text: fig1 is a schematic diagram showing reflections between a sea surface ( s ), sea floor ( w ) and a target reflector ( t ). various events that will be recorded in the seismogram are shown and are labelled according to the series of interfaces they are reflected at . the stars indicate the seismic source and the arrowheads indicate the direction of propagation at the receiver . events ending with ‘ s ’ were last reflected at the rough sea surface and are called receiver ghost events . down - going sea - surface ghost reflections are an undesirable source of contamination , obscuring the interpretation of the desired up - going reflections from the earth &# 39 ; s sub - surface . rough seas are a source of noise in seismic data . aside from the often - observed swell noise , further errors are introduced into the reflection events by ghost reflection and scattering from the rough sea surface . the rough sea perturbed ghost events introduce errors that are significant for time - lapse seismic surveying and the reliable acquisition of repeatable data for stratigraphic inversion . the effect of the rough sea is to perturb the amplitude and arrival time of the sea surface reflection ghost and add a scattering coda , or tail , to the ghost impulse . the impulse response can be calculated by finite difference or kirchhoff methods ( for example ) from a scattering surface which represents statistically typical rough sea surfaces . for example , a directional form of the pierson - moskowitz spectrum described by pierson , w . j . and moskowitz , l ., 1964 ‘ a proposed spectral form for fully developed wind seas based on the similarity theory of s . a . kitaigorodskii ’ j . geo . res ., 69 , 24 , 5181 - 5190 , ( hereinafter “ pierson and moskowitz ( 1964 )”), and hasselmann , d . e ., dunckel , m . and ewing , j . a ., 1980 ‘ directional wave spectra observed during jonswap 1973 ’, j . phys . oceanography , v10 , 1264 - 1280 , ( hereinafter “ hasselmann et al , ( 1980 )”). both the wind &# 39 ; s speed and direction define the spectra . the significant wave height (“ swh ”) is the subjective peak to trough wave amplitude , and is about equal to 4 times the rms wave height . different realisations are obtained by multiplying the 2d surface spectrum by gaussian random complex numbers . fig4 shows an example of rough sea impulses along a 400 m 2d line ( e . g . streamer ) computed under a 2 m swh 3d rough sea surface . the streamer shape affects the details of the impulses , and in this example the streamer is straight and horizontal . fig4 shows , from top to bottom : the ghost wavelet ( white trough ) arrival time , the ghost wavelet maximum amplitude , a section through the rough sea realisation above the streamer , and the computed rough sea impulses . the black peak is the upward travelling wave , which is unperturbed ; the white trough is the sea ghost reflected from the rough sea surface . the latter part of the wavelet at each receiver is the scattering coda from increasingly more distant parts of the surface . notice that the amplitude and arrival time ghost perturbations change fairly slowly with offset . the arrival time perturbations are governed by the dominant wavelengths in the sea surface , which are 100 - 200 m for 2 - 4 m swh seas , and the amplitude perturbations are governed by the curvature of the sea surface which has an rms radius of about 80 m and is fairly independent of sea state . the diffraction coda appear as quasi - random noise following the ghost pulse . the rough sea perturbations cause a partial fill and a shift of the ghost notch in the frequency domain . they also add a small ripple to the spectrum , which amounts to 1 - 2 db of error for typical sea states . in the post stack domain this translates to an error in the signal that is about − 20 db for a 2 m swh sea . fig5 shows an example of how such an error can be significant for time - lapse surveys . the panel on the top left shows a post - stack time - migrated synthetic finite difference seismic section . the top middle panel shows the same data but after simulating production in the oil reservoir by shifting the oil water contact by 6 m and introducing a 6 m partial depletion zone above this . the small difference is just noticeable on the black leg of the reflection to the right of the fault just below 2 s two - way travel - time . the panel on the right ( top ) shows the difference between these two sections multiplied by a factor of 10 . this is the ideal seismic response from the time - lapse anomaly . the left and middle bottom panels show the same seismic sections , but rough sea perturbations for a 2 m swh ( as described above ) have been added to the raw data before processing . note that different rough sea effects are added to each model to represent the