Attenuation of Multiples in Marine Seismic Data Geology 377 10/23/06
Marine Seismic Surveying Marine seismic surveying is performed to support marine environmental and civil engineering projects in the coastal zone. Also performed to evaluate and/or explore for new oil and gas reserves in the continental shelf.
Types of Surveys Single Channel Profiling: The source and receiver is contained in the same unit. Typically utilized in the upper 10 2 meters for high resolution engineering and environmental surveys. Unit can be deployed quickly and retrofitted to most any vessel. Measures the reflectivity of a wave with an angle of incidence of 0 degrees. Normal to a horizontal layer.
Types of Surveys Towed source and hydrophone arrays Utilized for exploration of oil and gas. Strong sources and hundreds of hydrophones peer deeper into the continental shelf.
Types of Surveys Multibeam acoustic seafloor mapping High frequency source sweeps the seafloor in a fan shape below the vessel. Not used for seafloor penetration. Used for seafloor contours.
Types of Surveys Side Scan Sonar peers laterally to determine relief of a feature, or the seafloor. Towfish flown in the water column. Try to keep the fish at a constant height above the seafloor.
Problem Multiples occur in marine seismic data due to the reflectivity of the water layer
Review of Reflection Phenomena Reflection Coefficient (R): Ratio of reflected and incident wave amplitudes R = Ar/Ai Ar = Amplitude of the reflected wave Ai = Amplitude of the incident wave Relates to the magnitude of reflection from the interface between two media with different physical properties
Acoustic Impedance (Z) Acoustic Impedance is a ratio of acoustic pressure to flow Z = ρv ρ=density of the material (kg/m 3 ) v=speed of the acoustic wave (m/s) Units of Rayles (kg/m 2 /s)
Reflection Coefficient (R) The full expression for the reflection coefficient : R=(Z 2 /Z 1 )- (1-(n-1)tan 2 α i ) (Z 2 /Z 1 )+ (1-(n-1)tan 2 α i ) Reflection occurs to some extent at every interface. n = the ratio of (v 2 /v 1 )^2, α is the angle of incidence of the wave ray
Reflection Coefficients in the Water Layer Let s take a look at the simple case of the reflection coefficient at the air/water interface and water/rock interface for a wave of normal incidence. air: ρ=1.3 kg/m 3 v=350 m/s seawater: ρ= 1027 kg/m 3 v=1500 m/s rock: ρ= 2650 kg/m 3 v=3000 m/s
Quick Computation of the Reflection Coefficients Air/Water Interface: 2 1 Consider the cartoon: The source is located in the water layer, so the wave is propagating toward the air/water interface upward R = (Z2-Z1)/(Z2+Z1) = (1.3*350)-(1027*1500) (1.3*350)+(1027*1500) = -0.999
Quick Computation of the Reflection Coefficients Water/Rock Interface: 1 2 Consider the cartoon: The source is located in the water layer, so the wave is propagating toward the rock/water interface downward R = (Z2-Z1)/(Z2+Z1) = (2650*3000)-(1027*1500) (2650*3000)+(1027*1500) = 0.67
Characteristics of the Reflection Coefficients The air/water boundary can be considered a perfect reflector. The -1 coefficient indicates that the reflection reverses polarity at the interface. The water/rock boundary has a coefficient of 0.67 indicating that the boundary allows transmission of energy, but can also be considered a relatively efficient reflector. The energy does not reverse polarity at this interface. So for any given shot, it is clear that there is energy in the water layer that represents primary reflections and multiple reflections.
Types of Multiples Multiples- Events that have undergone more than one reflection Short Path Multiples- Arrives so soon after the primary reflection that it adds a tail to the primary reflection. Long Path Multiples- Reflection between surface and water bottom over a great distance. Looks like a separate event/layer on the seismogram.
Short Path Multiples Peg-Leg Multiples- Short path multiples that have been reflected from the top and base of thin reflectors. Delays part of the energy from the primary reflection which effectively lengthens the wavelet. Have the same polarity as the primary. Path: Intuitively you can see that the path is slightly longer, so there is a delay in the arrival.
