Introduction to seismic body waves Tomography

Transcription

1 Introduction to seismic body waves Tomography Presented by Han Zhang Prepared by group work of Justin Wilgus and Han Zhang

2 Outlines 1. Velocity derived from wave equation 2. Two commonly used reference 1- D models 3. Naming seismic body waves at global scale 4. Basic ideas of body wave tomography 5. Possible improvement on inversion

3 Seismic wave equation we got in last class, ρu = λ + 2μ u μ u where dot above vector u denotes partial differential on time (. ), and two dots./ means do it twice (.0 ). And =.,.,. as we discussed. λ, μ and ρ are two./ lame parameters and density of media respectively. By dot timing operator from left side, we can get (using v = 0 ), ρ u = λ + 2μ 7 u which is compression wave equation since u denotes volume change of media. V 9 = λ + 2μ ρ Similarly, by taking the curl of two side, we can get, ρ u = μ 7 u for S wave since u is shear component of displacement. V ; = μ ρ

4 Seismic wave equation we got in last class, Despite the fact that density of materials inside Earth is ρu = λ + 2μ u μ u increasing with depth, lame parameters λ, μ are also increasing with depth and their enlargement ), factor and two is dots greater than ). that And of = density,.,.,. which as we discussed. finally leads λ, μ and to ρan are two./ increase function of wave velocity with depth inside mantle. where dot above vector u denotes partial differential on time (../ means do it twice (.0 lame parameters and density of media respectively. By dot timing operator from left side, we can get (using v = 0 ), ρ u = λ + 2μ 7 u which is compression wave equation since u denotes volume change of media. V 9 = λ + 2μ ρ Similarly, by taking the curl of two side, we can get, ρ u = μ 7 u for S wave since u is shear component of displacement. V ; = μ ρ

5 Reference Models PREM reference model km 400 km km Vs ρ Vp Depth (km) 2400 CMB ICB Velocity (km/s) or Density (g/cm ) 14

6 PREM reference model AK135 reference model km 400 km km km ρ Vs Vp km ρ Vs Vp Depth (km) CMB Depth (km) CMB ICB 5200 ICB Velocity (km/s) or Density (g/cm 3 ) Velocity (km/s) or Density (g/cm 3 )

7 PREM reference model AK135 PREM reference model km 400 km km 400 km km ρ Vs Vp km ρ Vs Vp Depth (km) CMB Depth (km) CMB ICB 5200 ICB Velocity (km/s) or Density (g/cm 3 ) Velocity (km/s) or Density (g/cm 3 )

8 Datasets of PREM and AK135 PREM 1. Geodetic data 2. Free oscillation and long- period surface wave 3. Body wave (primarily, direct P and S) Starting model of density derived from state equation. AK Body wave traveltime data (absolute traveltime of 18 seismic body waves and traveltime differences between them) Starting model can be regarded as IASP91

9 Phases Naming (global) source Generally, upgoing waves start with lower case letter while downgoing phases using upper case. Same rule for reflection/conversion points. Considering difference of sp and SP to see how to name different phases. (Shearer, Intro. to Seismology)

10 Phases Naming (exercise) Blue for P wave Red for S wave 60 Ray 1 Ray 2 ppcp 410 s 60 ScS 660 ScS CMB ICB 0 Animation available here

11 Body Wave Tomography Common assumptions of body wave traveltime tomography, 1. One good reference model available to make sure velocity perturbation of model region is small enough for linear inversion; 2. All blocks in our model region are well sampled by observations to reduce uncertainty of inversion result. Idea of body wave traveltime tomography based on ray theory, 1. Traveltime misfit between observed data and prediction of one certain ray is contributed by the blocks lie on this ray path only; 2. Total traveltime perturbation along the ray path can be summed from the product of traveltime in each block (calculated using reference model) with the fractional velocity perturbation within the block.

12 Body Wave Tomography r H = r 7 = r P = r Q = 0.01 sec; r S = r T = r U = r V = r W = 0 sec; a b c d e f 4 5 For each ray path, we can get one equation looks like this, r = = BCDEFG t =@ where r = is traveltime misfit between observation and prediction of ray path i, t =@ is absolute traveltime in block j of ray path i, and is velocity perturbation of block j. 8 g h i 6 7 For ray path 1 in the setting showed left, the equation should be, r H = P I + P J + P K by assuming t HI = t HJ = t HK = 1 sec. Rays are numbered aside their arrows and blocks are marked by low- case characters inside them. Simple traveltime (1 sec or 2 sec for one block) is assumed for practice.

13 Body Wave Tomography a b c d e f 4 5 r H = r 7 = r P = r Q = 0.01 sec; r S = r T = r U = r V = r W = 0 sec; 8 g h i 6 7 P I = P Y = 1%; P B = P E = P J = P [ = P K = P \ = P = = 0. White blocks have zero perturbation relative to reference model while black ones have perturbation of - 1%.

14 Body Wave Tomography Rewrite residual equation using matrix to get general form, d = Gm Residuals = Rays Model where d (1 x i) represents residuals of all the ray path, G (j x i) is related to ray path configuration and sensitive kernels adopted and m (1 x j) denotes perturbation model. In the case of i < j, which means the number of independent equations is less than parameters need to be recovered, we can get general solution only (infinitely many solutions). In the case of i = j, which means we have same amount of independent equations and model parameters, we will get exactly one solution. In the case of i > j, which means we have more independent equations than model parameters, theoretically we cannot get solution from above form. However, we are able to get least- square solution (L2 norm) of this kind of problem by solving G T d = G T Gm which theoretically have exactly one solution.

15 GAP_P4 Model (Fukao and Obayashi, 2013) Slab atop at 660 km discontinuity

16 GAP_P4 Model (Fukao and Obayashi, 2013) Slab cross 660 km discontinuity

17 Possible Improvements Body wave tomography works pretty well in helping us understand deep Earth structures, however, there are some defects of it, which may be overcame by joint inversion with other dataset. Some are listed here for checking, 1. L2 norm tend to results in a smooth model, which will average small- scale heterogeneity to its surroundings. Combining with converted wave modelling (such as receiver function). 2. Limited by the fact that most seismometers are deployed on continents and islands, which leads to poor ray path cover at crust and uppermost mantle beneath ocean. Combing with data from OBSs (ocean bottom seismometers) and/or electrical resistivity tomography (which is also expensive in ocean but is able to get very good shallow structures).

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19 Finite Frequency Kernels Banana- doughnut kernels showing the sensitivity of P- wave travel times at 60 epicentral distance to velocity perturbations in the mantle. (Hung et al, 2000)