H. Sadi Kuleli, W Rodi, Fuxian Song, M. Nafi Toksoz Department of Earth Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge,MA, August, 2011 TRAVEL TIME CALCULATIONS 1

Classification of Ray Tracing Methods for TT calculation 1 - Analytic solutions for 1-D models --Direct Analytical solution (Snell’s Law) -- Tau-P method Solutions expressed as integrals over depth (Buland and Chapman, 1983) 3D 2- Shortest (fastest) time methods (Solve: Travel time = minimum) Fermat’s Law -- Two -point ray bending ( Um and Thurber, 1987, Zhao et al, 1992; Zhao and Lei, 2004) -- Finite difference (propagate grid of times from source) (Ferra and Madariaga, 1987, Moser 1991, Podvin and Lecomte, 1991) 3- Eikonal solvers (Solve: | gradient of travel time | = slowness) -- Finite difference (grid of times) (Vidale, 1990, Podvin and Lecomte, 1991) -- Fast marching (grid of times) (Rawlinson and Sambridge, 2004.) 4- Ray shooters (Solve: ray equations) (differential equations for ray trajectory) -- Dynamic ray tracing (Cerveny, Molotkov and Psencik) 2

Direct Analytical Calculation and Microseismic Model 3 Depth(m) Distance(m) E E E

Calculate First Arrivals 1 D: Tau- P 3 D: Bending Method 4

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Bending (Pseudo Bending) ray tracing 6

Three-point perturbation Travel time by summation along the segments of ray path: Offset direction: 7

Illustration of pseudo-bending algorithm 8

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Dynamic Ray Tracing 10 Calculate ALL (required) Arrivals

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Definition of layers and Solution of Differential Equation System :, APPROXIMATION OF VELOCITY DISTRIBUTION. MM=0... BICUBIC SPLINE INTERPOLATION. MM=1... LINEAR VERTICAL INTERPOLATION BETWEEN ISOVELOCITY INTERFACES. MM=2... PIECE-WISE BILINEAR INTERPOLATION THE SHOOTING METHOD IS USED. TO PERFORM ITERATIONS, VARIOUS METHODS CAN BE APPLIED: METHOD=0... REGULA FALSI. METHOD=1... COMBINATION OF REGULA FALSI METHOD AND METHOD OF HALVING INTERVALS. METHOD=2... HALVING OF INTERVALS. METHOD=3... METHOD BASED ON PARAXIAL RAY APPROXIMATION. 12

Velocity Model 13

;5; 3,2,2,2, ;4; 3,3,2,,1

;5; 3,2,2,2,1 15 1;6; 1,2,2,2,2,1

Time MODE: 0, 1, 1 16

MODE: 0, 1, 1 and 1, 2, 1, 1 17

MODE: 0, 1, 1 / 1, 2, 1, 1 /

MODE: / / / /

Mode: 0 1 1/ / / / / : 20

Mode: 0 1 1/ / / / /

For Event 1 (E1) Dynamic Ray Tracing TT. 22

For Event E1 Analytical solution, Bending, Tau_P and DRT 23

PROGRAM SEIS88 IS DESIGNED FOR THE COMPUTATION OF RAYS OF SEISMIC WAVES WHICH ARRIVE AT A SYSTEM OF RECEIVERS DISTRIBUTED REGULARLY OR IRREGULARLY ALONG THE EARTH'S SURFACE, ALONG AN INTERFACE,OR ALONG A VERTICAL PROFILE. THE GENERATION OF WAVES IS SEMI-AUTOMATIC. AT RECEIVERS, CORRESPONDING TRAVEL TIMES ARE AUTOMATICALLY DETERMINED. OPTIONALLY, AMPLITUDES AND PHASE SHIFTS MAY BE EVALUATED. EFFECTS OF A SLIGHT ABSORPTION MAY BE ALSO CONSIDERED. ALL THESE QUANTITIES ARE STORED AND MAY BE OPTIONALLY PLOTTED OR USED FOR THE COMPUTATION OF SYNTHETIC SEISMOGRAMS. ALL THE DIRECT AND PRIMARY REFLECTED WAVES P AND S, INCLUDING THE CONVERTED WAVES AT THE POINT OF REFLECTION, CAN BE GENERATED AUTOMATICALLY. MULTIPLE REFLECTIONS OF ARBITRARY TYPE ARE OPTIONALLY GENERATED MANUALLY (BY INPUT DATA). 24 THE AMPLITUDES OF P, SV AND SH WAVES ARE EVALUATED BY STANDARD RAY FORMULAE. NO MODIFICATIONS ARE APPLIED IN SINGULAR REGIONS. THE PURE HEAD WAVES GENERALLY DO NOT EXIST IN LATERALLY INHOMOGENEOUS MEDIA WITH CURVED INTERFACES AND ARE NOT CONSIDERED IN SEIS88. THEY MIGHT BE, OF COURSE,SIMULATED BY SLIGHTLY REFRACTED WAVES. Calculation of Dynamic Ray Tracing “PROGRAM SEIS88”

GEOMETRICAL SPREADING IS DETERMINED BY SOLVING A SYSTEM OF TWO LINEAR ORDINARY DIFFERENTIAL EQUATIONS OF FIRST ORDER (SO CALLED DYNAMIC RAY TRACING SYSTEM) BY A MODIFIED EULER'S METHOD. SLIGHT ABSORPTION MAY BE CONSIDERED. IN THIS CASE, A GLOBAL ABSORPTION FACTOR(QUANTITY OBTAINED AS A TIME INTEGRAL OF 1/Q ALONG A RAY, WHERE Q IS A QUALITY FACTOR) IS ALSO EVALUATED. PROGRAM SEIS88 IS SUPPLEMENTED BY PROGRAMS FOR PREPARATION OF DATA FOR VELOCITY MODEL (PROGRAM SMOOTH), TO COMPUTE SYNTHETIC SEISMOGRAMS (PROGRAM SYNTPL), AND TO PLOT RAYS, TRAVEL-TIME AND AMPLITUDE-DISTANCE CURVES (PROGRAM RAYPLOT), SYNTHETIC SEIS-MOGRAMS (PROGRAM SEISPLOT) OR PARTICLE MOTION DIAGRAMS (PROGRAM POLARPLOT). 25

THANK YOU 26

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