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Appendix:  



"Recipe" to calulate a minimum 1D model



The following guide-lines for the calculation of a minimum 1D model have been developed 

through the application of equation (3) in many areas of both simple and complex crustal 

structure around the world [Reasenberg and Ellsworth, 1982; Kissling and Lahr, 1991; 

Maurer, 1993]. These guidelines do not guarantee convergence to an optimal solution. 

Rather, specific characteristics of the data set, and of the velocity structure may demand 

modifications of the procedure. The results also depend on the effectiveness of the data 

selection process [Kissling, 1988].

	Most of our modelling has been done with the program VELEST [Ellsworth, 1977; 

Roecker, 1981; and Kradolfer, 1989].  The programs of Crosson  [1976] and Pavlis and 

Booker  [1980] have also enjoyed considerable success for this purpose [Steppe and 

Crosson, 1978]. Scott  [1992] has recently conducted a thorough investigation of the 

problem.



1.	Establishing the a priori 1D model(s)

	Obtain all available a priori (prior to the 1D or 3D inversion) information regarding the 

stratification of the area under study (velocities, layer thicknesses, etc.). In general, use 

refraction seismic models, simplified where necessary to constant velocity layers. If no 

controlled-source seismology models are available, use phase correlations and cross over 

distances [e.g., Deichmann, 1987] from well-recorded earthquakes and/or infer the layered 

structure from geologic information. Define the media by several layers of increasing 

velocity with depth. Thicknesses of the layers in the upper crust should be about 2 km and in 

the lower crust about 4 to 5 km. Estimate layer velocities according to a priori information or 

a general crustal model. In case of incomplete or inconsistent information, or, if the area 

under consideration confines two or more distinctly different tectonic provinces, establish 

several 1D models. Choose a reference station with a continuous or nearly-continuous 

record of events. It must be a reliable station, preferably located toward the center of the 

network, and should not show extreme site effects. The model(s) and the reference station 

are called the a priori 1D model(s). If several significantly different a priori 1D models are 

established the following steps 2 through 5 are repeated for each 1D model seperately.



2.	Establishing the geometry and the velocity intervals of potential 1D model(s)

	Select about 500 of the best events in the data (i.e., those with the most high-quality P-

arrivals) that cover the entire area under consideration. Relocate them with routine VELEST 

using a damping coefficient of 0.01 for the hypocentral parameters and the station delays, 

and 0.1 for the velocity parameters. Invert for hypocenters every iteration and for station 

delays and velocity parameters every second iteration. Repeat this procedure several times 

with new (updated) velocities in the reference 1D model, with perhaps the new station 

delays, and with new hypocenter locations. Repeat the procedure also for reduced number of 

layers where possible by combining adjacent layers with similar velocities. Unless clearly 

indicated by the data, in most cases it is preferable to avoid low velocity layers, as they 

normally introduce instabilities.

	Our experience suggests that shot or blast data should not be included in the 1D-model 

inversion. Rather, such data should be used to set the near surface velocities, and to test the 

performance of the resulting minimum 1D model when used for locating hypocenters. This 

countrintuitive suggestion may be understood by considering that raypaths with both 

endpoints near the surface sample, on average, a much more heterogeneous part of the Earth 

than do raypaths from events in the seismogenic crust.

	The goal of this trial and error approach is to establish reasonable geometry of the crustal 

model and corresponding intervals for the velocity parameters and station delays. In 

addition, this approach provides valuable knowledge about the quality of the data. Procede 

to the next step when, a)  the earthquake locations, station delays, and velocity values do not 

vary significantly in subsequent runs; b)  the total RMS-value of all events shows a 

significant reduction with respect to the first routine earthquake locations; and c)  the 

calculated 1D velocity model and the set of station corrections make some geological sense 

(e.g., stations with negative traveltime residuals should lie in local high velocity areas with 

respect to the reference station, etc.) and do not violate a priori information. If all these 

requirements are satisfied, the result may be called the "updated a priori 1D model with 

corresponding station residuals".



3.	Relocation and final selection of events

	Relocate all events using the updated a priori 1D model with station residuals with a routine 

location procedure (HYPO71 [Lee and Lahr 1975]; HYPOINVERSE [Klein 1978]; 

HYPOELLIPSE [Lahr  1980]) or with VELEST in the single event mode (fixing the station 

and velocity parameters). Reselect the best (consider gap, number of observations, distance 

to next station) 500 or so events that should be well distributed over the volume under 

investigation. If more than one such subset of about 500 events can be extracted, proceed for 

each subset  separately with step 4 but try to obtain similar results.



4.	Calculation of minimum 1D model for one subset

	In general terms repeat step 2 with the updated a priori 1D model and station residuals and 

with a damping of 0.01 for the hypocentral, 0.1 for the station, and 1.0 for the velocity 

parameters. The goal of this step is to calculate the 1D model (velocity parameters and station 

residuals), that minimizes the total estimated location errors for a fixed geometry. Test the 

stability of the result by systematically and randomly shifting hypocenters and by 

underdamping the velocity parameters. If you are pleased with the performance of the 

solution fix the updated velocity parameters by overdamping and calculate the station 

residuals. The resulting velocity model with corresponding station residuals is called 

"minimum 1D model".



5.	Calculation of minimum 1D model for several subsets

	If several subsets of 500 events were extracted, test the dependence of your minimum 1D 

model on specific data. Find the 1D model and station residuals that will best fit the results 

from all subsets, mix data from different subsets, and repeat step 4. If the results are 

unsatisfactory, evaluate the best 1D model by the procedure described in step 6.