Short Path Multiples Ghosts- Occur in marine surveys when a submerged source is used. Energy travels up from source to the air/water interface, reflects, and continues down as a secondary source. As previously mentioned, the reflection is reversed in polarity from the original source.
Short Path Multiples Water Reverberation- Due to large reflection coefficients at the air/water and water/rock interface. This type of multiple also occurs as a long path multiple over long distances.
Filtering Basics Transforms- Operations that transform a function of certain variables to a related function of different variables (i.e. from time domain to the frequency domain by the Fourier transform). Transforms a vector to a different vector space for easy computation.
Filtering Basics Convolution- A mathematical operator which takes as inputs two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g in the time domain. Convolution is the multiplication of two functions in the Z domain or frequency domain.
Filtering Basics Correlation- Indicates the strength and direction of a linear relationship between the inputs of two functions, producing a third output function.
Transforms Z Transform- Converts a discretely sampled time function to a polynomial in Z the Z domain. Consider the continuous time function:
Z Transform
Z Transform The continuous function has been discretely sampled in time steps of 1 unit. The Z transform takes the amplitude of the value of the function as the coefficient of the term of a term in the polynomial. The time unit becomes the exponent of the polynomial term. x t 0.5 0 0.2702 1-0.2081 2-0.495 3-0.3268 4 0.1418 5 0.4801 6 0.377 7-0.0728 8-0.4556 9-0.4195 10
Z Transform The discretely sampled function can be viewed as: A vector: b t = (0.5000, 0.2702, -0.2081, -0.4950, -0.3268, 0.1418, 0.4801, 0.3770, -0.0728, -0.4556, -0.4195) A polynomial in the Z domain: B(Z)=0.5+0.2702Z-0.2081Z 2-0.495Z 3-0.3268Z 4 +0.1418Z 5 +0.4801Z 6 +0.377Z 7-0.0728Z 8-0.4556Z 9-0.4195Z 10 General form of the Z transform for a discretely sampled function b(t): B(Z)= t b t Z t
Features of the Z Transform The Z transform is also known as the unit delay operator. If B(Z)=1+2 Z +0Z 2 -Z 3 -Z 4 A time delay of n time units can be applied to the function by multiplying the polynomial by Z n.
Features of the Z Transform B(Z)=1+2Z+0Z 2 -Z 3 -Z 4 Time delayed function Delay by 1 time unit (Z 1 ) ZB(Z)=Z+2Z 2 +0Z 3 -Z 4 -Z 5
Features of the Z Transform The Z transform may also be used to create more complicated time functions from simple ones. Say B(Z) represents a source that you can see on a seismogram. B(Z) is said to be the impulse response of the source. We can add another source 5 time units later by simply creating a new function: C(Z)=B(Z)+Z 5 B(Z)
Features of the Z Transform If C(Z)=B(Z)+Z 5 B(Z) with B(Z)=0+2Z-1Z 2-1Z 3 +0Z 4 +Z 5 +Z 6 C(Z)= 0+2Z-1Z 2-1Z 3 +0Z 4 +Z 5 +3Z 6-1Z 7-1Z 8 +0Z 9 +Z 10 +Z 11 Notice that the signal has essentially repeated itself. However, look at the 3Z 6 term. Two terms were combined here, Z 5 (2Z)+Z 6. If pulses overlap with each other, the waveforms add together. This is called the linearity assumption.
Features of the Z Transform The waves do not interfere with each other despite traveling through heterogeneous materials. Nonlinearity arises from very large amplitude disturbances- not in the case of engineering and exploration seismology.
