T-K Lau (NOAA/ Oregon State University, CIMRS, Hatfield Marine Science Center, Newport, Oregon 97365, USA)
C.G. Fox  (NOAA Pacific Marine Environmental Laboratory, Newport, Oregon 97365, USA)

(http://www.pmel.noaa.gov/vents/staff/lau/error_report.html)

T-K. Lau (NOAA/Oregon State University, CIMRS, Hatfield Marine Science Center, Newport, OR 97365 USA)

C. G. Fox (NOAA, Pacific Marine Environmental Laboratory, Newport, OR 97365 USA)

NOAA/PMEL has deployed an array of autonomous, hydrophone moorings in the eastern equatorial Pacific, designed to continuously record low-frequency acoustic energy in the SOFAR channel for extended periods. Low-level seismicity (m_b > 2) can be detected at long ranges since hydroacoustic T-waves propagate efficiently in the oceanic sound channel. Relatively accurate locations can be derived from the T-waves by timing their arrivals at each hydrophone, modelling the propagation via ocean-basin-wide sound speed models, and performing a non-linear least squares minimization on origin time. Detectability and location accuracy vary widely over the Pacific basin, and the array geometry is less than ideal since it is colocated with surface weather buoys. Direct calculation of location errors for the array is difficult due to the limited number of sensors and therefore statistical degrees of freedom. A method was formulated to estimate location error throughout the Pacific region by assuming random timing errors for each sensor and analyzing multiple realizations for each grid cell in the region. A known eruptive site at Loihi Seamount offshore Hawaii was then used to calibrate the level of timing error and extend the error field to the entire study area. The resulting analysis indicates location accuracies for individual epicenters of less than 1 km within the array (8^oN-8^oS, 110^oW-95^oW) and about 5 km for the superfast spreading center at 17^oS on the East Pacific Rise. In the case of volcanic swarms with multiple events, more accurate locations are possible by averaging several epicenters.

Purpose of the Error Analysis

The error analysis will provide an indirect method to compute the standard error of the T-Phase locations based on several assumptions. An indirect method is needed because there are not sufficient Degrees of Freedom (df) to calculate the standard errors using a least-square procedure. The procedure will solve for 3 unknown variables: Latitude, Longitude and the origin time of the earthquake epicenter. Currently there are only 6 hydrophones available for event detection making the df to be at most 3 ( = 6 observations - 3 variables ). A df<3 or even df=0 will occur because the same event may not be recorded by all the hydrophones. With a small or 0 df, the standard errors computed from the least-square procedure will not give very accurate results or the errors will be very large. This method can provide an improved estimate of errors in each variable over the entire field.

Formulation for Computing the T-Phase Location by a Least-Square Method

The Nonlinear Least-Squares Method for estimating the T-Phase origin is as follows:
Minimize Residual =
SUM_{all i} W_i( A_i - B_i )²
where W_i = Weight and 1 can be used for unweighted calculations,
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/S_i = Travel time,
D_i = Distance between Hydrophone location i and the assumed T-Phase origin in (latitude,longitude), and
S_i= Sound Speed i between the path of the hydrophone and the assumed T-Phase origin.

The W_i can be computed using the travel times: D_i/S_i as follow
w_i = 1/travel time_i = S_i/D_i and Normalized it by W_i = (S_i/D_i) /max( w_i ) so that the hydrophone with the shortest travel time or the first arrival time will receive greater weight. Its weight will be 1 and the rest will be < 1 but > 0. The D_i's are the shortest distances between 2 points on earth, computed by a geodesic algorithm with a spheroidal earth assumption. The S_i 's are computed from a GDEM data base provided by the U.S. Navy.

Calculations of Errors

The objective is to obtain sampling distributions of the estimated variables: Latitude, Longitude and Origin Time. This is done by simulating a sampling experiment many times, and calculating the estimators each time. This builds up the distribution of the estimators, to show how close they are to the true values.

Given the latitude, longitude, and the hydrophone locations, the experiment will be as follows: Random Errors with a normal distribution will be added to either sound speeds, arrival times, or both; weights will be calculated from the arrival times if it is asked for; the least-square procedure is applied to determine the origin; and save the results.

The experiment is repeated, e.g. N=100 times per simulation.
Let x_i = the computed variable: either latitude, longitude, or origin time,
X = average of x_i for i=1 to N, and T = the true value of either latitude, longitude, or origin time, the following statistics will be calculated:

The Observed Bias = X - T which shows how close the computed value is to the true target. For example,
Observed Bias of Latitutde = Average of the computed latitudes - True latitude.

