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Chapter 5

Static corrections

Elevation (field) statics

• Elevation statics involve the computation and removal of the effect of different source

and receive elevations.

• This involves bringing the source and receiver to a common datum, usually below the

elevation of the lowest source or receiver.

• For this, we need a replacement velocity (Vr) for the material between the datum and

the source or receiver.

• The replacement velocity is either assumed from prior knowledge of the area or can

be estimated from uphole times or direct arrivals.

• The elevation static correction (tD) is given by:

tD = [(ES – ZS - ED) + (ER – ZR - ED)]/Vr, (5.1)

where, ES: ground elevation at shot location (from mean sea level),

ZS: depth of shot (= 0 for a surface source),

ER: ground elevation at receiver location (from mean sea level),

ZR: depth of receiver (= 0 for a surface geophone), and

ED: datum elevation (from mean sea level).

• tD is subtracted from the two-way traveltime of the trace belonging to that particular

source-receiver pair.

• Figure.

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Near-surface (weathering) corrections

• After elevation statics correction, it is important to correct for the effect of variable

thickness and lateral velocity variation of the weathering layer.

• The main methods used to correct for these effects are:

Ø Uphole surveys.

Ø Refraction statics.

Ø Residual statics.

Uphole survey

• A deep hole that penetrates below the weathering layer is used for this purpose.

• Several geophones are placed at various (known) depths in the hole. The geophone

locations must span the weathering and sub-weathering layers.

• A shot is fired at the surface near the hole and the direct traveltimes to the geophones

are recorded.

• A plot of the direct traveltimes versus the geophone depths can be used to compute

the velocities of the weathering and sub-weathering layers as well as the thickness of

the weathering layer at that location.

• This method attempts to construct a model of the weathering layer by estimating the

velocity and thickness of the weathering layer at several locations and interpolating

between these locations.

• Figure.

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Refraction statics

• This method is especially effective in estimating long-wavelength statics.

• Wavelength of statics refers to the width of the lateral (velocity or thickness) change

in the weathering layer relative to the spread length (maximum offset).

• This method is used to construct a model of the weathering layer by estimating the

velocity and thickness of the weathering layer.

• The following are some of the methods used for refraction statics calculation:

Ø Delay-time methods.

Ø The generalized reciprocal method (GRM).

Ø Least-squares methods.

• The first two methods involve picking first breaks, which is difficult, and require

specific raypath geometries, which might not be available.

• The least-squares methods employ the same concepts used for the residual-statics

method, but use refraction rather than reflection data.

Residual statics

• This method is especially effective in estimating short-wavelength statics.

• The most widely used method is the surface-consistent method.

Surface-consistent residual statics corrections:

• The basic assumption of this method is that the static shifts are time delays that only

depend on the source and receiver locations on the surface, not on raypaths in the

subsurface.

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• This assumption is valid only if all raypaths, regardless of source-receiver offset, are

vertical in the near surface.

• The surface-consistent assumption is generally good because the weathered layer

usually has a low velocity and refraction towards the normal at its base tends to make

raypaths vertical.

• The total residual time shift, tijk, can be expressed as:

tijk = ri + sj + Gk + Mk xij2, (5.2)

where ri: is the residual static time shift associated with the ith receiver,

sj: is the residual static time shift associated with the jth source,

Gk: is the difference in two-way traveltime at a reference CMP and the traveltime at

the kth CMP, and

Mk xij2: is the residual moveout that accounts for the imperfect NMO correction.

• Gk is a structural term, while Mk is a hyperbolic term.

• The purpose is to determine the unknowns ri, sj, Gk, and Mk from the known variables

tijk and xij.

• Usually, there are more equations than unknowns; hence, we use least-squares

approach to minimize the error energy:

E = ∑ijk [(ri + sj + Gk + Mk xij2) - tijk]2. (5.3)

Residual statics correction in practice

• In general, residual statics correction, in practice, involves the following three phases:

(1) Picking (calculating) the time shifts tijk.

(2) Decomposition of tijk into receiver, source, structural, and residual terms.

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(3) Application of derived source and receiver terms to traveltimes on pre-NMO-

corrected CMP gathers.

(1) Picking:

Ø It means estimating the time shifts tijk from the data.

Ø The most widely used method is the pilot trace method, which consists of the

following steps:

(1) A CMP with good S/N ratio is gained and NMO-corrected using a preliminary

velocity function.

(2) A specific horizon (reflection, event) is selected.

(3) The CMP gather is stacked.

(4) Each individual trace in the gather is crosscorrelated with the stack trace.

(5) Time shifts tijk(1), which correspond to maximum crosscorrelations, are picked.

(6) Shift each original trace by its corresponding time shift tijk(1).

(7) A preliminary pilot trace is constructed by stacking the time-shifted traces in

the gather.

(8) This pilot trace is, in turn, crosscorrelated with the shifted traces in the gather

and new time shifts tijk(2) are computed.

(9) Shift each once-shifted trace by its corresponding new time shift tijk(2).

(10) The total time shift is given as: tijk = tijk(1) + tijk

(2).

(11) A final pilot trace is constructed again by stacking the twice-shifted traces.

(12) This final pilot trace is crosscorrelated with the traces of the next gather to

construct the preliminary pilot trace for that gather.

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(13) The process is performed this way on all CMP gathers moving to left and right

from the starting (reference) CMP.

(14) The picked total time shifts (tijk) are passed to the next phase (decomposition).

Ø The following parameters are important when picking the time shifts in practice:

(a) Maximum allowable shift:

v It should be greater than all possible combined shot and receiver shifts at

any given location along the profile.

v However, it should be less than the dominant period of the data in poor

S/N ratio conditions.

(b) Correlation window:

v It should be chosen in an interval with the highest possible S/N ratio.

v It should be as large as possible and outside the mute zone whenever

possible.

(c) Other considerations:

v The residual moveout variations should not be large within the correlation

window.

v In areas of significantly poor S/N ratio, a second pass of residual statics

corrections must be done.

v A second residual statics correction pass means:

1. Do velocity analysis.

2. Do residual statics correction.

3. Do velocity analysis again.

4. Do residual statics correction again.

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(2) Decomposition:

Ø It involves least-squares decomposition of the picked time shifts found in phase

(1) into source, receiver, structural, and residual terms using equation (5.3).

Ø The procedure most widely used for solving the resulting system of linear

equations is the Gauss-Seidel iterative procedure.

Ø (Not required): For more detail on the Gauss-Seidel iterative procedure, follow

this link.

(3) Application: The individual static shifts associated with each source and receiver

location are applied to the pre-NMO-corrected gather traces.

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Appendix A

Elevation Statics

r

DRRDSSD V

EZEEZEt )()( −−+−−=

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ED

ER

ES

Vr

Ground surface

ZR

ZS

ES - ZS - ED

ER – ZR - ED

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Appendix B

Uphole Survey

DEPTH (M) VELOCITY (M/S) TIME (S)10 1000 0.01020 1000 0.02030 1000 0.03040 2000 0.03550 2000 0.04060 2000 0.045

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