CN106094033B - The orientation seismic beam forming method of singular value decomposition - Google Patents
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Abstract
The present invention relates to a kind of orientation seismic beam forming method of singular value decomposition, extracts the strong linear disturbance of low frequency, energy using wicket singular value decomposition method, and filtered out from original seismic data；Then main beam direction is determined according to exploration targets, using beam-forming method computation delay parameter, obtains orientation earthquake record；Singular value decomposition method reconstruct singular value finally is recycled for orientation earthquake record, filters out the random background noise in orientation earthquake record, obtains the orientation earthquake record of high s/n ratio.Through experiment, the orientation seismic beam forming method of singular value decomposition can effectively suppress strong linear disturbance and background random noise in earthquake record, and effectively enhance the energy value of target echo, realize improves the signal to noise ratio of geological data in terms of noise reduction and enhancing signal two, improves the quality of data of seismic prospecting.Especially earthquake effect in deep is better than singular value decomposition method and orientation seismic beam forming method.
Description
Technical field：
The present invention relates to one kind to orient seismic beam synthetic method, is to be directed to deep earthquake data acquisition and humanity activities
The low situation of Earthquakes record signal to noise ratio, in order to suppress strong linear disturbance and background random noise and strengthen useful signal energy
Amount, improve S/N ratio of seismic records, it is proposed that a kind of orientation seismic beam forming method of singular value decomposition.
Background technology：
Beam-forming thought comes from phased array radar field earliest.Because beam-forming method effectively can be strengthened on objective body
Useful signal, field of seismic exploration is introduced into quickly.Disclosed in CN103984019A《Local correlation weighting seismic beam synthesis side
Method》, CN104570121A discloses《Orient seismic wave distorted signal removing method》Disclosed with CN104793243A《Based on N
The orientation seismic wave data processing method of secondary root superposition》Three kinds of distinct methods are taken to solve main ripple in seismic beam formation technology
The problem of Shu Fangxiang external signals distort；CN103984007A is disclosed《Orient seismic wave delay parameter Optimization Design》With
CN104570097A is disclosed《Orientation earthquake record synthetic method based on Discrete Particle Swarm algorithm》Two methods are taken to solve
How optimum option delay parameter, make different earthquake record in target echo the problem of realizing in-phase stacking.On address
The current Patents delivered and paper are directed to how research strengthens target echo energy, to how to suppress orientation
Strong linear disturbance and background random noise in earthquake record does not carry out being directed to Journal of Sex Research.
Singular value decomposition method is introduced into field of seismic exploration, is mainly used in suppressing strong linear disturbance and background and makes an uproar at random
Sound.CN102338886A is disclosed《The polarized filtering method of face ripple in a kind of effective attenuation three-component coefficient》、
CN102193107A is disclosed《A kind of seismic wave field separation and denoising method》Disclosed with CN105319591A《Based on radial direction road
The adaptive surface wave pressing methods of SVD of conversion》Realize using strong linear in singular value decomposition method compacting earthquake record
Interference；CN102854533A is disclosed《A kind of denoising method that seismic data signal to noise ratio is improved based on wave field separation principle》, have
The background random noise in earthquake record is suppressed to effect, CN104865603A is disclosed《A kind of SVD for dipping bed is filtered
Method and device》Solve the problems, such as that rear wave resistance feature is distorted dipping bed after filtering during denoising.It is above-mentioned
And currently existing related singular value decomposition document for singular value decomposition suppress noise research be in target echo and
On the basis of noise can separate considerably from one another, and point out that the signal to noise ratio for working as earthquake record is very low, target echo and noise
When can not efficiently separate, singular value decomposition method is no longer applicable.
The content of the invention：
The purpose of the present invention is that for deep earthquake data acquisition and humanity activities Earthquakes data signal to noise ratio
When low, with reference to singular value decomposition method compacting noise and the excellent of seismic beam forming method enhancing target echo energy is oriented
Point, it is proposed that the orientation seismic beam forming method of singular value decomposition.
