Data modeling using Cagniardde Hoop method Jingfeng Zhang

Data modeling using Cagniardde Hoop method Jingfeng Zhang and Arthur B. Weglein M-OSRP annual meeting University of Houston May 10 th – 12 th, 2006 1

Outline • Background and Motivation • Theory: – Data generation using Cagniard-de Hoop method • Numerical tests • Conclusions 2

Background and Motivation • Data modeling is important for: – Evaluation of new algorithms – Forward model matching methods • Conventional data processing techniques: – Arrival time; Amplitude 3

Background and Motivation • (Recently) developed new algorithms: – Deghosting – ISS free surface multiple removal method – ISS internal multiple attenuation and elimination – Imaging without the velocity – Nonlinear inversion 4

Background and Motivation • Reasons to choose Cagniard-de Hoop method for deghosting: – 1. 5 D medium data will suffice for initial tests – “Perfect” data: regular integrand on a finite integral range – Quality control each processing step: deghosting performed in two steps 5

Background and Motivation Primary and S-G Receiver deghosting + R-G and S-R-G Source deghosting Primary 6

Theory The 2 D acoustic constant density wave equation: The corresponding Green’s function equation: Relationship: 7

Theory Fourier Transform over and (layered medium): where Just need to solving 1 D wave equation and matching boundaries for layered medium. 8

Theory Even for the direct wave in homogeneous medium: 9

Caniard-de Hoop Fourier Transform over and Laplace transform over : 10

Strategy: Manipulate the integral ( ) Aki & Richards (2 nd Edition) 11

Theory Direct wave: Primary: Pre-critical Pos-critical 12

Theory (1) Evaluation of the integration (direct wave): 13

Theory (2) Sign of : 14

Numerical Tests 15

Numerical Tests 16

Numerical Tests Correct data 17

Incorrect data 18

Deghosting result using correct data 19

Deghosting result using incorrect data 20

Deghosting results Red Solid: Exact results; Blue Dash: Calculated results 21

FSMR results Red Solid: Before FSMR; Blue Dash: After FSMR 22

Conclusions and Acknowledgments • Very high quality of data can be generated using Cagniard-de Hoop method. • It is demonstrated that using the generated data deghosting and FSMR algorithms produce very good results. • We appreciate the help from Adrian de Hoop. • The support of M-OSRP sponsors is much appreciated. 23