Waveform tomography and non-linearity  L  38 m N s = 101 N r = 101  10 m 30 m 21 m10.5 m

Waveform tomography and non-linearity R. Gerhard Pratt 1,2 Fuchun Gao 2 Colin Zelt 2 Alan Levander 2 1 Queen’s University, Kingston, Ontario 2 Rice University, Houston, Tx

Traveltime (ray) tomography s

Wavefront healing 150 m 300 m Frequencies: Hz (Peak frequency 300 Hz) Velocities: 3000 m/s Small anomaly 500 m/s Large anomaly 2300 m/s Fresnel zone width  L  38 m (300 Hz) Wavelength  10 m

Waveform tomography U(r s,r g ) Scattered wavefield ω Frequency  s 2 ( r ) Scattering potential G(r,r') Free space Green’s Function (e.g., Wu and Toksöz, 1987)  s2(r) s2(r) ω

Waveform tomography  s2(r) s2(r) ω 150 m 300 m Frequency 300 Hz Wavelength = 10 m

Waveform tomography  s2(r) s2(r) ω 150 m 300 m Frequency 300 Hz Wavelength = 10 m

Waveform tomography “Seismic wavepath” (Woodward and Rocca, 1989, 1992) Fresnel zone width  L  38 m (300 Hz) 150 m 300 m  s2(r) s2(r) ω Frequency 300 Hz Wavelength = 10 m

Linearized forward problem (after discretization):  d = A  p ( p are the model “parameters”) For ray methods, A is a forward projection operator, d are the traveltimes. For waveforms, A is a forward propagation operator, d are the scattered arrivals. Practical inversion: linearization

Linearized forward problem (after discretization):  d = A  p Misfit functional E(p) =  d t  d Practical inversion: gradient scheme Gradient p E = A T  d A T is either a backprojection or backpropagation operator

Practical inversion: gradient scheme

Chequerboard models 150 m 300 m Fresnel zone width  L  38 m (300 Hz) 101 sources (3 m) 101 receivers (3 m) Dominant wavelength  10 m (300 Hz) Velocities: 2300 / 3000 m/s  L  38 m N s = 101 N r = 101  10 m 30 m 21 m10.5 m

Model data: large anomalies 30 m Depth Time (ms)

Model data: medium anomalies 21 m Depth Time (ms)

Model data: small anomalies 10.5 m Depth Time (ms)

Traveltime results  L  38 m N s = 101 N r = 101  10 m 30 m 21 m10.5 m

Wideband ( Hz) waveform tomography results  L  38 m N s = 101 N r = 101  10 m 30 m 21 m10.5 m

Wideband waveform tomography results: zoom view

Non-linearity

Misfit functional Low frequency Higher frequency

Narrow band ( Hz) waveform tomography results  L  38 m N s = 101 N r = 101  10 m 30 m 10.5 m21 m

Combined results  L  38 m N s = 101 N r = 101  10 m 30 m 10.5 m21 m

Results with a random model ( Hz)  L  40 m N s = 101 N r = 101 N  15  10 m 150 m 300 m True model Traveltime + Waveform tomography Waveform tomography True model Traveltime tomography m/s m/s

Conclusions Waveform tomography has far better resolution than traveltime tomography Waveform tomography has far better resolution than traveltime tomography Non-linear effects are critical Non-linear effects are critical Low frequencies help Low frequencies help