Features of the Z Transform The Taylor Series expansion of f(x) is given by the inverse of the Z transform: The Taylor series:
Linear System General form of a linear system: Y(Z) = X(Z)B(Z) where Y(Z) is the output, X(Z) is the input, and B(Z) is the impulse response (if the input is the unit impulse). Input Impulse Response Output
Convolution A convolution is an integral (or sum, if the function is discretely sampled) that expresses the amount of overlap of one function A as it is reversed and shifted over another function B. Numerically, the calculation for convolution of two functions can be performed by time reversing one function and shifting it relative to the other, summing the incident values. Take a look at an example:
Convolution Convolve A with B, or C = A*B A = (a 0, a 1, a 2, a 3 ) B = (b 0, b 1, b 2 ) a 0 a 1 a 2 a 3 b 2 b 1 b 0 c 0 = a 0 b 0
Convolution a 0 a 1 a 2 a 3 b 2 b 1 b 0 c 1 =a 0 b 1 +a 1 b 0 a 0 a 1 a 2 a 3 b 2 b 1 b 0 c 3 =a 1 b 2 +a 2 b 1 +a 3 b 0 a 0 a 1 a 2 a 3 b 2 b 1 b 0 c 2 =a 0 b 2 +a 1 b 1 +a 2 b 0 a 0 a 1 a 2 a 3 b 2 b 1 b 0 c 4 =a 2 b 2 +a 3 b 1
Convolution a 0 a 1 a 2 a 3 b 2 b 1 b 0 c 5 =a 3 b 2 The function C is: C = (c 0 c 1 c 2 c 3 c 4 c 5 ) c 0 = a 0 b 0 c 1 = a 0 b 1 +a 1 b 0 c 2 = a 0 b 2 +a 1 b 1 +a 2 b 0 c 3 = a 1 b 2 +a 2 b 1 +a 3 b 0 c 4 = a 2 b 2 +a 3 b 1 c 5 = a 3 b 2
Convolution Return to the Z transform briefly: A = a 0 + a 1 Z+ a 2 Z 2 + a 3 Z 3 B = b 0 +b 1 Z+ b 2 Z 2 Multiply the two polynomials: C(Z) = A(Z)B(Z) C=a 0 b 0 +a 0 b 1 Z+a 0 b 2 Z 2 +a 1 b 0 Z+a 1 b 1 Z 2 +a 1 b 2 Z 3 +a 2 b 0 Z 2 +a 2 b 1 Z 3 +a 2 b 2 Z 4 +a 3 b 0 Z 3 +a 3 b 1 Z 4 +a 3 b 2 Z 5
Convolution Collection of like terms: c 0 = (a 0 b 0 ) c 1 (Z) = (a 0 b 1 +a 1 b 0 )Z c 2 (Z 2 )= (a 0 b 2 +a 1 b 1 +a 2 b 0 )Z 2 c 3 (Z 3 )= (a 1 b 2 +a 2 b 1 +a 3 b 0 )Z 3 c 4 (Z 4) = (a 2 b 2 +a 3 b 1 )Z 4 c 5 (Z 5 )= (a 3 b 2 )Z 5
Convolution From the product of the multiplication of the two functions in the Z domain, we can see that convolution in the time domain is equivalent to multiplication in the Z domain.
Convolution Properties: Commutativity f*g = g*f Associativity f*(g*h) = (f*g)*h Associativity with scalar multiplication a(f*g) = (af)*g = f*(ag) Distributivity f*(g+h) = f*g + f*h Differentiation (f*g) = f *g = f*g
Correlation The correlation, or cross-correlation, is a measure of similarity of two signals as a function of the relative time shift between them, commonly used to find features in an unknown signal by comparing it to a known one. Auto-correlation is the correlation of a data set with itself. Numerically, the correlation procedure is similar to convolution, except the signals are not time reversed with respect to each other. One signal is shifted with respect to the second signal. The amplitudes of the two signals are scaled with respect to each other and summed. Take a look at an example:
Correlation Correlate A with B, or Φ = A B A = (a 0, a 1, a 2, a 3 ) B = (b 0, b 1, b 2 ) Continuing to shift the function by the specified unit of time yields: a 0 a 1 a 2 a 3 b 0 b 1 b 2 Φ ab (0) = a 0 b 0 +a 1 b 1 +a 2 b 2 Φ ab (2) = a 2 b 0 +a 3 b 1 Φ ab (3) = a 3 b 0 a 0 a 1 a 2 a 3 b 0 b 1 b 2 Φ ab (1) = a 1 b 0 +a 2 b 1 +a 3 b 2
Correlation The general form of the cross correlation of two data sets, x t and y t : Φ xy (τ) = k x k y k+τ Φ is a function of the relative time between two signals Τ (tau) is the relative shift of the signal y K is the element of each signal that we are looking at
Correlation Intuitively, we can see that two similar data sets will have positive products with large sums. Conversely, two dissimilar data sets will have both positive and negative products with small sums. If two data sets are correlated by having a large negative sum, it means the data sets would be similar if one was inverted (the two data sets are similar, but out of phase with respect to each other).
