Reflections/diffractions from refractor terminations

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Problem 6.15a

A horizontal refractor is located under a north-south seismic line at a depth of 1200 m. The overburden velocity is 2500 m/s and the refractor velocity is 4000 m/s. The refractor is terminated by a linear vertical fault (HF in Figure 6.15a) 3500 m from the source point. Determine the traveltime curves when the fault strikes: (i) east-west, (ii) north-south, (iii) N30Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): ^{\circ} W.

Background

The traveltime curve for a horizontal refractor is given by equation (4.18a). The critical angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta _{c} ={\rm sin}^{-1} (V_{1} /V_{2}) . In Figure 6.15a the refraction does not exist in the interval SQ; the distance SQ is the critical distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x^{\prime} where


Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} x^{\prime} =2h{\rm \; tan\; }\theta_{c}. \end{align} (6.15a)
Figure 6.15a.  Events involving refractor termination.

Solution

We require the values of $ \theta _{c} $ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x^{\prime}/2 :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \theta_{c} ={\rm sin}^{-1} (2.50/4.00)=38.7^{0}; x^{\prime}/2=1200{\rm \; tan\; }38.7^{0} = 0.96\ {\rm km}. \end{align}

The fault is located at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): H which is more than 960 m from the source in all three cases, so refracted waves are involved.

Case (i). Fault perpendicular to the north-south line.

Events are the following (refer to Figure 6.15a):

  1. the direct wave; a straight line through the source with slope (1/2.50) s/km;
  2. an inline refraction, a straight line beginning at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q with slope (1/4.00) s/km; it is tangent at $ Q $ to the reflection hyperbola (curve 5);
  3. a reflected head wave between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): G from reflection point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): H with paths such as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): STHMP , a straight line extending from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): G with slope opposite to (2) and a larger intercept time;
  4. a diffraction generated at $ H $, a curve through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): F symmetrical about the vertical and tangent to the head-wave curve (2) at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): U (if prolonged beyond Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): F ) and to curve (3) at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): G ;
  5. a reflection, a hyperbola symmetrical about the vertical through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S and tangent to curve (2) at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): Q ;
  6. a reflected refraction such as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): STJKL (if the impedance contrast at the fault is large enough to produce a recognizable reflection), a straight line extension of curve (3) to the right of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): G .

Case (ii). Fault parallel to the seismic line.

A plan view of the fault is shown to the left of the “line” in Figure 6.15b. The observed events are

  1. the direct wave; a straight line passing through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S with slope (1/2.50) s/km;
  2. an inline refraction;
  3. a reflected refraction along paths such as STFMP shown in plan view in Figure 6.15b where the energy goes down from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): S at the critical angle until the refractor is reached at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T , then along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): TF to the fault at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): F where reflection occurs, after which the energy travels along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): FM until a ray peels off at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): M and travels up to the recorder at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): P , its traveltime curve is shown in Figure 6.15c; the curve is given by equation (4.18a) where we replace Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x with the distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} SF+FP = 2SF = 2[y^{2} +(x/2)^{2}]^{1/2} =2[3.50^{2} +x^{2} /4]^{1/2}, \end{align} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x here being the distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): SP ; the curve is a hyperbola (see Figure 6.15c);
  4. no diffractions will be observed because there is no point source;
  5. inline reflections; the reflection curves are the usual hyperbolas;
  6. there is no reflected refraction such as STJKL in Figure 6.15a in case (i) because it could only exist within a distance of 0.96 km from the fault.


Figure 6.15b.  Travelpaths for faults parallel and at angle to seismic line.


Figure 6.15c.  Arrival time curve for the reflected refractions.


Case (iii). Fault at an angle to the seismic line.

This case is similar to case (ii). The fault in the plan view shown to the right of the “line” in Figure 6.15b strikes at the angle N30Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): ^{\circ} W. The observed events are

  1. the direct wave;
  2. an inline refraction;
  3. a reflected refraction, a typical path being Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {ST}^{'}{F}^{'}{M}^{'}{P} ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T^{'} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): M^{'} being equivalents of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): T and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): M in Figure 6.15b; its traveltime curve is shown in Figure 6.15c; to derive the traveltime curve for this event, we use image point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): I for reflection in the fault, so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): IP replaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x in equation (4.18a). We get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t=IP/V_{2} +(2h{\rm \; cos\; }\theta _{c} )/V_{1}\\ =x^{2} +SI^{2} -2xSI{\rm \; cos\; }60^{\circ} )^{1/2} /4.00+2\times 1.20{\rm \; cos\; }38.7^{\circ} /2.50\\ =(1/4.00)(x^{2} -7.00x+49.00)^{1/2} +0.749. \end{align} Thus, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (4-2.96)^{2} =(x^{2} -7.00x+49.00) . The curve is a hyperbola (see Figure 6.15c);
  4. no diffraction event;
  5. a normal reflection;
  6. same as in case (ii).

