NMO for several layers with arbitrary dips

Seismic Data Analysis

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

Figure 3.1-15 shows a 2-D subsurface geometry that is composed of a number of layers, each with an arbitrary dip. We want to compute the traveltime from source location S to depth point D, then back to receiver location G, which is associated with midpoint M. Note that the CMP ray from midpoint M hits the dipping interface at normal incidence at D′, which is not the same as D. The zero-offset time is the two-way time along the raypath from M to D′.

Figure 3.1-15  Geometry for the moveout for a dipping interface in an earth model with layers of arbitrary dips. Adapted from .)

derived the expression for traveltime t along SDG as

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^2=t^2_0+\frac{x^2}{v^2_{NMO}}+higher\ order\ terms,}

(12)

where the NMO velocity 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 v^2_{NMO}=\frac{1}{t_0\cos^2\beta_0}\sum\limits^{N}_{i=1}v^2_i\Delta t_i \prod\limits^{i-1}_{k=1}\left(\frac{\cos^2\alpha_k}{\cos^2\beta_k}\right).}

(13)

The angles α and β are defined in Figure 3.1-15. For a single dipping layer, equation (13) reduces to equation (8). Moreover, for a horizontally stratified earth, equation (13) reduces to equation (4). As long as the dips are gentle and the spread is small, the traveltime equation is approximately represented by a hyperbola (equation 5), and the velocity required for NMO correction is approximately the rms velocity function (equation 4).

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 v^2_{rms}=\frac{1}{t_0}\sum _{i=1}^N v^2_i\Delta\tau_i,}

(4a)

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^2=t^2_0+\frac{x^2}{v^{2}_{rms}}.}

(4b)

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^2=t^2_0+\frac{x^2}{v^{2}_{rms}}+C_2x^4.}

(5a)

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=t_0\left(1-\frac{1}{S}\right)+\sqrt{\left(\frac{t_0}{S}\right)^2+\frac{x^2}{Sv^{2}_{rms}}}}

(5b)

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=(t_0-t_p)+\sqrt{t^2_p+\frac{x^2}{v_s^2}},}

(5c)

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 v_{NMO}=\frac{v}{\sin \alpha}.}

(8)

See also

• NMO for a flat reflector
• NMO in a horizontally stratified earth
• Fourth-order moveout
• NMO stretching
• NMO for a dipping reflector
• Moveout velocity versus stacking velocity
• Exercises
• Topics in moveout and statics corrections

External links

find literature about NMO for several layers with arbitrary dips