Traveltime Tomography

Introduction:

Seismic tomography, a methodology of estimating the Earth’s properties, has been applied to solid Earth sciences and exploration seismology. There are two main types of seismic data to be inverted by tomography: traveltime data and waveform data. Traveltime data is used to reconstruct Earth’s velocity models with lower resolution compared with waveform data. However, traveltime tomography is robust and easy to implement with fast computation ability.

I developed the first arrival traveltime tomography algorithm in 2D/3D TTI media. The derived analytical kernels represent the sensitivities of each anisotropic parameter in terms of traveltime. Such kernels directly indicate the observation of the structural parameters by analyzing their spatial distribution patterns. Several synthetic experiments show some combinations of anisotropic parameters can be invert successfully. The inversion results are necessary for depth migration.

 

The calculated Frechet derivatives can be used for a model parameterization explicitly and directly used in any local search minimization inversion algorithm, such as conjugate gradient or Gauss-Newton to yield the elements of the Jacobian matrix directly for arbitrary model parameterization. Each Frechet kernel presents the rates of change in the observations to perturbations in cell or medium properties, such as Thomsen’s anisotropic parameter. Therefore, the Frechet kernels are examined as sensitivity functions of the data to a particular parameter and indicate the sensitivity variations with various surveying configurations.

The sensitivity of traveltime to key TTI parameters as a function of the ray angles. Here, swp0 = 1 s/m, lenray = 1 m, ε= 0.15 and δ = 0.1 for calculating kernels using Equation (3-1) – (3-4). The kernel of sine function of tilt angle φ is calculated with assumption of 45° tilt angle.   

In traveltime tomography, each iteration consists of ray tracing in current reference model to compute the Frechet kernels (anisotropic kernels) and traveltime residuals, inverting for model updates and assessing data-fitting statistics and model variations. The calculations of the data fit provide the criteria to terminate the iteration and to select best model as output.

 

Synthetic examples:

1: Crosswell anisotropic tomography in multi-layer media with noise-free data

 

 

 

 

 

 

 

 

 

 

2D TTI crosswell test. (a) True model; (b) Reference model; (c) The result of inversion I with δ of 0.1, 0.04, 0.15 in each layer; (d) The result of inversion II with δ of 0.0, 0.0, 0.0 in each layer. Red arrows represent true symmetry axes in (a) and inverted symmetry axes in (b) and (d). Blue arrows denote the initial vertical symmetry axes.

2: Crosswell anisotropic tomography in multi-layer media with 5% Gaussian noise

3: VSP anisotropic tomography in multi-layer media with noise-free data

 

 

4: Anisotropic layered tomography by inverting for layer geometry, ε and δ

 

(a) Initial model; (b) Inversion result. In (b), the inverted ε are {0.167; 0.127; 0.109; 0.077} and inverted δ are {-0.163; -0.194; -0.113; -0.198} for the top to bottom layers. The parameter in true model are ε = {0.18; 0.16; 0.14; 0.12} and δ = {-0.11; -0.13; -0.15; -0.17}.