different seas at the time of acquisition . the difference obtained between the two sections is shown on the bottom right panel ( again multiplied by a factor of 10 ). the errors in the reflector amplitude and phase ( caused by the rough sea perturbations ) introduce noise of similar amplitude to the true seismic time - lapse response . to a great extent , the true response is masked by these rough sea perturbations . a method for correcting these types of error is clearly important in such a case , and with the increasing requirement for higher quality , low noise - floor data , correction for the rough sea ghost becomes necessary even in modest sea states . equation ( 1 ) gives the frequency domain expression for a preferred filter relating the up - going pressure field , p u ( x ), to the total pressure , p ( x ), and vertical particle velocity , v z ( x ). p u  ( x ) = 0 . 5  [ p  ( x ) + ρ   ω k z * v z  ( x ) ] ( 1 ) where k z is the vertical wavenumber for compressional waves in the water , ρ is the density of water and * denotes spatial convolution . the vertical wavenumber is calculated from k z 2 = k 2 − k x 2 for two - dimensional survey geometries , or k 2 2 = k 2 − k x 2 − k y 2 for three - dimensional survey geometries , with k 2 = ω 2 / c 2 , where c is the compressional wave speed in the water and k x is the horizontal wavenumber for compressional waves in the water . the regular sampling of typical towed streamer data allows k z to be calculated efficiently in the fk domain . fig7 shows an example of the filter response , ω / k z for dx = 6 m ( the filter is normalised for the display to an arbitrary value ). infinite gain poles occur when k z is zero . this corresponds to energy propagating horizontally ( at the compressional wave speed in water ). for towed streamer data , there is little signal energy with this apparent velocity , any noise present in the data with this apparent velocity should be filtered out prior to the filter application , or , the filter should be tapered at the poles prior to application to avoid amplification of the noise . by comparison to equation ( 1 ), we see that this is a normal incidence approximation , which occurs when k x is zero . this is implemented as a simple scaling of the vertical velocity trace followed by its addition to the pressure trace . equation ( 1 ) can also be formulated in terms of the vertical pressure gradient ( dp ( x )/ dz ). the vertical pressure gradient is proportional to the vertical acceleration : integrating in the frequency domain through division of iω , and substituting in equation ( 1 ) gives : p u  ( x ) = 0 . 5  [ p  ( x ) + 1    k z *  p  ( x ) /  z ] ( 4 ) fig6 a - 6 f show various embodiments for data acquisition set - ups and streamer configurations according to preferred embodiments of the invention . fig6 a shows a seismic vessel 120 towing a seismic source 110 and a seismic streamer 118 . the sea surface is shown by reference number 112 . in this example , the depth of streamer 118 is about 60 meters , however those of skill in the art will recognise that a much shallower depth would ordinarily be used such as 6 - 10 meters . the dashed arrows 122 a - d show paths of seismic energy from source 110 . arrow 122 a shows the initial down - going seismic energy . arrow 122 b shows a portion of the seismic energy that is transmitted through the sea floor 114 . arrow 122 c shows an up - going reflection . arrow 122 d shows a down - going ghost reflected from the surface . according to the invention , the down - going rough sea receiver ghost 122 d can be removed from the seismic data . fig6 b - 6 f show greater detail of acquisition set - ups and streamer configurations , according to the invention . fig6 b shows a multi - component streamer 124 . the streamer 124 comprises multiple hydrophones ( measuring pressure ) 126 a , 126 b , and 126 c , and multiple 3c geophones ( measuring particle velocity in three directions x , y , and z ) 128 a , 128 b , and 128 c . the spacing between the hydrophones 126 a and 126 b , and between geophones 128 a and 128 b is shown to be less than 12 meters . additionally , the preferred spacing in relation to the frequencies of interest is discussed in greater detail below . fig6 c shows a streamer 130 that comprises multiple hyrdophones 132 a , 132 b , and 132 c , and multiple pressure gradient measuring devices 134 a , 134 b , and 134 c . the spacing between the hydrophones 132 a and 132 b , and between pressure gradient measuring devices 134 a and 134 b is shown to be less than 12 meters . fig6 d shows a multi - streamer configuration that comprises hydrophone streamers 140 a and 140 b . the streamers comprise multiple hyrdophones 142 a , 142 b , and 142 c in the case of streamer 140 a , and 142 d , 142 e , and 142 f in the case