5
19 An approach to extract the parameters of source-heave dynamics for marine seismic applications Une approche pour extraire les paramètres de la dynamique du soulèvement de source dans les applications marines séismiques By Ferial El-Hawary, Technical University of Nova Scotia, Halifax, Nova Scotia. An important pre-processing step in shallow marine seismic data analysis involves compensation for an inherent source heave component. The Kalman filtering approach requires a model for the heave process. Identifying optimum model parameters is the subject of this paper. This paper discusses problem formulation based on the available frequency spectrum record of the heave process. The problem reduces to solving a set of non- linear equations. Here the Ncwton-Raphson method is applied to actual field data to obtain the optimal model parameters. Une étape importante dans le prétraitement de l'analyse des données reliées aux applications séismiques marines, comprend une compensation du composant due au soulèvement de source inhérent. Un approche utilisant le filtre Kalman requiert un modèle pour le procédé de soulèvement. Le but principal de cet article est d'identifier les paramètres optimaux du modèle. Cette communication établit l'énoncé du problème à partir du spectre disponible pour le procédé de soulèvement. Ce problème consiste à résoudre un système d'équations non linéaires. On utilise la méthode de Newton-Raphson à partir des données recueillies pour obtenir les paramètres optimaux du modèle. Introduction Deep towed systems (DTSs) with well-defined and repeatable broad-band output pulses have been developed to perform shallow marine seismic experiments. A typical example is the Huntec Deep Towed Seismic System, which has been used in the Seabed Project to develop a methodology for geological classification by acoustic remote sensing. The DTS (fish) contains an electrodynamic source (boomer) which produces an impulse-like output directed downwards. The boomer is fired one to four times per second. A hydrophone mounted on the fish picks up the echoes from the sea floor. The fish is towed by a vessel at speeds of up to 2 m/sec and depths of up to 300 m. 1 To improve resolution and signal levels and to reduce noise effects and unwanted sea-surface echoes, the fish is deployed closer to the sea floor. Heaving of the towed vessel leads to changes in fish depth which can significantly advance or delay an echo, as compared with the previous echo. This heaving can mask the actual topography of the sea floor. To correct the problem, fish depth is continuously sensed with an accuracy of the order of 0.1 m and the boomer firing time is delayed or advanced accordingly. 2 3 The sub-bottom profiler echoes are used to obtain seismic cross- sections of the seabed. Ocean sediment characterization is an im- portant activity that relies on sub-bottom profiler echoes obtained from a DTS system. 4 ' 7 The requirement to remove any residual ef- fects of the tow-fish motion is an important pre-processing task. This arises from (successful) attempts to characterize sea-floor sediments in terms of their acousic scattering and reflecting parameters, obtained by inversion modelling of the motion- corrected seismic data. 1 ' 7 An earlier paper 8 discusses the use of Kalman filtering to com- pensate for source-heave effects. A similar application of the prin- ciples involved to buoy-wave data filtering is treated in reference 9. The procedure requires a model of the heave process based on the frequency spectrum of the heave record. This model identification problem is discussed in reference 10 from a theoretical point of view. In that paper it was concluded that the identification problem can be formulated and solved using Newton's method and the suc- cess rate of that method is presented for a number of simulated records. The parameter estimation problem also arises in structural dynamics. A recent contribution, 11 utilizes a least p h squares op- timization technique to identify model parameters. The problem is formulated as a non-linear parameter estimation with the object of minimizing the (root mean square) error between the actual measurements and the predicted values in the frequency domain. The result is à set of non-linear equations whose solutions are sought using the Newton-Raphson method. This paper discusses aspects of actual computational results including effects of band- width and initial estimates of the solution. The model The literature in marine hydrodynamics, such as in references 12, 13 and 14, provides details on modelling of heave hydrodynamics. On the basis of the frequency spectrum of the heave record, a model for heave dynamics consistent with those found in the area of hydrodynamics is postulated. To model the heave motion, we construct a record Z(t) using available reflection records (received signals). A train of source pulses fired at regular intervals is used. Each pulse is associated with a reflection record. The time delay from the instant of pulse in- itiation of the first peak on the record (as a water sediment interface reflection) is converted into a distance by multiplying by the speed of sound in water. The sequence Z(r) exhibits a random com- ponent superimposed on the sea-bed profile as shown in Figure 1. Construction procedure is explained in reference 8. Figure 1 covers most of the important information about the subsurface strata. This record z(t) for the experimental data was obtained from 512 reflection records, initiated at 7 , = 0.5 sec intervals. The Fourier transform Z(f) of the record z(t) gives the frequency spectrum of Can. Elect. Eng. J., Vol. 12 No. 1, 1987

