# Flat refractor

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

Consider the refraction wavefront and raypath geometry in Figure 3.4-11a associated with a single-layer nearsurface model. On top, we see a plot of first-breaks. For simplicity, consider a flat surface and flat refractor. For the head wave to form, and thus the refraction to occur, the requirement is that the overburden velocity vw be smaller than the substratum velocity vb.

The traveltime profile depicts the first breaks seen on the shot record in Figure 3.4-11b. Note that to the left of the crossover offset xc (also known as critical distance) are the first breaks associated with the the direct arrivals. Also note that to the right of offset xc are the first breaks associated with the refracted arrivals. From the refraction theory [1]; [2], the inverse of the slope of the line associated with the refracted wave arrivals is equal to the bedrock velocity vb. Also note that the inverse of the slope of the line associated with the direct wave arrivals is equal to the velocity of the weathering layer vw.

By picking the first breaks, the weathering and bedrock velocities, vw and vb are estimated. By extending the line associated with the refracted arrivals to zero-offset, intercept time ti, the time at x = 0, is estimated. From these three parameters, it is easy to show that depth to the bedrock zw is given by

 ${\displaystyle z_{w}={\frac {v_{b}v_{w}t_{i}}{2{\sqrt {v_{b}^{2}-v_{w}^{2}}}}}.}$ (41a)

We assume that vb > vw. Derivation of this formula is left to Section C.5.

Alternatively, we can measure the critical distance corresponding to the change from the direct arrival to the refracted arrival on the traveltime plot and use it in computing the depth to the bedrock. Equation (41a), in terms of the critical distance xc, takes the form

 ${\displaystyle z_{w}={\frac {1}{2}}{\sqrt {\frac {v_{b}-v_{w}}{v_{b}+v_{w}}}}x_{c}.}$ (41b)

It may not be easy to measure the critical distance when depth to bedrock is small. In such cases, it is better to use the intercept time to compute the depth to the bedrock by way of equation (41a).

After computing zw, the total static correction ΔτD to the specified datum level can be applied by

 ${\displaystyle \Delta \tau _{D}=-{\frac {2z_{w}}{v_{w}}}+{\frac {2(E_{D}-E_{S}+z_{w})}{v_{b}}},}$ (42)

where ES is the surface elevation. If there is a difference between the elevations of shot and receiver stations, then an additional elevation correction using the bedrock velocity is required. Moreover, if the shots are located in boreholes, then the measured uphole time also must be incorporated into equation (42). The estimated statics correction given by equation (42) is an average value over a distance that can range from the critical distance to the spread length, depending on the number of traces used in estimating the bedrock velocity. Nevertheless, more than one shot-point is within a spread length. Therefore, an adequate definition of the near-surface model can be achieved and datum corrections can be computed for the entire profile.

Figure 3.4-11  (a) Geometry for refracted arrivals. Here, vw = weathering velocity, vb = bedrock velocity, zw = depth to the refractor equivalent to the base of the weathering layer, θc = critical angle, and xc = crossover distance. The direct wave arrival has a slope equal to 1/vw and the refracted wave arrival has a slope equal to 1/vb. (b) A shot record that exhibits the direct wave and the refracted wave depicted in (a). (c) Geometry for a dipping refractor with forward traveltime profile associated with the direct wave and refracted wave arrivals, and (d) with both forward and reverse traveltime profiles. See text for details.