NOAA/ OERD T-Phase Project: Location Analysis

Location Analysis is one of most used operations for the 2 T-Phase data processing programs. It is used to compute the original location in terms of Latitude, Longitude, and Origin time of the selected T-Phase events. When the arrival times of all the selected events and the locations of the hydrophones that recorded the all selected events are gethered, a Nonlinear Least-Square method is used to calculate the estimated location and origin time.

The Location Analysis allows users to select the T-Phase events. From there on, users can choose one of many options such as locate the origin, or examine the error surface in 3-D or 2-D contours plot which is useful to locate local solution if there is one. Also there is an option to allow users to locate the arrival times of T-Phase events. When given a latitude and longitude position and an assumed origin time as the inputs, the program will search where the T-Phase events could be. If they are found, they will be displayed synchronously at the main display window. This option is useful to confirm whether a particular event has happened or not.

The other major operations for the T-Phase programs are Signal Analysis and identify T-Phase signals by mathematical filters.

Procedure for Computing T-Phase Locations

In order to obtain a good result from locating the origin, a better sound speed model for the ocean is required. The U.S. Naval Oceanographic Office provided NOAA/OERD the Digital Environmental Model (GDEM) Version 3. It contains sound speeds at different depths and in 4 different seasons of the Pacific Ocean. The GDEM data were processed to extract the sound speeds at the specified depth of all 4 seasons. Since the specified depth is not available at every points in GDEM, spline interpolation is used to obtain the result. The final extracted results are stored into a grid in 0.5 degree spacing and may have data points missing because of islands or continents.

When sound speeds are required for the Least-Square Model to compute the T-Phase origin, an initial location is given and the sound speeds between the initial and the hydrophone locations are computed from the processed GDEM data base. This process is calculation intensive; therefore, it is only being done once at the beginning and asuming the initial location is close to the ture location and the sound speeds will not vary too much.

The Nonlinear Least-Square Model for estimating the T-Phase origin is follow:
Minimize SUM_{all i} W_i( A_i - B_i)²
where W_i = Weights, A_i = Recorded arrival time i, B_i = O_i + D_i/S_i
B_i = Estimated arrival time i, O_i = Estimated Origin time i, D_i = Distance bewteen Hydrophone location i, and the assumed T-Phase origin in (latitidue,longitude), and S_i
= Sound Speed i bewteen the path of the hydrophone and the assumed T-Phase origin. The D_i's are the shortest distances between 2 points on earth. They are computed by a geodesics routine with spheroidal earth assumption.

Once the users provided the parameters: A_i's and the hydrophone locations. An initial guess of the T-Phase location and the origin time are either given or predefined. The latitude, longitude and the origin time of the T-Phase event can be calculated by least-square estimation.

The standard errors of the computed T-Phase locations has been studied. They are calculated by simulations due to the small degrees of freedom <= 3.

An IDL library routine: CURVE_FIT uses the Gradient-Expansion (Marquardt) algorithm to do least-square estimation. This method combines the best features of the gradient search with the method of linearizing the fitting function. I have modified CURVE_FIT to compute the T-Phase event's origin.