Appendix D: Resolution

What is the lateral and vertical resolution associated with a refraction tomography experiment? That is, what is the smallest feature resolvable in a refraction tomogram? According to the discussion on covariance matrices, the model covariance matrix should provide the answer to this question. However, the covariance matrix is based on the wrong physics! That is, it assumes infinitely short wavelengths so that it predicts, in the possible case, resolution on the scale of atomic distances. Something must be wrong!

The problem is that the wavelengths are finite and so interference effects, which are ignored by ray theory, prevent perfect resolution. To take into account interference effects we need to consider Fresnel zones in the far-field. Using a lot of algebra I showed in one of our annual reports that, under the far-field approximation, the best possible horizontal resolution dxand vertical resolution dz in a refraction tomogram is given by:

$$\cong ~ \lambda_2$$ dx

$$\cong cos(\theta)\lambda_1/4$$ dz (2.19)

where $\lambda_2$ is the wavelength in the lower medium and and $\lambda_1$ is the wavelength in the upper medium.

A simple geometrical argument about Fresnel zones can be made to justify the use of equations C1 and C2 to estimate resolution. The dx resolution estimates can be derived by setting the arrival-time difference associated with the raypaths in the upper figure above to be equal to 1/2 the period, i.e.,

T _SABC - T_SAR = 0.5 T, (2.20)

where T is the period of the wavelet. Solving for the path length AB yields the horizontal resolution dx. This value of AB can be set to dx because it is the location of the scatterer that can be laterally offset from A and still contribute to the refraction arrival at R within 1/2 period of the initial onset.

The vertical resolution dzcan be derived in a similar manner, except this is the vertical offset a scatterer can be lifted from the refractor and still contribute to the arrival within 1/2 T of the initial onset.

 

Figure 2.12: Raypaths associated with horizontal resolution (top figure) and vertical resolution (lower figure). Note that as the critical angle theta goes to zero the vertical resolution approaches lambda₁/4, that of reflection migration.

Therefore the resolution estimates are the best possible case. Worst cases arise when the geophone and source spacings are wider than the wavelength.