale error is necessary to use the 1975-1986 preseismic rate of elevation change to correct coseismic elevation changes for the long-term subsidence rate and to verify our scale error assignment for the 1986 LACO rods. Fortunately, the Engineers Office of LACO (incorporated into the Department of Public Works in 1985) conducted a standard 1st Order leveling survey in the Whittier Narrows area during April-July 1975, coincident with the Road Department survey. Since the Engineers Office used calibrated rods in their survey, their leveling offers an independent means to assess the 1975 LACO Road Department rod error. The difference between the Engineers Office and Road Department surveys is shown in Figure A6a for the northern most 20 km in line 1. Direct comparison of this difference with topography along leveling route (Figure A6c) reveals a strong correlation. A tilt-versus-slope plot indicates a 284+-58 ppm (>95% confidence level) of rod scale error in the Road Department survey (Table A2). When this error is removed, the difference between the two 1975 surveys is small (Figure A6b).

If the 120 ppm of 1986 rod scale error is combined with the 285 ppm of 1975 rod scale error, a 405 ppm of rod scale error is expected in the 1986-1975 elevation changes. This is consistent with a direct search for elevation-dependent error in 1986-1975 profiles (Figures A7a and A8a). A tilt-versus-slope plot for the same section of line 1 shows a 411+-112 ppm correlation at >99.5% confidence level, while a similar plot shows a 316+-116 ppm error for line 2, at 98% confidence level. A -400 ppm correction is thus applied to the 1975 leveling.

Atmospheric refraction error. Atmospheric refraction error arises due to the refraction of the line of sight through a vertically stratified air mass. The error is proportional to the vertical temperature gradient, the height difference between adjacent setups, and the square of the sight length (the distance between the instrument and rod) [Stein et al., 1986]. Because the mean sight lengths of the 1986 and 1975 surveys were nearly identical (31.3 and 32.9 m, respectively) and both surveys were conducted in the winter, during which typical southern California temperature gradients are about 1.0°C over the 0.5-2.5 m height difference [Stein et al., 1986], the cumulative refraction error for both surveys is 7-8 mm. Since only differential refraction is important for the elevation changes, refraction corrections are unnecessary.

Random error. Random leveling error propagates as the square root of the leveled distance and is independent of leveling path. A good measure of the random error can be obtained by examining the misclosures around circuits of different lengths or by comparing multiple surveys of a subset of BMs. By summing elevation changes around each circuit, it is possible to test the internal consistency of the leveling data. If the survey is free of error, then the sum of elevation changes around any circuit would be equal to zero.

All 1986 LACO leveling circuits that connect with lines 1 and 2 are used to assess error in the 1986 leveling; the results are shown in Figure A9a. The observed misclosure in each circuit is plotted as a function of the leveling length around the circuit in Figure A9b. Also plotted in Figure A9b is a reference line displaying a theoretical relationship between random error and leveling distance: m = (distance)^1/2, with =1.2 mm(km)^-1/2 as an example. The mean observed value of for the 1986 LACO survey is ₁₉₈₆= 1.2 mm(km)^-1/2.

Since no circuits were leveled for 1987 survey, the standard deviation of a double-run section () is used instead. In double-run leveling, the mean difference for all (forward-backward) runnings must not differ significantly from zero, and the standard deviation of the difference gives a good indication of the random error per section. If follows a Gaussian distribution, then ₁₉₈₇==0.74 mm(km)^-1/2 The propagation of random errors in 1987-1986 elevation changes is then determined by _net=(₁₉₈₆²+₁₉₈₇²)^1/2 =1.4 mm(km)^-1/2, predicting 9.5 mm of cumulative error over the 40-km-long line 1 and 8.5 mm over the 35-km-long line 2. We take random errors to propagate from the junction of lines 1 and 2 (BM CG 4113; cross in Figure 7), since we solve for the elevation change of this point as a free parameter in this inversion.

Subsidence Error

To correct for nontectonic subsidence or uplift during the 1.7-year coseismic period (January 1986 to November 1987), we use the average subsidence rate obtained for the 1986-1975 preseismic period. By doing so, we have assumed that the non tectonic subsidence in the Whittier Narrows area was constant during the 11-year period between 1975 and 1986. This assumption gains some validation by the relatively stable fluid withdrawal rates of the oil fields (Figure A10). The typical subsidence correction for the coseismic period is +-3 mm and the uncertainty of the correction is assumed to be 25%.

The principal feature of the preseismic deformation on line 1 is a confined subsidence hole in Pico Rivera, at leveling distance of 22-35 km (Figure A7b). There is no apparent correlation between the subsidence and water table changes (compare Figures 5a and 5b). Because the Pico Rivera hole is also seen when differencing the 1987 NGS survey and the 1975 LACO Engineers Office 1st Order survey, we are certain that it is not an artifact of either the 1975 or 1986 LACO Road Department surveys.

Metropolitan Los Angeles is actively pumped for groundwater, and line 1 bisects the Whittier Narrows Flood control basin (Figure 1b), which is an infiltration facility designed to stabilize long-term water table drawdown. Line 1 also crosses both the Montebello and Santa Fe Spring oil fields and skirts the Montebello gas storage facility. The net oil+water+gas withdrawal at the Montebello oil field has been steady at 7000 Millions of band per year since 1974 (Figure A10a). Recharge to the Santa Fe Springs oil field has exceeded net liquid withdrawal during some periods; since 1982 the net rate of fluid and gas withdrawal has averaged 2500 Million of bands per year (Figure A10b). The Montebello Gas storage facility lies at a depth of 2.3 km and is centered 3.5 km west of line 1 (Figure 1b). It contains 1.2 x 10⁹ m³ of gas at a reservoir pressure of 2 x 10⁷ dyne cm^-2 (20 bar). To account for possible elevation changes due to the non tectonic activities in these locations, we assigned 50% greater uncertainty to observations within these areas.

The final coseismic elevation changes, corrected for leveling and subsidence errors, is given in TABLE A3. Note that a constant can be freely added to all observations.

Acknowledgments. We are grateful to Donald Huff, William Voge, and Richard Mitchell of the Los Angeles County Department of Public Works and Bob Martine of the National Geodetic Survey for providing the leveling data used in this study. Lucile Jones made copies of the CIT/USGS southern California network data available. Discussions with Robert Yeats and our colleagues at the USC Whittier Narrows Earthquake briefing and reviews by Malcolm Johnston, Frank Wyatt, Goran Ekstrom, and Greg Hirth greatly improved our understanding. This work was carried out while Jian Lin was a visiting scientist at the U.S. Geological Survey.