Main idea is that：By the orientation seismic beam forming method of singular value decomposition, realize to strong linear
Interference and background random noise filter out and the enhancing of target echo energy.This method uses wicket singular value first
The strong linear disturbance of decomposition method extraction low frequency, energy, and filtered out from original seismic data；Then determined according to exploration targets
Main beam direction, using beam-forming method computation delay parameter, obtain orientation earthquake record；Finally for orientation earthquake record
Singular value decomposition method reconstruct singular value is recycled, the random background noise in orientation earthquake record is filtered out, obtains high s/n ratio
Orientation earthquake record.This method can strengthen target echo while strong linear disturbance and background random noise is removed
Energy, realize from noise reduction and enhancing signal two in terms of improvement S/N ratio of seismic records.
The present invention is achieved by the following technical solutions：
The orientation seismic beam forming method of singular value decomposition, comprises the following steps：
A, pending earthquake record U is chosen in existing earthquake record0(t, x), choose with U0Company centered on (t, x)
Continue the earthquake record collection U={ U of N number of adjacent shot point-(N-1)/2(t,x),…,U-1(t,x),U0(t,x),U1(t,x),…,U(N-1)/2
(t, x) }, N be earthquake record number, wherein N >=3, t be earthquake record time collection, x be earthquake record trace gather, U0(t,
X) shot point earthquake record centered on, if Ug(t, x) be U in any earthquake record, then-(N-1)/2≤g≤(N-1)/2, in Ug(t,
X) required according to denoising, choose any two points A and B on linear disturbance direction to be filtered out, linear disturbance includes sound wave, face
Ripple, refracted wave, direct wave, the coordinate for remembering A are (t1×fs,x1), B coordinate is (t2×fs,x2), t1And t2Respectively A, B point are sat
Corresponding to mark then, x1And x2Taoist monastic name, f respectively corresponding to A, B point coordinatessIt is linear dry according to formula (1) estimation for sample rate
The slope disturbed
B, on the basis of the k of estimation, chosen at equal intervals before and after k respectively m value composition slope collection M, gap size l according to
Formula (2) calculates, and m calculates according to formula (3)：
Wherein, t' is the time width that is read in earthquake record of linear disturbance, the slope collection M=of selection k-ml ...,
K-il ..., k-l, k, k+l ..., k+il ..., k+ml } ,-m≤i≤m；
C, as i=0, along slope k+il directions, n data point, composition data body W, per number are respectively chosen before and after A points
The abscissa y at strong pointjCalculated according to formula (4)：
Wherein ,-n≤j≤n, data volume W={ U are definedg(y-n,x1-n),…,Ug(y-1,x1-1),Ug(y0,x1),Ug(y1,
x1+1),…,Ug(yn,x1+n)}；
D, the amplitude difference sum Δ E of data volume W adjacent datas is calculated according to formula (5)i
E, i=-m is made respectively ..., -2, -1, during 1,2 ..., m, calculated successively according to step c~d remaining in slope collection M
Amplitude difference sum corresponding to data volume in slope direction, composition set S={ Δ E-m,…,ΔE-i,…,ΔE-1,ΔE0,Δ
E1,…,ΔEi,…,ΔEm, calculate minimum value Δ E in S according to formula (6)min, by Δ EminCorresponding slope direction k+il
It is designated as the direction of linear disturbance：
ΔEmin=min { Δ E-m,…,ΔE-i,…,ΔE-1,ΔE0,ΔE1,…,ΔEi,…,ΔEm}；(6)
F, centered on A points, by the use of it is parallel with the slope direction k+il of selection and at a distance of t sampling point of Δ two straight lines as
Data intercept body X it is upper and lower when window border, orderThe wicket number of linear disturbance is included according to formula (7) interception
According to body X：