An approach to extract the parameters of source-heave dynamics for marine seismic applications

  • Upload
    ferial

  • View
    212

  • Download
    0

Embed Size (px)

Citation preview

Page 1: An approach to extract the parameters of source-heave dynamics for marine seismic applications

19

An approach to extract the parameters of source-heave dynamics for marine seismic

applications

Une approche pour extraire les paramètres de la dynamique du soulèvement de source dans les

applications marines séismiques

By Ferial El-Hawary, Technical University of Nova Scotia, Halifax, Nova Scotia.

An important pre-processing step in shallow marine seismic data analysis involves compensation for an inherent source heave component. The Kalman filtering approach requires a model for the heave process. Identifying optimum model parameters is the subject of this paper. This paper discusses problem formulation based on the available frequency spectrum record of the heave process. The problem reduces to solving a set of non­linear equations. Here the Ncwton-Raphson method is applied to actual field data to obtain the optimal model parameters.

Une étape importante dans le prétraitement de l'analyse des données reliées aux applications séismiques marines, comprend une compensation du composant due au soulèvement de source inhérent.

Un approche utilisant le filtre Kalman requiert un modèle pour le procédé de soulèvement. Le but principal de cet article est d'identifier les paramètres optimaux du modèle. Cette communication établit l'énoncé du problème à partir du spectre disponible pour le procédé de soulèvement. Ce problème consiste à résoudre un système d'équations non linéaires. On utilise la méthode de Newton-Raphson à partir des données recueillies pour obtenir les paramètres optimaux du modèle.

Introduction

Deep towed systems (DTSs) with well-defined and repeatable broad-band output pulses have been developed to perform shallow marine seismic experiments. A typical example is the Huntec Deep Towed Seismic System, which has been used in the Seabed Project to develop a methodology for geological classification by acoustic remote sensing. The DTS (fish) contains an electrodynamic source (boomer) which produces an impulse-like output directed downwards. The boomer is fired one to four times per second. A hydrophone mounted on the fish picks up the echoes from the sea floor. The fish is towed by a vessel at speeds of up to 2 m/sec and depths of up to 300 m. 1 To improve resolution and signal levels and to reduce noise effects and unwanted sea-surface echoes, the fish is deployed closer to the sea floor. Heaving of the towed vessel leads to changes in fish depth which can significantly advance or delay an echo, as compared with the previous echo. This heaving can mask the actual topography of the sea floor. To correct the problem, fish depth is continuously sensed with an accuracy of the order of 0.1 m and the boomer firing time is delayed or advanced accordingly.2 3

The sub-bottom profiler echoes are used to obtain seismic cross-sections of the seabed. Ocean sediment characterization is an im­portant activity that relies on sub-bottom profiler echoes obtained from a DTS system. 4' 7 The requirement to remove any residual ef­fects of the tow-fish motion is an important pre-processing task. This arises from (successful) attempts to characterize sea-floor sediments in terms of their acousic scattering and reflecting parameters, obtained by inversion modelling of the motion-corrected seismic data. 1 ' 7

An earlier paper8 discusses the use of Kalman filtering to com­pensate for source-heave effects. A similar application of the prin­ciples involved to buoy-wave data filtering is treated in reference 9. The procedure requires a model of the heave process based on the frequency spectrum of the heave record. This model identification

problem is discussed in reference 10 from a theoretical point of view. In that paper it was concluded that the identification problem can be formulated and solved using Newton's method and the suc­cess rate of that method is presented for a number of simulated records. The parameter estimation problem also arises in structural dynamics. A recent contribution, 1 1 utilizes a least ph squares op­timization technique to identify model parameters. The problem is formulated as a non-linear parameter estimation with the object of minimizing the (root mean square) error between the actual measurements and the predicted values in the frequency domain. The result is à set of non-linear equations whose solutions are sought using the Newton-Raphson method. This paper discusses aspects of actual computational results including effects of band­width and initial estimates of the solution.