Multiple Removal Deterministic Predictive Filtering
Deterministic Predictive Filtering Deghosting with predictive filters Recall the geometry of the ghost:
Deterministic Predictive Filtering The ghost creates the equivalent of a second apparent shot. The second source follows the initial shot by τ milliseconds, which is a function of the shot depth. Since the ghost is reflected off the free surface (air/water interface) it exhibits 180 degree shift in polarity.
Deterministic Predictive Filtering The simplified impulse response of the ghost is shown to the right. The first element is the unit impulse. The second element is the ghost. τ is the time lag of the ghost k is the amplitude of the reflected energy
Deterministic Predictive Filtering The objective is to define a filter such that the input is reduced to the desired output, which is the unit spike of energy with no multiples. Filter 1 0
Deterministic Predictive Filtering x is the input y is the desired output f is the filter that gets from x to y In the time domain we convolve the input with the filter to get the desired output x*f = y
Deterministic Predictive Filtering Recall that convolution in the time domain is multiplication in the Z domain X(z)F(z) = Y(z) Filter design: F(z) = Y(z)/X(z)
Deterministic Predictive Filtering F(Z) = Y(Z)/ X(Z) X(Z) = 1 kz τ Y(Z) = 1 F(Z) = 1/(1 kz τ ) Note that the Taylor series is the inverse of the Z transform, so Filter 0
Deterministic Predictive Filtering F(Z) = 1/(1 kz τ ) F(Z) = 1 + kz τ k 2 z 2τ +k 3 z 3τ This filter has infinite length, so let s truncate it after the first 2 points and convolve it with X(Z)
Deterministic Predictive Filtering Y(Z) = F(Z)*X(Z) f o x o = 1 f t x o + f o x t = k + -k = 0 The multiple has been attenuated, but there is still one more term left to deal with
Deterministic Predictive Filtering The ghost has been eliminated but a new false event has been created at 2τ. The false event has amplitude of k 2. By increasing the number of terms in the filter, the amplitude of the false event gets smaller and smaller. Last term of the convolution
Deterministic Predictive Filtering Predictive Filtering to Attenuate Reverberation Effects After the source stops putting energy into the water layer, the reflections continue and decrease in amplitude. Due to the large acoustic impedances at the interfaces (i.e. air/water and water/rock).
Deterministic Predictive Filtering What is actually happening: The energy from the source propagates toward the air/water interface, reflects with a 180 degree phase shift, propagates toward the seafloor, transmits some energy and reflects again. The energy in the water layer now has an amplitude of k, which is a function of the reflection coefficient and impulse amplitude. The wave now reflects off the sea surface boundary and is phase shifted again. This continues for several cycles of the water layer.
Deterministic Predictive Filtering Reverberation has the form: 1,-2k,3k 2 at the receiver All of the terms with the exception of the first one represent multiples. Design a filter similar to that used in the deghosting technique.
Deterministic Predictive Filtering Up until now we have only looked at multiples from the source point of view. Multiples occur at both the source and receiver locations. First order reverberation (two types).
Deterministic Predictive Filtering Second Order Reverberation (three types). The goal of the filter is to take out the effect of the water layer and maintain the information of the trace below the water/rock interface. As can be seen in the accompanying cartoon, the energy travels through the water column two round trips. 2 nd Order Reverb
Deterministic Predictive Filtering Besides the two round trips through the water column, the energy also goes into the earth. The information from the sub-bottom areas is preserved. 2nd Order Reverb
Deterministic Predictive Filtering The term for the second order reverberation becomes 3k 2 z 2τ. The coefficient 3 because there are three options for the wave energy to follow. 2 nd Order Reverb
Deterministic Predictive Filtering The effect of the filter term 3k 2 z 2τ is: Which removes the information from the red boxes only and preserves information about the sub-bottom.