G, the correction amount delta t by inclined linear interference correction in X into horizontal lineups is calculated according to formula (8)j, note correction
Data volume afterwards is X'：
Δtj=yj-y0 (8)
H, singular value decomposition is carried out to X' according to formula (10)：
Wherein, upIt is X'X'ΤCharacteristic value corresponding to eigenvector composition matrix, vpIt is X'ΤX' characteristic value pair
The matrix for the eigenvector composition answered, σpIt is X'X'ΤOr X'ΤThe non-negative square root of X' characteristic values, i.e. X''s is strange
Different value, singular value arrange according to the order successively decreased, σ1＞ σ2＞ ... ＞ σp＞ ... ＞ σr, r is matrix X' order, 1≤p≤r, Τ
For transposition symbol；
I, singular value is reconstructed, by σ2,σ3,…,σp,…,σr0 is set to, retains first singular value σ1, according to formula (11)
Recover only data volume X ", X " comprising linear disturbance and do not include background random noise：
X "=σ1u1v1 Τ (11)
Wherein, u1For X'X'ΤDominant eigenvalue corresponding to eigenvector composition matrix, v1For X'ΤX''s is maximum intrinsic
The matrix that eigenvector corresponding to value forms；
J, data volume X " is corrected back into inclined linear interference according to formula (12) is counter, data volume Y is obtained, from earthquake record Ug
Y is subtracted in (t, x)：
K, to earthquake record UgAll linear disturbances are handled one by one by step a~j in (t, x), by UgInstitute is wired in (t, x)
Property interference remove, obtain earthquake record Rg(t,x)；
Earthquake record collection is carried out to step a~k processing per big gun earthquake record, then per big gun earthquake record center line in U l,
Property interference be removed, obtain new earthquake record collection：
R={ R-(N-1)/2(t,x),…,R-1(t,x),R0(t,x),R1(t,x),…,R(N-1)/2(t,x)}；
M, a point coordinates (t is chosen in target echo center position3×fs,x3), if focus point coordinates is (t0×
fs,x0), calculate earthquake main beam direction θ according to formula (13)max, according to formula (14) computation delay parameter τ：
Wherein, D is detector interval, and d is big gun spacing, and v is reflecting interface overlying interval velocity, is obtained by surveying area's data；
N, the earthquake record collection after delay is：R'={ R-(N-1)/2'(t-(N-1)τ/2,x),…,R-1'(t-τ,x),R0'
(t,x),R1'(t+τ,x),…,R(N-1)/2' (t+ (N-1) τ/2, x), N big guns earthquake record in R' is superimposed according to formula (15),
Synthesis orientation earthquake record R " (t, x)：
Wherein ,-(N-1)/2≤e≤(N-1)/2；
O, Seismic Traces number is set as H, and per pass signal in R " (t, x) is done into related place to first of signal using correlation method
Reason, obtain delay difference set Δ t "={ Δ t' of the per pass signal relative to first of signal1,Δt'2,…,Δt'h,…,Δ
t'H, 1≤h≤H, according to formula (16) by target echo school in R " (t, x) into horizontal lineups signal, the ground after smoothing
Shake record is designated as r (t, x)：
R (t, x)=R " (t- (Δ th-Δt1),xh) (16)
P, r (t, x) is subjected to singular value decomposition according to formula (17)：
Wherein, r' be r (t, x) order, 1≤q≤r', σ 'qFor r (t, x) r (t, x)ΤOr r (t, x)ΤR (t, x) characteristic value
Non-negative square root, i.e. r (t, x) singular value, u'qFor r (t, x) r (t, x)ΤCharacteristic value corresponding to eigenvector composition
Matrix, v'qFor r (t, x)ΤThe matrix that eigenvector corresponding to r (t, x) characteristic value forms；
Q, singular value is reconstructed, by σ '3,σ'4,…,σ'r'0 is set to, retains the first two singular value σ '1,σ'2, according to formula
(18) the data volume r'(t, x for only including horizontal lineups are recovered)：
R, according to formula (19) by r'(t, x) in horizontal lineups return the same phase of hyperbolic according to gathering the anti-schools of Δ t " in step n
Axle, obtain newly orienting earthquake record r " (t, x)：
R " (t, x)=r'(t+ (Δ t'h-Δt'1),xh) (19)
Wherein, r " (t, x) is to be ultimately oriented earthquake record.
