Open-File Report 96-517
To assign appropriate weight to each leveling measurement in the modeling process it is necessary to asses the accuracy of the data. This is accomplished by examining the difference between forward and backward leveling runs of each section between bench marks., or misclosures of the entire circuit of bench marks. Leveling surveys contain both systematic and random errors. Standards and practices for precision leveling are designed to reduce both types of errors [Federal Geodetic Control Committee, 1984]. The NGS classifies three standards of leveling; first, second and third-order. The most precise leveling is first-order. First and second order leveling has three subdivisions class 0, class 1, and class 2, class 1 being of higher precision than class 2. Procedures, equipment details, and precision are defined for each order and class.
The random error allowed to accumulate along a leveling line is expressed as , where L is the length of the line in kilometers. The value of a can be calculated in two ways. First, through agreement of the forward and backward measurements of the height differences between BMs and secondly through the misclosure of a leveling circuit. Using the former method, a is the standard deviation computed from several double run sections and is calculated as
(1)
where N is the number of double run sections, D j is the length in kilometers of the jth double-run section, and var j is the variance of that section. Double run sections are measured twice or more in a forward and backward direction. The variance is calculated as
(2)
where n is the number of measurements of the jth double-run section, hi is the observed height and h (h-bar) is the average height of the observed runnings.
An alternative way to assess the accuracy of a leveling survey is to determine the cumulative height difference for the measurements associated with a loop in a survey. This is referred to as the misclosure of a leveling loop. Here a is calculated as , where E is the misclosure in millimeters. In the absence of systematic errors the values of a calculated from the forward and backward differences and from misclosures should be about equal.
The principle sources of systematic errors in leveling data arise from incorrectly calibrated leveling rods and from atmospheric refraction effects. All surveys were tested for rod miscalibration by regressing the gradient of the topography against the elevation distance per kilometer [Stein, 1981]. If the survey was found to be contaminated by rod miscalibration errors then it was not used in this study. For example, a 1978 survey from Chatsworth to Woodland Hills showed a significant correlation between topography and elevation change. A 1974 survey of this route was therefore used to for the coseismic elevation changes along this line. All surveys in this analysis contained corrections for known systematic errors which occur in leveling, i.e., refraction, rod miscalibration, thermal expansion of the rods, collimation, and orthometric corrections.
For the first Order Class II leveling surveys made in 1994, the a values calculated using equation (1) ranged from 0.69 to 1.29 mm/km1/2 (Table 1). The a values for the preseismic data set where calculated from the difference between the forward and backward leveling measurements (equation 1) and ranged from 0.93 to 1.89 mm/km1/2 (Table 1).
The uncertainty in elevation change at a BM, , is given as
(3)
where and are the preseismic and postseismic a values respectively, and is the subsidence rate at that point. The uncertainty in relative elevation change between two BMs, is then
(4)
where L is the distance between BMs i and i+1.
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U.S. Geological Survey
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