Interferometric Traveltime Tomography with Closure Phase

The Interferometric Traveltime Tomography (ITT) method was previously shown to be both effective and stable in eliminating either shotor receiver-static errors using synthetic and field data example. In this report, we apply the ”closure phase” theorem to traveltime tomography and test its effectiveness in eliminating both the shotand receiver-static errors simultaneously with synthetic crosswell data. Results show that the ITT method with ”closure phase” theorem, denoted as the CP method, can successfully eliminate both the shotand receiver-timing shifts. However, the cost is a loss of resolution that sometimes is untolerable. INTRODUCTION The ITT method was previously tested (Schuster and Zhou, 1999; Zhou, 1999) on synthetic crosswell data, VSP+crosswell data, surface refraction data, and field data with either random or systematic timing shifts. Results showed that the ITT method can eliminate either large random/systematic source or receiver timing shifts that are unacceptable by the standard method, provided the master trace for each shot gather is associated with the shortest raypath within the shot gather, and the initial velocity model contains the DC component of the slowness field. However, this ITT method can only deal with cases where there are only shot or receiver static errors, but not both. In contrast, field seismic data contain both shot and receiver static errors which should be considered together. The possible remedy is to apply the so-called ”closure phase” theorem to traveltime tomography which was proposed in earlier reports (Schuster and Zhou, 1999; Zhou, 1999). In this report, I will develop the CP method, discuss the validation of the inversion operator in the CP method, and then test the CP method on synthetic crosswell data with both shot and receiver timeshifts. METHODOLOGY We have compared the performance of the standard tomography method to that of the ITT method in the previous report (Zhou, 1999). To combine the ”closure phase” theorem with the ITT method, we need to modify the formulas for the misfit function and gradient. In the CP method, traveltimes are picked and 6 rays from 3 shots and 3 receivers are properly chosen to form closure, and the time difference of these 6 rays are then calculated (Zhou and Schuster, 2000). This time difference is denoted as the closure time. These time differences between observed and calculated closure times are then smeared along the corresponding rays. So the CP misfit function is given by