Beneficial effect：Through experiment, the orientation seismic beam forming method of singular value decomposition disclosed by the invention can be effectively
Strong linear disturbance and background random noise in earthquake record are suppressed, and effectively enhances the energy value of target echo, it is real
Showed improves the signal to noise ratio of geological data in terms of noise reduction and enhancing signal two, improves the quality of data of seismic prospecting.For
Deep and strong man's text Earthquakes data signal to noise ratio are low, and strong linear disturbance development, random noise disturbance is serious, target reflection letter
Number low feature of energy, the orientation seismic beam forming method of singular value decomposition can effectively improve such Earthquakes data
Quality, effect is better than singular value decomposition method and orientation seismic beam forming method.
Brief description of the drawings：
Two layers of layer-cake model of Fig. 1
The beam-forming earthquake record quality improvement of Fig. 2 singular value decompositions
A, original seismic data；
B, earthquake record after the orientation seismic beam forming method processing based on singular value decomposition
Embodiment：
It is described in further detail with reference to the accompanying drawings and examples：
In the present embodiment with d=10m, D=10m, v1=1500m/s, v2=2000m/s, fsCarried out exemplified by=1000 deep
Portion's earthquake data acquisition and humanity activities area improve S/N ratio of seismic records processing, but deep earthquake data acquisition and people
The parameter that literary activity area does not provide using the orientation seismic beam forming method of singular value decomposition in by embodiment is limited.
A, pending earthquake record U is chosen in existing earthquake record0(t, x), choose with U0Company centered on (t, x)
Continue the earthquake record collection U={ U of 7 adjacent shot points-3(t,x),…,U-1(t,x),U0(t,x),U1(t,x),…,U3(t, x) }, t
For the time collection of earthquake record, x is the trace gather of earthquake record, U0Shot point earthquake record centered on (t, x), if Ug(t, x) is in U
Any earthquake record, then -3≤g≤3, in UgIn (t, x) on the direction of face ripple 1 arbitrarily choose two point coordinates A and B, note A for (269,
40), B is (288,41).According to the slope of formula (1) estimation linear disturbance
B, on the basis of the k of estimation, m value is chosen at equal intervals before and after k respectively, gap size l counts according to formula (2)
Calculate, m calculates according to formula (3)：
Wherein, t'=0.1s, it is computed, slope collection M={ k-50l ..., k-il ..., k-l, k, k+l ..., the k+ of selection
Il ..., k+50l }, -50≤i≤50；
C, as i=0, along slope 0.053+0.003i directions, 50 data points, composition data are respectively chosen before and after A points
Body W, the abscissa y of each data pointjCalculated according to formula (4)：
Wherein, -50≤j≤50, data volume W={ Ug(y-50,x1-50),…,Ug(y-1,x1-1),Ug(y0,x1),Ug(y1,
x1+1),…,Ug(y50,x1+50)}；
D, the amplitude difference sum Δ E of data volume W adjacent datas is calculated according to formula (5)i
E, i=-50 is made respectively ..., when -2, -1,1,2 ..., 50, calculate in slope collection M and remain successively according to step c~d
Amplitude difference sum corresponding to data volume in remaining slope direction, composition set S={ Δ E-50,…,ΔE-i,…,ΔE-1,ΔE0,
ΔE1,…,ΔEi,…,ΔE50, calculate minimum value Δ E in S according to formula (6)min, by Δ EminCorresponding slope direction
0.03 is designated as the direction of linear disturbance：
0.45=min { Δ E-50,…,ΔE-i,…,ΔE-1,ΔE0,ΔE1,…,ΔEi,…,ΔE50} (6)