All traveltime curves are normal except (3).

Problem 6.15b

Repeat for the east-west fault for a refractor that dips Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 10^{\circ} to the north with the source to the south.

Solution

In Figure 6.15d, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta_{c} =38.7^{\circ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): SB=1.20 km, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): BH=3.50 km. We make frequent use of the law of sines to calculate distances:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} BT=(1.20/{\rm \; sin\; }51.3^{\circ} ){\rm \; sin\; }38.7^{\circ} =0.74\ {\rm km},\\ ST=1.2{\rm \; sin\; }100^{\circ} /{\rm \; sin\; }51.3^{\circ} =1.51\ {\rm km},\\ TH=QC=3.50-BT=2.76\ {\rm km},\\ SQ=ST{\rm \; sin\; }77.4^{\circ} /{\rm \; sin\; }41.3^{\circ} =2.23\ {\rm km},\\ TQ=ST{\rm \; sin\; }61.3^{\circ} /{\rm \; sin\; }41.3^{\circ} =2.01\ {\rm km},\\ QU=QC{\rm \; sin\; }128.7^{\circ} /{\rm \; sin\; }41.3^{\circ} =3.26\ {\rm km},\\ SU=SQ+QU=5.49\ {\rm km},\\ CU=QC{\rm \; sin\; }10^{\circ} /{\rm \; sin\; }41.3^{\circ} =0.73\ {\rm km},\\ HU=HC+CU=TQ+CU=2.74\ {\rm km},\\ HG=HU{\rm \; sin\; }41.3^{\circ} /{\rm \; sin\; }61.3^{\circ} =2.06\ {\rm km},\\ HD= 1.20+3.50 {\rm \; sin\; }10^{\circ} =1.81\ {\rm km},\\ SG=SU-GU=SU-(HU{\rm \; sin\; }77.4^{\circ} /{\rm \; sin\; }61.3^{\circ} )\\ =5.49-3.05=2.44\ {\rm km},\\ SD=3.50{\rm \; cos\; }10^{\circ} =3.45\ {\rm km}. \end{align}

Figure 6.15d.  Events involving termination of a dipping refractor (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta = critical angle).

For the refraction,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{Q} =(ST-TQ)/2.50=1.41\ {\rm s},\\ t_{U} =(ST+HU)/2.50+TH/4.00=2.39\ {\rm s}. \end{align}

For the reflected refraction,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{G} =(ST+HG)/2.50+TH/4.00=2.12\ {\rm s},\\ t_{S} =2(ST/2.50+TH/4.00)=2.59\ {\rm s}. \end{align}

The diffraction from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): H has its minimum traveltime at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): D :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{D} =(ST+HD)/2.50+TH/4.00=2.02\ {\rm s}. \end{align}

For the reflection,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} t_{0} =2(1.20{\rm \; cos\; }10^{\circ} )/2.50=0.95\ {\rm s},\\ t_{Q} = {\rm same\ as}\ t_{Q}\ {\rm for\ the\ refraction}\ =1.41\ {\rm s}. \end{align}

Problem 6.15c

What effect will the manner of terminating the refractor have, that is, how will the amplitude of the reflected refraction depend on the dip of the terminating fault?

Solution

Provided the impedance contrast across the fault is large enough, any abrupt termination of the refractor will generate a reflected refraction. The attitude of the terminating fault will have a relatively small effect on the amplitude provided that the dip of the fault is such that the angle of incidence is close to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 90^{\circ} .

Problem 6.15d

Most commonly a faulted refractor terminates against rock of lower acoustic impedance, but the opposite situation can also happen. What differences will this make?

Solution

The nature of refractors is that they have high velocity, hence usually terminate against rocks of lower acoustic impedance and a reflected refraction has opposite phase to the incident refraction. However, if a refractor terminates against a higher impedance, they will have the same phase.

Problem 6.15e

Extend the profile for part (a), case (i), an appreciable distance beyond the fault so as to plot the diffraction from the refractor termination. Assume uniform 2.50-km/s material beyond the refractor termination.

Solution

The extensions of curves (1), (2), (4), and (5) are shown in Figure 6.15a. Events (3) and (6) do not exist to the right of the fault.

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