The model

The literature in marine hydrodynamics, such as in references 12, 13 and 14, provides details on modelling of heave hydrodynamics. On the basis of the frequency spectrum of the heave record, a model for heave dynamics consistent with those found in the area of hydrodynamics is postulated.

To model the heave motion, we construct a record Z(t) using available reflection records (received signals). A train of source pulses fired at regular intervals Γ is used. Each pulse is associated with a reflection record. The time delay from the instant of pulse in­itiation of the first peak on the record (as a water sediment interface reflection) is converted into a distance Ζ by multiplying by the speed of sound in water. The sequence Z(r) exhibits a random com­ponent superimposed on the sea-bed profile as shown in Figure 1. Construction procedure is explained in reference 8. Figure 1 covers most of the important information about the subsurface strata. This record z(t) for the experimental data was obtained from 512 reflection records, initiated at 7 ,= 0.5 sec intervals. The Fourier transform Z(f) of the record z(t) gives the frequency spectrum of

Can. Elect. Eng. J., Vol. 12 No. 1, 1987

Page 2: An approach to extract the parameters of source-heave dynamics for marine seismic applications

20 CAN. ELECT. ENG. J., VOL. 12 NO. I, 1987

the heave dynamics, which is required to model the phenomenon of source heave. This is assumed to be caused by purely random ex­citation (white noise) due to current and wave effects on the towed body, towed cable, and ship.

In order to avoid aliasing, the Fourier transform of the record with 0.5 sec time spacing is obtained through time scaling of 4 χ 10 r S ratio. This time scaling gives a 20/is sampling period for signal processing and corresponds to a sampling frequency of 50 kHz which is much higher than the aticipated frequency components in the record. Figure 2 shows the magnitude of the Fourier transform Z{f) with a maximum frequency of 0.96 Hz in the original record or 0.48 cycles/m spatial frequency. More details are given in reference 18.

Inspection of the frequency-spectrum characteristic reveals the presence of a number of dominant frequency bands. This suggests that the power spectral density (PSD) can be modelled by passing a purely radom input through a narrow-band filter with drifting centre frequencies corresponding to the observed dominant fre­quencies. This particular idealization is shown in Figure 3. The narrow-band filter's response is similar to that of a simple resonant electric circuit (RLQ, which can be modelled using a second-order transfer function as shown in Figure 4.

The Kalman filter requires a heave model in state-space form, preferably of the lowest order, so as to reduce the computational effort.

For the purposes of this study only, a narrow-band filter cor­responding to the envelope of the heave spectrum is assumed to represent the heave dynamics. This can be visualized as a reduced-order model for a complex physical phenomenon. The transfer function of the heave response model in terms of the Laplace operation (s=ju) is chosen as

Z(5) = Κ

Lcdo s 1

(D

The natural (or centre) frequency of the system is ω<>, while Q is the quality factor obtained from the half-power points (/i and / 2 ) .

Q = / o / ( W i ) (2)

Parameter estimation problem

The problem of finding the model parameters Kt ω<>, and Q can be formulated such that the following least-squares error criterion is minimized:

J=i2[Zm(<»i)-Zc(X,u>i)V (3)

The value of Zm (ω,) is available for each frequency ω, for the number of experiments N. The expression for the model Zc is a function of ω, and X, the parameter vector

The objective functional given in equation (3) is convex in the do­main of interest. In Figure 5, the family of curves J versus Q for a selected number of ω values is presented.

10" V » Or

ϊ I t ι ι

- Out lo Seabed

- Due to Source Heave

0 0.16 0.32 0.48 0.64 0.80 0.96 Frequency Band Spatial f(H 2 Lin)

Figure 2: The frequency response of heave dynamics.

Figure 3: Narrow band filter with drifting centre frequencies.

z( t )

m 5 . 0

4 . 0

3 . 0

2 . 0

i 0-01 ' I I I I I 1 1 1 L

25 50 75 100 «25 150 175 2 0 0 225 2 5 0 » SEC

z(t)

, n p u t Damped System „ 0 u , p uJ Λ Impulse Sequence Narrow Band

Figure 1: Peak delay time record z(t) corresponding to the water sediment interface. Figure 4: Heave model.