F, centered on coordinate points A, with parallel with the slope direction 0.03 of selection and 100 sampling points apart two straight lines
As data intercept body X it is upper and lower when window border, include the wicket data volume X of linear disturbance according to formula (7) interception：
G, the correction amount delta t by inclined linear interference correction in X into horizontal lineups is calculated according to formula (8)j, note correction
Data volume afterwards is X'：
Δtj=yj-y0 (8)
H, singular value decomposition is carried out to X' according to formula (10)：
I, singular value is reconstructed, by σ2,σ3,…,σp,…,σr0 is set to, retains first singular value σ1, press
Only data volume X ", X " comprising linear disturbance, which are recovered, according to formula (11) does not include background random noise：
X "=σ1u1v1 Τ (11)
Wherein, u1For X'X'ΤDominant eigenvalue corresponding to eigenvector composition matrix, v1For X'ΤX' is most
The matrix of eigenvector composition corresponding to big characteristic value；
J, data volume X " is corrected back into inclined linear interference according to formula (12) is counter, data volume Y is obtained, from earthquake record Ug
Y is subtracted in (t, x)：
K, to earthquake record UgFace ripple 2 and sound wave are handled one by one by step a~j in (t, x), by UgInstitute is linear in (t, x)
Interference removes, and obtains earthquake record Rg(t,x)；
L, step a~k processing will be carried out in earthquake record collection U per big gun earthquake record, then per in big gun earthquake record in U
Linear disturbance is removed, and obtains new earthquake record collection R={ R-3(t,x),…,R-1(t,x),R0(t,x),R1(t,x),…,R3
(t,x)}；
M, a point coordinates (1020,60) is chosen in target echo center position, if focus point coordinates is (0,0),
Earthquake main beam direction θ is calculated according to formula (13)max, according to v1=1500m/s, according to formula (14) computation delay parameter τ：
N, the earthquake record after being delayed integrates as R'={ R-3'(t-3τ,x),…,R-1'(t-τ,x),R0'(t,x),R1'(t+
τ,x),…,R3' (t+3 τ, x), 7 big gun earthquake records in R' are superimposed according to formula (15), synthesis orientation earthquake record R " (t,
x)：
Wherein, -3≤e≤3；
O, Seismic Traces number is set as 101, is done per pass signal in R " (t, x) to first of signal using correlation method related
Processing, obtain delay difference set Δ t "={ Δ t' of the per pass signal relative to first of signal1,Δt'2,…,Δt'h,…,Δ
t'101, 1≤h≤101,
According to formula (16) by target echo school in R " (t, x) into horizontal lineups signal, the earthquake note after smoothing
Record is designated as r (t, x)：
R (t, x)=R " (t- (Δ th-Δt1),xh) (16)
P, r (t, x) is subjected to singular value decomposition according to formula (17)：
Wherein, r'=101 be r (t, x) order, 1≤q≤101, σ 'qFor r (t, x) r (t, x)ΤOr r (t, x)Τr(t,x)
The singular value of the non-negative square root, i.e. r (t, x) of characteristic value, u'qFor r (t, x) r (t, x)ΤCharacteristic value corresponding to eigenvector
The matrix of composition, v'qFor r (t, x)ΤThe matrix that eigenvector corresponding to r (t, x) characteristic value forms；
Q, singular value is reconstructed, by σ '3,σ'4,…,σ'r'0 is set to, retains the first two singular value σ '1,σ'2, according to formula
(18) the data volume r'(t, x for only including target echo are recovered)：
R, according to formula (19) by r'(t, x) in target echo return hyperbolic according to the anti-schools of set Δ t " in step o
Lineups, obtain newly orienting earthquake record r " (t, x)：
R " (t, x)=r'(t+ (Δ t'h-Δt'1),xh) (19)
Wherein, r " (t, x) is to be ultimately oriented earthquake record.