of streamer 140 b . the spacing between the hydrophones is shown to be less than 12 meters . the separation between steamers 140 a and 140 b in the example shown in fig6 d is less then 2 meters . although the preferred separation is less than 2 meters , greater separations are contemplated as being within the scope of the invention . fig6 e shows a cross sectional view of a dual streamer arrangement . fig6 f shows a multi - streamer configuration comprising three hydrophone streamers 140 a , 140 b , and 140 c . adequate spatial sampling of the wavefield is highly preferred for the successful application of the de - ghosting filters . for typical towed streamer marine data , a spatial sampling interval of 12 m is adequate for all incidence angles . however , to accurately spatially sample all frequencies up to 125 hz ( for all incidence angles ), a spatial sampling interval of 6 . 25 meters is preferred . these spacings are determined according to the nyquist spatial sampling criterion . note that if all incidence angles are not required , a coarser spacing than described above can be used . the filters can be applied equally to both group formed or point receiver data . fig8 is a flow chart illustrating some of the steps of the de - ghosting method for the combination of pressure and vertical velocity data to achieve separated pressure data , according to a preferred embodiment of the invention . in step 202 , spatial filter coefficients are calculated . the coefficients are preferably dependent on the characteristics of the acquisition parameters 203 ( such as the temporal sample interval of the pressure and particle motion data , the spatial separation of the vertical particle motion measuring devices , and the spatial aperture of the filter ), the density of the fluid medium 206 , and the speed of the compressional wave in the fluid medium ( or velocity of sound ) 204 . vertical particle motion data 208 and pressure data 212 are received , typically stored as time domain traces on a magnetic tape or disk . in step 210 , the vertical particle motion data 208 are convolved in with the spatial filter to yield filtered vertical particle motion data . in step 214 the filtered vertical particle motion data are added to pressure data 212 to give the downward propagating component of the separated pressure data . alternatively , in step 216 the filtered vertical particle motion data are subtracted from pressure data 212 to give the upward propagating component of the separated pressure data . finally , in step 218 the upward component is further processes and analysed . the processing described herein is preferably performed on a data processor configured to process large amounts of data . for example , fig9 illustrates one possible configuration for such a data processor . the data processor typically consists of one or more central processing units 350 , main memory 352 , communications or i / o modules 354 , graphics devices 356 , a floating point accelerator 358 , and mass storage devices such as tapes and discs 360 . it will be understood by those skilled in the art that tapes and discs 360 are computer - readable media that can contain programs used to direct the data processor to carry out the processing described herein . fig1 shows a shot record example , computed under a 4 m significant wave height ( swh ) sea and using the finite - difference method described by robertsson , j . o . a ., blanch , j . o . and symes , w . w ., 1994 ‘ viscoelastic finite - difference modelling ’ geophysics , 59 , 1444 - 1456 ( hereinafter “ robertsson et al . ( 1994 )”) and robertsson , j . o . a ., 1996 ‘ a numerical free - surface condition for elastic / viscoelastic finite - difference modelling in the presence of topography ’, geophysics , 61 , 6 , 1921 - 1934 ( hereinafter “ robertsson ( 1996 )”). the streamer depth in this example is 60 m . the left panel shows the pressure response and the right panel shows the vertical velocity response scaled by the water density and the compressional wave speed in water . a point source 50 hz ricker wavelet was used and the streamer depth was 60 m in this example . the choice of streamer depth allows a clear separation of the downward travelling ghost from the upward travelling reflection energy for visual clarity of the de - ghosting results . the trace spacing on the plot is 24 m . a single reflection and its associated ghost are shown , along with the direct wave travelling in the water layer . perturbations in the ghost wavelet and scattering noise from the rough sea surface are evident . fig1 shows the results of de - ghosting the shot record shown in fig1 . the left panel shows the result using the normal incidence approximation and the right panel shows the result using the exact solution . the exact solution