Hu Jing Wang Wenming Yao Jiaqi Finite-Difference Calculation of Travel Times John Vidale ,1988 Group 1 Hu Jing Wang Wenming Yao Jiaqi

Outline 1 Introduction 2 Calculation Method 3 Raytracing Test 4 Application 5 Conclusion

Introduction 1 Raytracing medium uniform horizontally layered Shooting Bending

Introduction 1 local minimum Finite difference Difficulties: 1. for strongly varying velocity fields 2. expensive 3. in shadow zones local minimum Finite difference

Calculation Method 2 B1 ~ B4 𝑡 𝑖 = ℎ 2 𝑠 𝐵 𝑖 + 𝑠 𝐴 h C1 ~ C4 𝑡 𝑖 = ℎ 2 𝑠 𝐵 𝑖 + 𝑠 𝐴 C1 ~ C4 Type 1: flat wavefronts Type 2: curved wavefronts

Calculation Method 2 Flat wavefronts 𝒕 𝟐 (0,h) 𝒕 𝟑 (h,h) 𝒕 𝟏 (h,0) 𝒕 𝟎 (0,0) 𝜕𝑡 𝜕𝑥 2 + 𝜕𝑡 𝜕𝑧 2 =𝑠 𝑥,𝑧 2 (1) 𝜕𝑡 𝜕𝑥 = 1 2ℎ 𝑡 0 + 𝑡 2 − 𝑡 1 − 𝑡 3 (2a) 𝜕𝑡 𝜕𝑧 = 1 2ℎ 𝑡 0 + 𝑡 1 − 𝑡 2 − 𝑡 3 (2b) 𝑡 3 = 𝑡 0 + 2 ℎ𝑠 2 − 𝑡 2 − 𝑡 1 2 (3) A ,B1 ,B2 ⇒ C1

Calculation Method 2 Curved wavefronts 𝒕 𝟐 (0,h) 𝒕 𝟑 (h,h) 𝒕 𝟏 (h,0) 𝒕 𝟎 (0,0) 𝑡 0 = 𝑡 𝑠 +𝑠 𝑥 𝑠 2 + 𝑧 𝑠 2 (4a) 𝑡 1 = 𝑡 𝑠 +𝑠 𝑥 𝑠 +ℎ 2 + 𝑧 𝑠 2 (4b) 𝑡 2 = 𝑡 𝑠 +𝑠 𝑥 𝑠 2 + 𝑧 𝑠 +ℎ 2 (4c) virtual source 𝑥 𝑠 , 𝑧 𝑠 , 𝑡 𝑠 𝑡 3 = 𝑡 𝑠 +𝑠 𝑥 𝑠 +ℎ 2 + 𝑧 𝑠 +ℎ 2 (5)

Calculation Method 2 Flat wavefronts 𝐸= 𝑡 3 − 𝑡 3 𝑐 ℎ𝑠 𝐸= 𝑡 3 − 𝑡 3 𝑐 ℎ𝑠 Flat wavefronts 𝑡 3 = 𝑡 0 + 2 ℎ𝑠 2 − 𝑡 2 − 𝑡 1 2

Calculation Method 2 Curved wavefronts Round-off error 𝑡 3 = 𝑡 𝑠 +𝑠 𝑥 𝑠 +ℎ 2 + 𝑧 𝑠 +ℎ 2 Round-off error

Calculation Method 2 ‘mixed’ scheme ‘simple’ scheme

Calculation Method 2 𝒕 𝟐 (0,h) 𝒕 𝟑 (h,h) 𝒕 𝟏 (h,0) 𝒕 𝟎 (0,0)

Calculation Method 2 𝒕 𝟑 𝜕𝑡 𝜕𝑥 = 1 2ℎ 𝑡 2 − 𝑡 1 𝒕 𝟏 𝒕 𝟎 𝒕 𝟐 𝜕𝑡 𝜕𝑥 = 1 2ℎ 𝑡 2 − 𝑡 1 3 1 2 𝒕 𝟏 𝒕 𝟎 𝒕 𝟐 𝜕𝑡 𝜕𝑧 = 1 ℎ 𝑡 3 − 𝑡 0 𝑡 3 = 𝑡 0 + ℎ𝑠 2 +0.25 𝑡 2 − 𝑡 1 2

Calculation Method 2

Calculation Method 2

Raytracing test 3 Velocity model

Raytracing test 3 Finite difference vs. ray tracing Simple scheme: error 0.1 per cent Mixed scheme: reduced to 0.03 per cent

Application 4 1. Earthquake location large computer costs Raytracing limitations： large computer costs problems with multipathing no arrivals in the shadow zones FDM: Find global minimums travel time from a source to all receivers No problems with multipathing and lower velocity zone lower comuter costs

Application 4 2. Tomography Ray path can be traced by following the gradient in travel time from B back to A. With known first arrivals and ray path ,we can set up a tomograhic inversion for the velocity.

Application 4 3. Finite difference speed-up 4. Kirchhoff migration Knowledge of the travel time to every point in a numerical grid can aid in schemes like finite-difference wave simulation that also use a numerical grid. If one is interested only in the first arrivals, there is no need to compute the wave field more than a few seconds behind the first arrival. 4. Kirchhoff migration

Conclusion 5 Pros A computationally quick, accurate method to calculate a field of travel times. (2) It may be used in numerous applications, and has several advantages over raytracing methods. (3) The scheme naturally follows diffractions if they are the first arrivals, even through shadow zones. Cons Velocity varying (v2>v1)

Thank you for your attention!