Claims (1)
1. the orientation seismic beam forming method of a kind of singular value decomposition, it is characterised in that comprise the following steps：
A, pending earthquake record U is chosen in existing earthquake record0(t, x), choose with U0It is continuous N number of centered on (t, x)
The earthquake record collection U={ U of adjacent shot point-(N-1)/2(t,x),…,U-1(t,x),U0(t,x),U1(t,x),…,U(N-1)/2(t,
X) }, N be earthquake record number, wherein N >=3, t be earthquake record time collection, x be earthquake record trace gather, U0(t,x)
Centered on shot point earthquake record, if Ug(t, x) be U in any earthquake record, then-(N-1)/2≤g≤(N-1)/2, in Ug(t,x)
It is middle to be required according to denoising, choose any two points A and B on linear disturbance direction to be filtered out, linear disturbance include sound wave, face ripple,
Refracted wave, direct wave, the coordinate for remembering A are (t1×fs,x1), B coordinate is (t2×fs,x2), t1And t2Respectively A, B point coordinates
It is corresponding then, x1And x2Taoist monastic name, f respectively corresponding to A, B point coordinatessFor sample rate, linear disturbance is estimated according to formula (1)
Slope
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<mo>(</mo>
<mn>2</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>m</mi>
<mo>=</mo>
<mfrac>
<mrow>
<msup>
<mi>t</mi>
<mo>&prime;</mo>
</msup>
<mo>&times;</mo>
<msub>
<mi>f</mi>
<mi>s</mi>
</msub>
</mrow>
<mn>2</mn>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, the time width that t' reads for linear disturbance in earthquake record, slope collection M={ k-ml ..., the k- of selection
Il ..., k-l, k, k+l ..., k+il ..., k+ml } ,-m≤i≤m；
C, as i=0, along slope k+il directions, n data point, composition data body W, each data point are respectively chosen before and after A points
Abscissa yjCalculated according to formula (4)：
<mrow>
<msub>
<mi>y</mi>
<mi>j</mi>
</msub>
<mo>=</mo>
<mfrac>
<mi>j</mi>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mi>i</mi>
<mi>l</mi>
</mrow>
</mfrac>
<mo>+</mo>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
<mo>&times;</mo>
<msub>
<mi>f</mi>
<mi>s</mi>
</msub>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein ,-n≤j≤n, data volume W={ U are definedg(y-n,x1-n),…,Ug(y-1,x1-1),Ug(y0,x1),Ug(y1,x1+
1),…,Ug(yn,x1+n)}；
D, the amplitude difference sum Δ E of data volume W adjacent datas is calculated according to formula (5)i
<mrow>
<msub>
<mi>&Delta;E</mi>
<mi>i</mi>
</msub>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>j</mi>
<mo>=</mo>
<mo>-</mo>
<mi>n</mi>
</mrow>
<mrow>
<mi>n</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</munderover>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mn>2</mn>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mo>(</mo>
<mrow>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mn>2</mn>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mi>j</mi>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mi>j</mi>
<mo>)</mo>
</mrow>
<mo>;</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
E, i=-m is made respectively ..., -2, -1, during 1,2 ..., m, remaining slope in slope collection M is calculated successively according to step c~d
Amplitude difference sum corresponding to data volume on direction, composition set S={ Δ E-m,…,ΔE-i,…,ΔE-1,ΔE0,ΔE1,…,
ΔEi,…,ΔEm, calculate minimum value Δ E in S according to formula (6)min, by Δ EminCorresponding slope direction k+il is designated as line
Property interference direction：
ΔEmin=min { Δ E-m,…,ΔE-i,…,ΔE-1,ΔE0,ΔE1,…,ΔEi,…,ΔEm}； (6)
F, centered on A points, interception is used as by the use of parallel with the slope direction k+il of selection and at a distance of t sampling point of Δ two straight lines
Data volume X it is upper and lower when window border, orderThe wicket data volume of linear disturbance is included according to formula (7) interception
X：
<mrow>
<mi>X</mi>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msub>
<mo>-</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mn>0</mn>
</msub>
<mo>-</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mi>n</mi>
</msub>
<mo>-</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mn>0</mn>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mi>n</mi>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mn>0</mn>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mi>n</mi>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
G, the correction amount delta t by inclined linear interference correction in X into horizontal lineups is calculated according to formula (8)j, after note correction