shows a consistent response over all offsets , whereas the normal incidence approximation starts to break down at incident angles greater than about 20 degrees , and shows a poorer result at the near offsets . note that the direct wave is not amplified by the exact filter application even though the poles of the filter lie close to its apparent velocity . the exact filter is tapered before application such that it is has near unity response for frequencies and wavenumbers corresponding to apparent velocities of 1500 m / s and greater . the weak event just below the signal reflection is a reflection from the side absorbing boundary of the model . it is upward travelling and hence untouched by the filter . fig1 shows details of the de - ghosted results for a single trace from fig1 . the trace offset is 330 m corresponding to a 37 degree incidence angle . the upper panel shows the normal incidence approximation , and the lower panel shows the exact solution . not only does the exact solution provide a superior result in terms of the de - ghosting , but also in terms of amplitude preservation of the signal reflection — the upper panel shows loss of signal amplitude after the de - ghosting . the filters described herein are applicable to , for example , measurements of both pressure and vertical velocity along the streamer . currently , however , only pressure measurements are commercially available . therefore , engineering of streamer sections that are capable of commercially measuring vertical velocity is preferred in order to implement the filters . fig1 a - b illustrate two possible examples of multi - component streamer design . fig1 a shows a coincident pressure and single 3 - component geophone . in this design , the 3 - component geophone is perfectly decoupled from the streamer . fig1 b shows a coincident pressure and twin 3 - component geophones . in this design , one of the 3 - component geophones is decoupled from the streamer , the other is coupled to the streamer ; measurements from both are combined to remove streamer motion from the data . in an alternative formulation , the filters make use of vertical pressure gradient measurements . an estimate of vertical pressure gradient can be obtained from over / under twin streamers ( such as shown in fig6 d and 6 e ) and multiple streamers ( such as shown in fig6 f ) deployed in configurations analogous to that described in robertsson ( 1998 ), allowing the filters to be directly applied to such data . however , for the results to remain sufficiently accurate , the streamers should not be vertically separated by more than 2 m for seismic frequencies below approximately 80 hz . an important advantage of multiple streamer configurations such as shown in fig6 f is that their relative locations are less crucial than for over / under twin streamer geometries , where the two streamers are preferably directly above one another . the filters described here are applied in 2d ( along the streamer ) to data modelled in 2d . the application to towed streamer configurations naturally lends itself to this implementation , the cross - line ( streamer ) sampling of the wavefield being usually insufficient for a full 3d implementation . application of these filters to real data ( with ghost reflections from 3d sea surfaces ) will give rise to residual errors caused by scattering of the wavefield from the cross - line direction . this error increases with frequency though is less than 0 . 5 db in amplitude and 3 . 6 ° in phase for frequencies up to 150 hz , for a 4 m swh sea . these small residual noise levels are acceptable when time - lapse seismic surveys are to be conducted . invoking the principle of reciprocity , the filters can be applied in the common receiver domain to remove the downward travelling source ghost . reciprocity simply means that the locations of source and receiver pairs can be interchanged , ( the ray path remaining the same ) without altering the seismic response . fig1 can also be used to define the source ghost if the stars are now regarded as receivers and the direction of the arrows is reversed , with the source now being located at the arrow . this application is particularly relevant for data acquired using vertical cables , which may be tethered , for example , to the sea floor , or suspended from buoys . in the case of fig6 a , those of skill in the art will understand that as the seismic vessel 120 travels though the water , the firing position of source 110 will change . the different positions of source 110 can be then be used to construct data in the common receiver domain as is well known in the art . while preferred embodiments of the invention have been described , the descriptions and figures are merely illustrative and are not intended to limit the present invention .