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Chapter 6

Static corrections

The objective of static corrections is to account for near-surface effects.

These corrections include:

o Elevation static correction, which accounts for variable elevations of the

sources and receivers.

o Residual static correction, which accounts for lateral variations in the velocity

and thickness of the weathering layer.

The weathering layer is the shallowest low-velocity layer. It is composed of

unconsolidated and loosely consolidated sediments.

Common weathering layers in arid areas include: sand dunes, sabkhas, gravel plains,

karsts, and valley fills.

The weathering layer velocity is usually much less than those of the underlying sub-

weathering layer (bedrock) and deeper layers.

Therefore, the weathering layer produces large contribution to the overall traveltime

of rays.

Figure 6.1 shows the effect of noise versus statics on T-X curves.

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Elevation (field) static correction

It involves the computation and removal of the effect of different source and receiver

elevations.

This involves bringing the sources and receivers to a common datum, preferably

below the elevation of the lowest source or receiver.

For this, we need a replacement velocity (Vr) for the material between the elevation

of the datum and that of 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:

r

DRRDSS

DV

EZEEZET

)()( , (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 always subtracted from the two-way traveltime of the trace belonging to that

particular source-receiver pair.

Figure 6.2 shows elevation profiles in land and marine surveys.

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Residual static correction

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

thickness and lateral velocity variation of the weathering layer.

Figure 6.3 shows the difference between elevation and residual static corrections.

The main methods used to correct for these effects are:

Uphole surveys.

Refraction statics.

Surface-consistent statics.

Uphole surveys

A fairly shallow hole (100-200 m) that penetrates the weathering layer and the upper

part of the sub-weathering layer is drilled 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 the uphole location.

Figure 6.4 shows a typical uphole geometry.

Figure 6.5 shows an example of a real uphole survey and its analysis.

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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.

This method has the advantage of providing highly accurate near-surface velocities

and thicknesses. However, it is very costly if high lateral resolution is required.

Therefore, this method is usually good 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).

Refraction statics

Most versions of this method are effective in estimating short-wavelength statics.

This method is used to construct a model of the weathering layer (WL) by estimating

the velocity and thickness of the WL.

The following are standard methods used for refraction statics calculation:

1. Delay-time Method:

It uses the slopes of the direct and head waves as well as the head wave’s

intercept time of many shot records along the profile to estimate the WL

velocity and thickness under each receiver.

This method requires picking first breaks, which is difficult. It also requires

reversed raypath geometries, which might not be available.

Figure 6.6 gives more details on the delay-time method.

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2. Generalized reciprocal Method (GRM):

It uses reversed refraction profiles to estimate the optimum WL thickness

under each receiver.

This method requires picking first breaks, which is difficult. It also requires

reversed raypath geometries, which might not be available.

3. Least-squares Method:

It uses least square analysis to find the best-fit WL velocity-thickness model

to the first arrivals (i.e., direct and head waves).

It uses tomographic methods to invert the observed first arrivals for the best-

fit velocity-thickness model of the WL.

Surface-consistent residual static correction method

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

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

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

the subsurface.

This assumption is valid only if all raypaths are vertical in the near surface, regardless

of source-receiver offset and refractor depth (Figure 6.7).

The surface-consistent assumption is generally good because the WL usually has a

low velocity and refraction towards the normal at its base tends to make raypaths

vertical. What other conditions would make this assumption more realistic?

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The total residual static time shift on any trace can be expressed as:

2

ijkkjiijk XMGSRT , (2)

where:

Ri: is the residual static time shift associated with the ith

receiver position (i = 1, …, I,

where I is the number of receivers used in the survey),

Sj: is the residual static time shift associated with the jth

source position (j = 1, …, J,

where J is the number of sources used in the survey),

Gk: is the difference in two-way traveltime (due to structure) at a reference CMP and

the traveltime at the kth

CMP (k = 1, …, K, where K is number of CMPs in the

survey), and

kM : is the residual NMO associated with the trace generated by the jth

source

and recorded by the ith

receiver and it accounts for possible imperfect NMO

correction due to using imperfect NMO velocities for the kth

CMP. Why is Mk

multiplied by Xij2?

Generally, we have more equations than unknowns for typical seismic surveys (Is this

true for your line?). This is a typical least-squares problem.

Our objective is to find those Ri, Sj, Gk, and Mk that will minimize the following error-

energy between observed and calculated time shifts using model parameters in

equation (2):

2 2

1 1 1

[( ) ]I J K

i j k k ij ijk

i j k

E R S G M X T

. (3)

calculated Tijk observed Tijk

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Residual static correction in practice

Residual static correction, in practice, involves the following three phases:

(1) Picking (calculating) the observed time shifts Tijk.

(2) Decomposition (distribution) of Tijk into receiver, source, structural, and

imperfect-NMO terms.

(3) Application of only source and receiver terms to the traces of pre-NMO-corrected

CMP gathers.

(1) Picking:

It means estimating the observed time shifts Tijk from the data.

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) The CMP gather is stacked to produce the first pilot trace.

(3) Each individual trace in this CMP gather is crosscorrelated with the first pilot

trace.

(4) Time shifts '

ijkT , which correspond to maximum crosscorrelations, are picked.

(5) Shift each original trace by its corresponding time shift '

ijkT .

(6) A second pilot trace is constructed by stacking the once-shifted traces in the

gather.

(7) The second pilot trace is, in turn, crosscorrelated with the once-shifted traces

in the gather and new time shifts ''

ijkT are computed. Naturally, ''

ijkT < '

ijkT .

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(8) Shift each once shifted-trace by its corresponding new time shift ''

ijkT .

(9) The total time shift is given as: Tijk = '

ijkT + ''

ijkT .

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

(11) The third pilot trace is crosscorrelated with the third pilot trace of the next

CMP gather to calculate the structural term Gk.

(12) The process is performed this way on all CMP gathers moving to left and/or

right from the starting (reference) CMP gather.

(13) The picked total time shifts (Tijk) are passed to the next phase

(decomposition).

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

(a) Maximum allowable shift:

It is the maximum shift allowed for crosscorrelations.

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

any given location along the profile.

It should be less than the dominant period of the data in poor S/N ratio

data (why?).

A value between 30 and 40 ms is reasonable.

(b) Correlation window:

It is taken around a prominent primary reflection.

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

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

possible.

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(c) Other considerations:

The residual-NMO term (Mk) should not be large within the correlation

window. Therefore, you should have done your best effort with the NMO

correction.

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

correction must be done.

A second residual static correction pass means:

1. Do velocity analysis.

2. Do residual static correction.

3. Do velocity analysis again.

4. Do residual static correction again.

(2) Decomposition:

It involves least-squares decomposition of the picked time shifts (Tijk) found in

phase (1) into source, receiver, structural, and residual-NMO terms using equation

(3).

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

equations is the Gauss-Seidel iterative procedure.

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

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

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Appendix A

Elevation Statics

r

DRRDSSD

V

EZEEZET

)()(

Back

ED

ER

ES

Vr

Ground surface

ZR

ZS

ES - ZS - ED

ER – ZR - ED