Page 3: An approach to extract the parameters of source-heave dynamics for marine seismic applications

EL-HAWARY: SOURCE-HEAVE DYNAMICS

It is clear that sub-optimal conditions are given by where, for simplicity, we set

21

dJ dx

= 0 (5) * ' = ^ [ Ζ " < « < ) - £ ΐ (15)

As a result, we have

Ν ^ Ζ ^ , ω ( ) - ^ = 0

The unknowns are ω0 and Q. The Newton-Raphson method (see reference 17) is proposed in reference 10 to solve the optimality conditions iteratively. Theoretical details of the algorithm are also

(6) discussed in reference 10.

where the error in Ζ is Computational results

ΖΕ(Χ,ω{) = Zm(co,) - Ζ 0(Ζ,ω,) A computer program to implement the Newton-Raphson

( 7 ) algorithm has been written and successfully tested for a number of simulated records in conjunction with actual heave records obtain-

In essence we have three equations in three unknowns. The form e d in field trials. Details of the tests are given in reference 1. Figure 6 of the equations is detailed in the Appendix.

Recalling the expression for ZE, we thus need to solve:

(8)

shows the frequency spectrum of the heave record in solid lines, while the envelope result is represented by the dotted line. The model parameters obtained are:

ω0 = 1.1823247 rad/s,

Κ = 0.86043 and Q = 2.508

[ w £ ] = ο 1?

(9)

Y? [ Ζ „ ( ω , ) - ! ] = θ (10)

The above equations are non-linear in Q and ωο but Κ appears linearly.

We can exploit the fact that Κ appears linearly to reduce the com­putational effort. We thus require the solution of the two equations (9) and (10), with Κ given by

K = si

Figure 5: Family of curves J versus Qfor a selected number of frequency values.

(11)

Note also that

(12)

We are thus interested in solving (9) and (10) rewritten as

N 2 ΗΛωο,Ο) = Σ [ ( - ) - ψ = 0 ( 1 3 >

2·5· 2(F)

1 5 +

Η 2(ωο,ρ) = £ [ ( 2 ) 2 - ψ = 0 (14) Figure 6: Solid lines represent the frequency spectrum of the heave record. Dot­ted lines represent the frequency spectrum envelope result of the estimated heave record model.

Page 4: An approach to extract the parameters of source-heave dynamics for marine seismic applications

22 CAN. ELECT. ENG. J., VOL. 12 NO. 1, 1987

The minimum value of the sum of the square of the errors is given by:

j m i n = 9.7447

The performance of the algorithm depends on the initial guess values. Table 1 lists the required number of iterations for con­vergence, to a tolerance of 10"3, for different initial guesses. As ex­pected, the closer the initial guess is to the actual solution, the fewer iterations are required. Table 2 shows the progress in Q and ω 0

values with iterations, while Table 3 shows the relative error values. The function values at convergence are:

// ι(ω 0 ,β) = 0.0011

Hi(o>o,Q) = 0.00001132

If

— and — dQ θω0

are neglected as reported in reference 10, the method is slower to converge. It is clear that the implementation with a full Jacobian shows a superior convergence pattern. The savings in computa­tional time per iteration neglecting some terms are far outweighed by the doubling of the number of iterations.

$M · . . ·Λ . Via ' - . « 5

Conclusions

This paper has detailed the application of the Newton-Raphson method to solve for the optimal estimates (in the least-square sense) of the parameters of the heave-dynamics model, based on frequency-response records. Sample computational results related to actual field records have been given to demonstrate the effec­tiveness of the method. Our computational approach is faster than the method proposed in reference 11 as Newton's method is well known for its quadratic convergence properties in the neighbourhood of the solution point. Extension of the method to higher order models and a study of the trade-offs in model sophistication versus accuracy are the subjects of future work.

References

1. Parrot, D.R., Dodds, D.J., King, L.H. and Si m ρ kin, P.G., "Measurement and Evaluation of the Acoustic Reflectivity of the Sea Floor," Canadian Journal of Earth Sciences, Vol. 17, No. 6, pp. 722-737, 1980.

2. Hutchins, R.W., Dodds, D. J., Parrot, R. and King, L.H., "Characterization of Sea Floor Sediments by Geo-Acoustic Scattering Models using High Resolution Seismic Data", Oceanology Internationalt March, 1982.