6.	Evaluation of different minimum 1D models for same area

	If several significantly different a priori 1D models were established and steps 2 through 5 

were successfully completed for each of them, you may base your choice of one minimum 

1D model on the result of the following performance test: Select all traveltime data from 

quarry blasts or shots (i.e., from sources of known location) and relocate these events for 

the different minimum 1D models without fixing the depth during the location process. If the 

near surface velocities for several station locations are known, compare the station residuals 

with the differences between the average layer velocity and the local velocities. Finally, 

select the minimum 1D model that best resembles the a priori information.





References



Crosson, R.S., Crustal structure modelling of earthquake data, 1, Simultaneous least squares 

estimation of hypocenter and velocity parameters, J. Geophys. Res., 81, 3036-3046, 1976.



Deichmann, N., Focal depths of earthquakes in northern Switzerland, Ann. Geophysicae, 4, 395-

402, 1987.



Ellsworth, W.L., Three-dimensional structure of the crust and mantle beneath the island of 

Hawaii.  PhD thesis, Massachusetts Institute of Technology, 1977.



Kissling, E., Geotomography with local earthquake data, Rev. Geophys., 26, 659-698, 1988.



Kissling, E., and J. C. Lahr, Tomographic image of the Pacific slab under southern Alaska, 

Eclogae Geol. Helv., 84/2, 297-315, 1991.



Klein, R.W., Hypocenter location program HYPOINVERSE, I, Users guide to versions 1,2,3, 

and 4, U.S. Geol. Surv. Open-file rep., 78-694, 1978.



Kradolfer, U., Seismische Tomographie in der Schweiz mittels lokaler Erdbeben. PhD thesis, 

ETH Z焤ich, 109p., 1989.



Lahr, J.C., HYPOELLIPSE/MULTICS: a computer program for determining local earthquake 

hypocentral parameters, magnitude, and first motion pattern, U.S. Geol. Surv. Open-file 

rep., 80-59, 1980.



Lee, W.H.K., and J. C. Lahr, HYPO71: A computer program for determining hypocenter, 

magnitude, and first motion pattern of local earthquakes, U.S. Geol. Survey Open-file rep., 

75-311, 1975.



Maurer, H.R., Seismotectonics and upper crustal structure in the western Swiss Alps. PhD thesis, 

ETH Z焤ich, 133p., 1993.



Pavlis, G.L., and J. R. Booker, The mixed discrete-continuous inverse problem: Application to 

the simultaneous determination of earthquake hypocenters and velocity structure, J. 

Geophys. Res., 85, 4801-4810, 1980.



Reasenberg, P., and W.L. Ellsworth, Aftershocks of the Coyote Lake, California, earthquake of 

August 6, 1979: a detailed study, J. Geophys. Res., 87, 10637-10655, 1982.



Roecker, St., Seismicity and tectonics of the Pamir-Hindu Kush region of central Asia, Ph.D. 

thesis, Mass. Inst. Technol., 1981.



Scott, J.S., Microearthquake studies in the Anza seismic gap, Ph.D. thesis, University of 

California, San Diego, Scripps Inst. of Oceanography, 277p., 1992.



Steppe, J. A., and R. S. Crosson, P-velocity models of the southern Diablo Range, California, 

form inversion of earthquake and explosion arrival times, Bull. Seis. Soc. Am., 68, 357-

367, 1978.



Thurber, C.H.: Earth structure and earthquake locations in the Coyote Lake area, central 

California,  Ph.D. thesis, Mass. Inst. Technol., 1981.



Thurber, C.H., Hypocenter-velocity structure coupling in local earthquake tomography, Phys. 

Earth Planet. Int., 75, 55-62, 1992.



GAP-Abstract



7294 Seismology techniques

Initial reference models in seismic tomography

E. Kissling (Institute of Geophysics, ETH-Hoenggerberg, 8093 Zuerich, 

Switzerland), W. L. Ellsworth, D. Eberhart-Phillips, U. Kradolfer

  The inverse problem of 3D local earthquake tomography is 

formulated as a linear approximation to a non-linear function. Thus, 

the solutions obtained and the reliability estimates depend on the initial 

reference model. Inappropriate models may result in artifacts of 

significant amplitude. Here, we advocate the application of the same 

inversion formalism to determine hypocenters and 1D velocity model 

parameters, including station corrections, as the first step in the 3D 

modelling process. We call the resulting velocity model the minimum 

1D model. For test purposes, a synthetic data set based on the velocity 

structure of the San Andreas fault zone in central California was 

constructed. Two sets of 3D tomographic P-velocity results were 

calculated with identical traveltime data and identical inversion 

parameters. One used an initial 1D model selected from a priori 

knowledge of average crustal velocities, and the other used the 

minimum 1D model. Where the data well-resolve the structure, the 3D 

image obtained with the minimum 1D model is much closer to the true 

model than the one obtained with the a priori reference model. In 

zones of poor resolution, there are fewer artifacts in the 3D image 

based on the minimum 1D model. Although major characteristics of 

the 3D velocity structure are present in both images, proper 

interpretation of the results obtained with the a priori 1D model is 

seriously compromised by artifacts that distort the image and that go 

undetected by either resolution or covariance diagnostics. (Seismic 

tomography, initial reference models.)

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