Methods for estimating location using signal with varying signal speed

Abstract

Robust methods are developed to provide bounds and probability distributions for the locations of objects as well as for associated variables that affect the accuracy of the location such as the positions of stations, the measurements, and errors in the speed of signal propagation. Realistic prior probability distributions of pertinent variables are permitted for the locations of stations, the speed of signal propagation, and errors in measurements. Bounds and probability distributions can be obtained without making any assumption of linearity. The sequential methods used for location are applicable in other applications in which a function of the probability distribution is desired for variables that are related to measurements.

1 - 19 . (canceled) 
   
   
       20 . A method for analytically estimating at least one location of at least one primary point using isodiachronic location comprising the steps of:
 (a) estimating differences in travel time from at least one said primary point and a constellation of stations;   (b) estimating speeds of a signal between said primary point and each said station where at least one said speed is different than others;   (c) obtaining the analytical solution for said isodiachronic location of said primary point;   (d) stopping if there is no said solution for said location of said primary point;   whereby at least one said location for at least said primary point is estimated analytically using said isodiachronic location.   
   
   
       21 . The method of  claim 20  further including iterating said analytical solution for convergence when effects of advection significantly alter at least one said difference in travel time. 
   
   
       22 . A method of obtaining at least one function of a probability distribution for location of at least one primary point using isodiachronic location comprising the steps of:
 (a) estimating differences in travel time from at least number of stations in a constellation;   (b) establishing prior bounds for said location of said primary point and for all arguments of a primary function for said location of said primary point;   (c) assigning probability distributions to said arguments;   (d) generating realizations of said arguments to compute realizations of said primary function for said location within its said prior bounds using a Monte-Carlo technique;   (e) forming said function of said probability distribution for said primary point from said realizations;   (f) stopping when there are less than an acceptable number of said realizations of said primary point;   whereby at least one said function of said probability distribution for said location of at least one said primary point using said isodiachronic location is estimated.   
   
   
       23 . The method of  claim 22  further including obtaining at least one bound for location of at least one said primary point from at least one database of synthesized travel time differences. 
   
   
       24 . The method of  claim 22  further including forming a function of a probability distribution of at least one said argument from its posterior realizations. 
   
   
       25 . The method of  claim 23  further including forming a function of a probability distribution of at least one said argument from its posterior realizations. 
   
   
       26 . The method of  claim 22  further including using Sequential Bound Estimation to estimate at least one function of a probability distribution of at least one variable from at least one of its posterior realizations when the number of said stations exceeds said number in said constellation. 
   
   
       27 . The method of  claim 23  further including using Sequential Bound Estimation to estimate at least one function of a probability distribution of at least one variable from at least one of its posterior realizations when the number of said stations exceeds said number in said constellation. 
   
   
       28 . A method for analytically estimating a location of at least one primary point using isosigmachronic location comprising the steps of:
 (a) estimating sums of travel times for a constellation of stations and said primary point;   (b) estimating speeds of a signal between said primary point and each said station where at least one said speed is different than others;   (c) utilizing at least one said analytical solution for said location of said primary point;   (d) stopping if there is no said solution for said location of said primary point;   whereby said location of said primary point is estimated with at least one said analytical solution using said isosigmachronic location.   
   
   
       29 . The method of  claim 28  further including iterating at least one said analytical solution for convergence when effects of advection significantly alter at least one of said sums of travel times. 
   
   
       30 - 57 . (canceled)