Deterministic Predictive Filtering Also third, fourth, etc. Need to account for these in filter design because they rescale (make the coefficients of the k term larger) the amplitudes of the subsequent reverberations
Deterministic Predictive Filtering The signal that we will have to filter is actually: X(Z) = 1-2kz τ +3k 2 z 2τ
Deterministic Predictive Filtering The first unit spike still represents the primary energy The filter for dereverberation effects: X(Z) F(Z) = Y(Z) X(Z) = 1-2kz τ +3k 2 z 2τ F(Z) = 1/ (1-2kz τ +3k 2 z 2τ ) F(Z) = 1+2kz τ +k 2 z 2τ
Deterministic Predictive Filtering F(Z) = 1+2kz τ +k 2 z 2τ f is: This is called the three point Backus filter
Deterministic Predictive Filtering The complete water reverberation function X(Z) that we just defined can be viewed from a different standpoint: Since the energy goes up and down through the water layer: Convolve the unit spike with the simple water layer model that we defined earlier for the downgoing wave. Convolve again for the upgoing wave energy to the surface.
Deterministic Predictive Filtering Y*X = (1)*(1,-k,k 2,-k 3,k 4,-k 5,k 6 ) Convolve the result with itself, since it is going back through the water column. (1,-k,k 2,-k 3 )* (1,-k,k 2,-k 3 ) and transform to the Z domain We get 1-2kz τ +3k 2 z 2τ -4k 3 z 3τ which is the complete effect of reverberation in the water layer.
Deterministic Predictive Filtering Similarly, if we take the filter in the Z domain for the simple water layer and square it, we get the filter for the complete water layer. Remember that multiplication in the Z domain is equivalent to convolution in the t domain.
Performance of the Predictive Filter To check the performance of the filter, convolve the reverberation water layer model with the filter in steps of τ, the time reverberation time lag. X*F=Y Y= (1,-2k,3k 2,-4k 3,5k 4,-6k 5 )*(1,2k,k 2 )
Performance of the Predictive Filter y0 = 1(1) = 1 y1 = 1(-2k)+2k(1) = 0 y2 = 1(3k 2 )+2k(-2k)+k 2 (1) = 0 y3 = 1(-4k 3 )+2k(3k 2 )+k 2 (-2k) = 0.. So we see that the filter removes the effect of the reverberation related multiples.
Determining Filter Parameters To determine the time lag, τ, between the primary and the ghost, correlate a trace with respect to itself. This form of correlation is called auto correlation. The auto correlation of a trace will determine the time lag, τ, of the reverberation. After performing the operation, the correlation will exhibit a large trough. Remember that correlation is the multiplication and summation of the amplitudes of two signals, while one function is being time shifted past the other. The time lag between the primary and the reverberation coincides with the time shift that occurred to get the large trough.
Determining Filter Parameters To determine k, the amplitude of the reflection, you must determine the reflection coefficient of the multiple energy at the seafloor. The inverted amplitude of the first reflection can be divided by the amplitude of the impulse spike to get the reflection coefficient. k = (amplitude of the impulse)(r)
Determining Filter Parameters For simplicity, consider a system with a time step of 1 unit. Consider the trace to the right representing a very simple signal.
Determining Filter Parameters Find the time lag of the reverberation by autocorrelation of the simple signal seen in the previous slide. (3-1 0 0 0 0 0-2 1 0 0 0 0) (3-1 0 0 0 0 0-3 1 0 0 0 0) It appears that the time lag is 7 time units between the source and the reverberation. (3-1 0 0 0 0 0-2 1 0 0 0 0) (3-1 0 0 0 0 0-3 1 0 0 0 0)
Determining Filter Parameters Now determine k Amplitude of the impulse is 3, the inverted value of the first multiple is 2 R = 2/3 k = 3R
Determining Filter Parameters Since this is deterministic filter prediction we can find the parameters by the methods above, but this gets to be time consuming to evaluate from trace to trace. We can get best fit values of the two parameters by implementing a least squares inversion and applying them to the stacked data.
Assignment Define a filter and implement on the following stacked profile: 3 3 3 3 0 0 0 0-2 -2-2 -2 0 0 0 0