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

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

3. Introduction 1 Raytracing medium uniform horizontally layered Shooting
Bending

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

5. 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

6. Calculation Method 2 Flat wavefronts 𝒕 𝟐 (0,h) 𝒕 𝟑 (h,h) 𝒕 𝟏 (h,0) 𝒕 𝟎
(0,0)
𝜕𝑡 𝜕𝑥 𝜕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 ℎ𝑠 2 − 𝑡 2 − 𝑡 1 2
(3)
A ,B1 ,B2 ⇒ C1

7. 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 𝑧 𝑠 +ℎ 2
(4c)
virtual source 𝑥 𝑠 , 𝑧 𝑠 , 𝑡 𝑠
𝑡 3 = 𝑡 𝑠 +𝑠 𝑥 𝑠 +ℎ 𝑧 𝑠 +ℎ 2
(5)

8. Calculation Method 2 Flat wavefronts 𝐸= 𝑡 3 − 𝑡 3 𝑐 ℎ𝑠
𝐸= 𝑡 3 − 𝑡 3 𝑐 ℎ𝑠
Flat wavefronts
𝑡 3 = 𝑡 ℎ𝑠 2 − 𝑡 2 − 𝑡 1 2

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

10. Calculation Method 2
‘mixed’ scheme
‘simple’ scheme

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

12. 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 ℎ𝑠 𝑡 2 − 𝑡 1 2

13. Calculation Method 2

14. Calculation Method 2

15. Raytracing test 3
Velocity model

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

17. 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

18. 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.

19. 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

20. 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)

21. Thank you for your attention!