The Observed Mean Square Error (MSE) = [SUM_{all i} ( x_i - T )²]/N
which measures on average how far X is off its target. It also measures the square-root of the MSE = Bias Root Mean Square (RMS).

The Observed Variance =
[SUM_{all i} ( x_i - X )²]/N
which indicates how much X fluctuates.
Also, the square-root of Observed Variance = Standard Error. This will be the result for the study. The Standard Error will be accepted if the Observed Bias is small and the MSE is very close to the Observed Variance.

Calibration for Error Distribution

The error analysis is performed via Monte Carlo simulations using normally distributed random numbers to represent measurement errors. These are added to arrival or travel times. The distribution of these errors will influence the results of error analysis; therefore, the standard deviation (STD) of the error distribution should be set as realistic as possible. This is done by using events with a known origin as the calibration.

Loihi is a seamount located at approximately 18.9^oN and 155^oW. It is off the SE coast of the big island of Hawaii. Continuous eruptions were recorded by the Hawaii Volcano Observatory from Loihi between Julian days 199 to 210, 1996. They were recorded by the NOAA hydrophones as well as other seismic stations in Hawaii. From all the events recorded only by the 5 NOAA hydrophones (their locations are shown in figures 1, 2, & 3 except the one at 0^o, 95^oW), 1653 of them were chosen. All their locations were computed and are shown in Figure 4. Assuming a common point of origin, the scatter of the epicenter locations gave the following statistics:

The same statistics for the origin time cannot be independently extracted from the Loihi events.

Assuming the Standard Deviations from the computed Loihi locations are representation of the location error at that site, simulations were run to determine what arrival time measurement error should be used to obtain approximately the same Standard Deviation from the Loihi statistics, i.e., determine the standard deviation (STD) of the normal distribution for generating random errors.

STD = 0.75 was selected as a conservative choice. It gave the standard errors in degrees of 0.111 for the latitude and 0.459 for the longitude. They are higher than the Loihi statistics especially the longitude.

Results from the Simulations

Using normally distributed errors with mean=0, and standard deviation (STD)=0.75 added only to the predicted arrival times, two simulations were performed. They cover the NE and SE of the Pacific Ocean with 6 hydrophones. They were deployed by NOAA, positioned at the 2 meridians 110^oW and 95^oW, spread 8 degrees both North and South from the equator to form a rectangular array on the map as shown in the following 3 figures. Weighted Error Analyses were also run and the weighted results were not significantly different from the unweighted results.

Figures 1 and 2 show the contours of standard errors in minutes for latitudes and longitudes when computing earthquake epicenter locations. For example, if an event is located at 20^oN and 150^oW, the standard errors for this position will be approximately 6 minutes in latitude and 20 minutes in longitude, i.e., the ranges of the computed location will be 20^oN +/- 6 minutes and 150^oW +/- 20 minutes. By the Empirical Rule in statistics, these are the approximately 68% confidence intervals. For 95.5%, double the errors, e.g., 20^oN +/- 2x(6 minutes).  For 99.7%, triple the errors. Again this is assuming the error of arrival times is +/- 0.75 seconds, and the error distribution is normal with the mean = 0 and standard deviation = 0.75.

Summary

An error analysis has provided esitmates of the standard error of earthquake epicenter locations derived from the NOAA/PMEL hydrophone array as shown in Figures 1, 2, and 3 with the following assumptions:

It is assumed that the random errors from a normal distribution are a good approximation of the real-time errors.  The normality assumption is a neutral one to use since the event distribution from any given T-Phase origin is unknown in general. The standard deviation: STD computed from the calibration, used for the error distribution is assumed to be a realistic one. The errors are added to the arrival times for calculating the standard errors. Bear in mind that the results are inferenced by the selection of STD. The results should be used with care.

The calculations of the sound speeds are based on July to September season, and the speeds are modify by the correction factor: 0.998. No errors are added to the sound speeds for this study. It is assumed that the errors of the sound speeds are negligible and can be ignored. The locations of 6 hydrophones shown in figure 1 to 3 are used for the study. If any one of these assumptions is altered, the experiment must be repeated to reflect the changes.

The report of this study is available on the web:
http://www.pmel.noaa.gov/vents/staff/lau/tphase/error_report.html