Data volume is X'：
Δtj=yj-y0 (8)
<mrow>
<msup>
<mi>X</mi>
<mo>&prime;</mo>
</msup>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>&Delta;t</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msub>
<mo>-</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>&Delta;t</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mn>0</mn>
</msub>
<mo>-</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<msub>
<mi>&Delta;t</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mi>n</mi>
</msub>
<mo>+</mo>
<msub>
<mi>&Delta;t</mi>
<mi>n</mi>
</msub>
<mo>-</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>&Delta;t</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>&Delta;t</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mn>0</mn>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<msub>
<mi>&Delta;t</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mi>n</mi>
</msub>
<mo>+</mo>
<msub>
<mi>&Delta;t</mi>
<mi>n</mi>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>&Delta;t</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>&Delta;t</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mn>0</mn>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<msub>
<mi>&Delta;t</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mi>n</mi>
</msub>
<mo>+</mo>
<msub>
<mi>&Delta;t</mi>
<mi>n</mi>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
H, singular value decomposition is carried out to X' according to formula (10)：
<mrow>
<msup>
<mi>X</mi>
<mo>&prime;</mo>
</msup>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>p</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>r</mi>
</munderover>
<msub>
<mi>&sigma;</mi>
<mi>p</mi>
</msub>
<msub>
<mi>u</mi>
<mi>p</mi>
</msub>
<msup>
<msub>
<mi>v</mi>
<mi>p</mi>
</msub>
<mi>T</mi>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, upIt is X'X'ΤCharacteristic value corresponding to eigenvector composition matrix, vpIt is X'ΤIt is intrinsic corresponding to X' characteristic value
The matrix of vector composition, σpIt is X'X'ΤOr X'ΤThe non-negative square root of X' characteristic values, i.e. X' singular value, singular value is according to passing
The order arrangement subtracted, σ1＞ σ2＞ ... ＞ σp＞ ... ＞ σr, r is matrix X' order, and 1≤p≤r, Τ are transposition symbol；
I, singular value is reconstructed, by σ2,σ3,…,σp,…,σr0 is set to, retains first singular value σ1, recover according to formula (11)
Go out the data volume X " for only including linear disturbance, wherein X " does not include background random noise：
X "=σ1u1v1 Τ (11)
Wherein, u1For X'X'ΤDominant eigenvalue corresponding to eigenvector composition matrix, v1For X'ΤX' dominant eigenvalue pair
The matrix for the eigenvector composition answered；
J, data volume X " is corrected back into inclined linear interference according to formula (12) is counter, data volume Y is obtained, from earthquake record Ug(t,x)
It is middle to subtract Y：
<mrow>
<mi>Y</mi>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msub>
<mo>-</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>-</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mn>0</mn>
</msub>
<mo>-</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mi>n</mi>
</msub>
<mo>-</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mn>0</mn>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mi>n</mi>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
<mtd>
<mrow></mrow>
</mtd>
<mtd>
<mo>.</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mo>-</mo>
<mi>n</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mn>0</mn>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>...</mn>
</mtd>
<mtd>
<mrow>
<msup>
<msub>
<mi>U</mi>
<mi>g</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>y</mi>
<mi>n</mi>
</msub>
<mo>+</mo>
<mi>&Delta;</mi>
<mi>t</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>+</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>12</mn>
<mo>)</mo>
</mrow>
</mrow>
K, to earthquake record UgAll linear disturbances are handled one by one by step a~j in (t, x), by UgInstitute is linear dry in (t, x)
Removal is disturbed, obtains earthquake record Rg(t,x)；
Earthquake record collection is carried out to step a~k processing per big gun earthquake record, then linearly done in every big gun earthquake record in U l,
Disturb and be removed, obtain new earthquake record collection：
R={ R-(N-1)/2(t,x),…,R-1(t,x),R0(t,x),R1(t,x),…,R(N-1)/2(t,x)}；
M, a point coordinates (t is chosen in target echo center position3×fs,x3), if focus point coordinates is (t0×fs,