3. Hutchins, R.W., "Removal of Tow Fish Motion Noise From High Resolution Seismic Profiles" presented at SEG-US Navy Symposium on Acoustic Imaging Technology and On-Board Data Recording Processing Equipment, Bay St. Louis, Mississippi, 1978.

4. Cochrane, N.A. and Dunsiger, A.D., "Seabed Roughness Characterization by Broadband Acoustic Echo Sounding," in Proc. Fifth International Conference on Port and Ocean Engineering Under Arctic Conditions, Trondheim, Norway, Aug. 13-17, 1979.

5. Dunsiger, D.A., Cochrane, N.A. and Vetter, W.J., "Seabed Characterization from Broadband Acoustic Echosounding with Scattering Models," IEEE Jour­nal of Oceanic Engineering, Vol. ΟΕ-6, No. 3, July, 1981.

6. Dodds, D.J., "Attenuation Estimates from High Resolution Sub-bottom Pro­filer Echoes," in Bottom-Interacting Ocean Acoustics New York: Plenum Publishing Corporation, 1980.

7. Dunsiger, A.D., Chari, T.R., Fader, G.B., Peters, G.R., Simpkin, P.G. and Zielinski, Α., "Ocean Sediments—A Study Relating Geophysical Geotechnical, and Acoustic Properties," Can. Geotech. J., Vol. 18 No. 4, pp. 492-501, 1981.

8. El-Hawary, F., "Compensation for Source Heave by use of a Kalman Filter," IEEE Journal of Oceanic Engineering, Vol. ΟΕ-7, No. 2, pp. 89-96, April 1982.

9. Severance, R.W., "Optimum Filtering and Smoothing of Buoy Wave Data," Journal of Hydronautics, Vol. 9, pp. 69-74, April 1975.

10. El-Hawary, F., "Optimal Parameter Estimation for a Heave Response Model," in Proceedings of IEEE Oceans Conference '82, Washington, D.C., September 1982.

11. Lu, X. and Vandiver, J.K., "Damping and Natural Frequency Estimation Using the Least Pth Optimization Technique' ' , Paper OTC 4283, presented at 14th Off­shore Technology Conference, May 1982.

12. Bhattacharya, R., Dynamics of Marine Vehicles, New York: Wiley-Interscience, 1978.

13. Price, W.G. and Bishop, R.E.D., Probabilistic Theory of Ship Dynamics, New York: Wiley-Interscience, 1974.

14. McCormick, M.E., Ocean Engineering Wave Mechanics, New York: Wiley-Interscience, 1973.

15. Gelb, Α., Applied Optimal Estimation, The Analytic Sciences Corporation, U.S.A., 1974.

16. Meditch, J.S., Stochastic Optimal Linear Estimation and Control, New York: McGraw-Hill, 1969.

17. Ortega, J.M. and Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, New York: Academic Press, 1970.

18. El-Hawary, F., "Modeling and signal processing for identification of ocean sub­surface features from acoustic reflections," Ph.D. dissertation, Memorial Univ. of Newfoundland, ch. 4, May 1981.

APPENDIX

1-1

1-2

The derivative of Z c with respect to Κ is

Page 5: An approach to extract the parameters of source-heave dynamics for marine seismic applications

EL-HAWARY: SOURCE-HEAVE DYNAMICS 23

0ZC I Observing that K, Q, and ω<> appear independently of the summa-~dK = Y(Q <*) ω) * ^ ** ο η > w e c a n r e w " t e t n e optimally conditions as

The derivative of Zc with respect to β is ^v'^ = 0 1 - 9

3 β ^ 3 ( β , Ω Ο , Ω , ) L COO Ω , J * 2

The derivative of Zc with respect to <*> is

^ = i S t a ) ' - e ) ' ] « έ - ^ - [ ( ΐ ) " - ( ί ) ] -

Having obtained the expressions for the derivatives, we now substitute in the optimality conditions to obtain the equations for A . 4 . r . . . . . . . / t ~ . / Λ Q< _ A „ ,

o l .· J; J An alternative form is obtained by adding (17) to (18) and subtract-optimal estimation / , o \ r · υ v mg (18) from (17) to yield Ν

Σ Ζ Γ Α = 0 1-6 y, ΖΕ(ΛΓ,ω,) η

Σ = ^ 1 - - - ] Ό 1-7 f Ζ ^ , ) Γ ^ γ _ ι

~ ι? L « , <*J TO l u i ' J U L L "