Weathering and elevation (near-surface) corrections

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Problem 8.18a

Show that when the source is below the LVL, weathering and elevation corrections for a geophone at the source are given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta t_{0} =2\left(E_{S} -D_{S} -E_{D} \right)/V_{H} +t_{uh}, \end{align} }

Figure 8.18a.  Calculation of weathering corrections.

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E_{S}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E_{D}} are the elevations of the source point and datum, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D_{S}} is the depth of the source, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_{uh}} is the uphole time.

Background

Corrections are necessary to eliminate the effect of changes in the elevation of the surface and in the thickness of and velocity in the LVL. The corrections in effect reduce the traveltimes to those that would be observed if the source and geophones were located on a reference datum, usually a horizontal plane below the base of the low-velocity layer. These corrections are called static corrections because they are the same for all reflections regardless of their arrival times.

Solution

In Figure 8.18a, the correction to the traveltime for a wave going from the source at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} down to the datum is


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta t_{S} =\left(E_{S} -D_{S} -E_{D} \right)/V_{H}. \end{align} } (8.18a)

The correction for travel from the datum up to a geophone at the source point is


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta t_{g} =\Delta t_{S} +t_{uh}, \end{align} } (8.18b)

so the total correction for the traveltime for a geophone at the source point is


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta t_{0} =2\left(E_{S} -D_{S} -E_{D} \right)/V_{H} +t_{uh}. \end{align} } (8.18c)

Problem 8.18b

If the split spread Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle ACB} in Figure 8.18a is used to find the dip, what correction must be applied to the dip moveout?

Solution

The dip moveout is obtained by subtracting traveltimes at sources Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} in Figure 8.18b. If traveltimes have not been corrected for weathering and elevation, the dip moveout must be corrected; this is the differential weathering correction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta t_{d}} . Using equation (8.18b) we have


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta t_{d} =(\Delta t_{g} )_{B} -(\Delta t_{g} )_{A} =(\Delta t_{S} +t_{uh} )_{B} -(\Delta t_{S} +t_{uh} )_{A}. \end{align} } (8.18d)
Figure 8.18b.  Differential weathering correction.

Problem 8.18c

Derive an expression to correct the traveltime for a geophone at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C} in Figure 8.18b Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} being source points.

Solution

We use the first-break traveltimes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_{AC}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_{BC}} that correspond to the paths Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A'C''C} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B'C'C} . The velocity contrast at the base of the LVL is usually large enough that the paths Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C'C} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C''C} are so close to vertical that the distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C'C''} is very small. Hence the sum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \left(t_{AC} +t_{BC} \right)} is given by


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left(t_{AC} +t_{BC} \right)\approx \left(A'B'\right)/V_{H} +2t_{w},\\ \mbox{so} \qquad\qquad t_{w} \approx \frac{1}{2} [\left(t_{AC} +t_{BC} \right)-(AB/V_{H})], \end{align} } (8.18e)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_{w}} is the traveltime through the LVL at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D_{W} =V_{W} t_{w}} . Therefore the correction that effectively places the geophone at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C} on the datum is


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta t_{C} =t_{w} +\left(E_{C} -D_{W} -E_{D} \right)/V_{H}. \end{align} } (8.18f)

We must add to this the correction that locates the source on the datum, namely Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta t_{S}} , given by equation (8.18a). Thus the total correction for a traveltime recorded at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C} is


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta t_{C} =t_{w} +\left(E_{C} -D_{W} -E_{D} \right)/V_{H} +\Delta t_{S}. \end{align} } (8.18g)

Problem 8.18d

The weathering and elevation corrections given by equations (8.18a) to (8.18g) assume that the source is below the base of the LVL. What changes are required if the source is within the LVL?

Solution

When the source is within the LVL, the wave traveling down to the datum is in the LVL for the distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (D_{W} -D_{S}} , hence equation (8.18a) is changed to


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta t_{S} &=\left(D_{W} -D_{S} \right)/V_{W} +\left(E_{S} -D_{W} -E_{D} \right)/V_{H} \\ &=\left(E_{S} -D_{W} -E_{D} \right)/V_{H} +\Gamma _{S}, \end{align} } (8.18h)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Gamma _{S} =\left(D_{W} -D_{S} \right)/V_{W}} . Equation (8.18b) is unchanged provided we use the value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta t_{S}} in equation (8.18h). Equation (8.18c) becomes


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta t_{0} &=\left(D_{W} -D_{S} \right)/V_{W} +2\left(E_{S} -D_{W} -E_{D} \right)/V_{H} +\left[\left(D_{W} -D_{S} \right)/V_{W} +t_{uh} \right]\\ &=2\Gamma _{S} +2\left(E_{S} -D_{W} -E_{D} \right)/V_{H} ]+t_{uh}. \end{align} } (8.18i)

Equation (8.18d) is unchanged provided we use equation (8.18h) for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta t_{S}} . Equation (8.18e) becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left(t_{AG} +t_{BG} \right)=AB/V_{H} +2t_{w} +\left(\Gamma _{A} +\Gamma _{B} \right), \end{align} }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Gamma _{A} =\Gamma _{SA}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Gamma _{B} =\Gamma _{SB}} . Thus, equation (8.18e) becomes


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} t_{w} =\frac{1}{2} [\left(t_{AG} +t_{BG} \right)-AB/V_{H} -(\Gamma _{A} +\Gamma _{B})]. \end{align} } (8.18j)

Equations (8.18f,g) are unchanged except that we must use the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_{w}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta t_{S}} from equations (8.18j) and (8.18h).

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