x0), calculate earthquake main beam direction θ according to formula (13)max, according to formula (14) computation delay parameter τ：
<mrow>
<msub>
<mi>&theta;</mi>
<mrow>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>a</mi>
<mi>r</mi>
<mi>c</mi>
<mi>t</mi>
<mi>a</mi>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mfrac>
<mrow>
<mo>(</mo>
<msub>
<mi>x</mi>
<mn>3</mn>
</msub>
<mo>-</mo>
<msub>
<mi>x</mi>
<mn>0</mn>
</msub>
<mo>)</mo>
<mo>&times;</mo>
<mi>D</mi>
</mrow>
<mrow>
<mo>(</mo>
<msub>
<mi>t</mi>
<mn>3</mn>
</msub>
<mo>-</mo>
<msub>
<mi>t</mi>
<mn>0</mn>
</msub>
<mo>)</mo>
<mo>&times;</mo>
<mi>v</mi>
<mo>/</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>13</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>&tau;</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>d</mi>
<mi> </mi>
<msub>
<mi>sin&theta;</mi>
<mrow>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
</mrow>
</msub>
</mrow>
<mi>v</mi>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>14</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, D is detector interval, and d is big gun spacing, and v is reflecting interface overlying interval velocity, is obtained by surveying area's data；
N, the earthquake record collection after delay is：R'={ R-(N-1)/2'(t-(N-1)τ/2,x),…,R-1'(t-τ,x),R0'(t,x),
R1'(t+τ,x),…,R(N-1)/2' (t+ (N-1) τ/2, x), N big guns earthquake record in R' is superimposed according to formula (15), synthesis is fixed
To earthquake record R " (t, x)：
<mrow>
<msup>
<mi>R</mi>
<mrow>
<mo>&prime;</mo>
<mo>&prime;</mo>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>e</mi>
<mo>=</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mi>N</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>/</mo>
<mn>2</mn>
</mrow>
<mrow>
<mo>(</mo>
<mi>N</mi>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
<mo>/</mo>
<mn>2</mn>
</mrow>
</munderover>
<msup>
<msub>
<mi>R</mi>
<mi>e</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>+</mo>
<mi>e</mi>
<mi>&tau;</mi>
<mo>,</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>15</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein ,-(N-1)/2≤e≤(N-1)/2；
O, Seismic Traces number is set as H, and per pass signal in R " (t, x) and first of signal are done into relevant treatment using correlation method,
Obtain delay difference set Δ t "={ Δ t' of the per pass signal relative to first of signal1,Δt'2,…,Δt'h,…,Δt'H,
1≤h≤H, according to formula (16) by target echo school in R " (t, x) into horizontal lineups signal, the earthquake note after smoothing
Record is designated as r (t, x)：
R (t, x)=R " (t- (Δ th-Δt1),xh) (16)
P, r (t, x) is subjected to singular value decomposition according to formula (17)：
<mrow>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>q</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<msup>
<mi>r</mi>
<mo>&prime;</mo>
</msup>
</munderover>
<msub>
<msup>
<mi>&sigma;</mi>
<mo>&prime;</mo>
</msup>
<mi>q</mi>
</msub>
<msub>
<msup>
<mi>u</mi>
<mo>&prime;</mo>
</msup>
<mi>q</mi>
</msub>
<msup>
<mi>v</mi>
<mo>&prime;</mo>
</msup>
<msup>
<msub>
<mrow></mrow>
<mi>q</mi>
</msub>
<mi>T</mi>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>17</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, r' be r (t, x) order, 1≤q≤r', σ 'qFor r (t, x) r (t, x)ΤOr r (t, x)ΤR (t, x) characteristic value it is non-negative
The singular value of square root, i.e. r (t, x), u'qFor r (t, x) r (t, x)ΤCharacteristic value corresponding to eigenvector composition matrix, v'q
For r (t, x)ΤThe matrix that eigenvector corresponding to r (t, x) characteristic value forms；
Q, singular value is reconstructed, by σ '3,σ'4,…,σ'r'0 is set to, retains the first two singular value σ '1,σ'2, according to formula (18)
Recover the data volume r'(t, x for only including horizontal lineups)：
<mrow>
<msup>
<mi>r</mi>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>q</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mn>2</mn>
</munderover>
<msub>
<msup>
<mi>&sigma;</mi>
<mo>&prime;</mo>
</msup>
<mi>q</mi>
</msub>
<msub>
<msup>
<mi>u</mi>
<mo>&prime;</mo>
</msup>
<mi>q</mi>
</msub>
<msup>
<mi>v</mi>
<mo>&prime;</mo>
</msup>
<msup>
<msub>
<mrow></mrow>
<mi>q</mi>
</msub>
<mi>T</mi>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>18</mn>
<mo>)</mo>
</mrow>
</mrow>
R, according to formula (19) by r'(t, x) in horizontal lineups return hyperbolic lineups according to gathering the anti-schools of Δ t " in step n, obtain
To new orientation earthquake record r " (t, x)：
R " (t, x)=r'(t+ (Δ t'h-Δt'1),xh) (19)
Wherein, r " (t, x) is to be ultimately oriented earthquake record.
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