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Application of Spectral Analysis in the RemoteSensing of Marine Geological Deposits and in the

Detection of Periodicities in the SedimentationProcesses in Continental Geological Systems..

PARTI.

Luc PEIRLINCKX,LeoP. VANBIESENANDElfiVANOVERLOOP

Vrije Universiteit BrusselEarth Technology Institute

Pleinlaan 2RJO50 BRUSSELS

Belgium

Abstract 1- Echosounders are used to investigate the presence and thecomposition of marine geological deposits and to enable the automaticgeneration of a sub bottom profiling record, when sailing along rivers,estuaries and in some coastal regions. The method is based on the time domainreflectometrical principal, where underwater acoustic pulses are used. Thefrequency content of the emitted pulses is determining the vertical resolution,Le. in the direction orthogonal to the surface of the sea, and the penetrationinto the sediments. Typically, frequencies ranging from a few Hz to somehundred thousands of Hz are used (seismic, sonic and ultrasonic signals). Ingeneral ceramic transducers are used to generate the emitted pulses and torecord the reflected echoes. In this paper the present noise sources corruptingthe measured echograms will be investigated. Firstly, the measurementprocedures and the set-ups for the noise measurement will be discussed. TheBelgian oceanographic vessel, the Belgica, was used for the measurements ofthe noise generated by propellers, engines, stabilizers, the fish incorporatingthe transducers, the waves and the marine background noise. Secondly, theestimation of the Power Spectral Density (PSD) of the correla,Jed noise sourcesis estimated. Different modern estimators will be compared and the nature ofthe different periodical components in the noise will be treated. An ARMA-and AR-model will be furthermore presented to model the most importantnoise components. These models will be used to validate some of the syntheticechograms, generated in the laboratory. The study will demonstrate the highdegree of correspondence between the model and the observed measurements.

1 Part I of this lecture was presented at the IEEE Instrumentation and MeasurementTechnology Conference, IMTC/90, February 13-15, 1990, San Jose, California, USA, and ispublished in the Conference Record of the IMTC/90 Conference, IEEE Catalog N° 90CH2735-9entitled: "On The Measurement and the Modelling of the Correlated Noise Sources Corrupting theMarine Geological Echosounding Experiments", Leo. P. Van Biesen, Luc Peirlinckx, Serge Masynand Stanislas Wartel, pp 353-359.

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1. INTRODUCTION

The acoustic reflectometry is an appropriate method to solve some of the problems stated in amarine environment. One is only interested here in the study of the sediment layers and thenear subbottom, and not in a geological prospecting of the underground of seas and rivers.This means that information up to a few meters of depth via a non-destructive or non-disturbing technique is envisaged, Le. without having the composition and location of thesediment layers altered while measuring. This implies that the use of probes towed along theseabottom is excluded. The acoustic echosounding method is for this remote sensing task a farbetter candidate then the techniques based on seismics. This is due to the fact that it is possibleto generate emitted pulses with a highpower content electronically (up to a few kW) and to theshort duration of an experiment, since only a low penetration is aimed, so that with a highrepetition frequency the bottom can be scanned (e.g. 5 records/second).Although from metrological point of view, the acoustic reflectometry seems to be a goodmeasurement method, it follows from practice that the interpretation of the echograms is not asimple nor a trivial task! Several argumentscan be used to explain this statement: the definitionof mud (but also other sediments), the used instrumentation and the physical environmentwhere the experiments have to be carried out The definition of mud is certainly one of themost difficult tasks because it is a very heterogeneous, cohesive sediment composed of a smallfraction of sand and an important fraction of clay and organic material. Furthermore, themodellisation of the present correlated noise sources corrupting the measured echograms isrequired in order to validate the used model for the acoustical wave propagation. It is alsonecessary to incorporate the effect of the used instrumentation in the model. Therefore, we willdescribe the measurement proceduresand set-up in the following paragraph.

2. MEASUREMENTPRQCEDURES AND_SET-UP

The currently used measurement set-up for subbottom profiling echosounders is shown infigure 1. A measurement probe is towed aside the ship. The probe contains a set oftransmitter/receiver transducers (severalpiezo-electric/ceramic sensors to receive and/or moreceramic transducers to transceive), which is located in a dynamic acceptable housing, called afish. The transmitter converts an electrical impulse into an acoustical waye. The power of theemitted signals is a function of the observedattenuation of the media and of the depth and has avalue ranging up to several kW. The emitted power can be seismic, sonic or ultrasonic (from afew mHz up to several hundreds of kHz). The actual emitters usually offer an impulsewindowed amplitude modulated signal with a fixed carrier, e.g. 3.5 kHz or 12 kHz in theaudioband or e.g. 300 kHz in the ultrasonic band. Their spectrum is therefore composed of thespectrum of the applied time window convolved with a Dirac-distribution located at the carrierfrequency, mostly yielding into narrowband signals due to the sinc approximation of thewindowed spectra [1]. It has been shown, however, that this point of view is in contradictionwith the optimal generation of echogramsby the acoustical generating unit [2]. The reason thatsuch signals are still popular is due to the fact that only a limited bandpass characteristic isneeded and the ability to generate such pulses with high power relatively easy. Furthermore, astrong resemblance with seismic pulses can be observed.In addition with the classical echosounder equipment as described above, a Dynamic SignalAnalyzer (DSA71O-SPINNOV)is incorporated in the measurement set-up in order to digitizethe measured echograms without aliasing problems (anti-alias filtering: 11th order Cauerelliptic fllter, pass band ripple of:f: 0.3 dB, stop band attenuation 96 dB with digital fllteredpassband equalisation [3]).The DSA-equipment offers furtheImOrethe advantage that estimation algorithms, which will bediscussed further on, can be implementedin a VME-environment in order to assure a fast modeof operation (FFT algorithms implementedin DSP-flrmware library).

I

I

I

II

I

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I

I

Fish towed by the ship

sediment layers,e.g. mud .. '

~- :tlom of the sea- "'i ~

fig. 1: A typical subbottom profiling echosounder measurement set-up: the transceiverpart.

LESTIMA nON OF TIll PSD OF TIll CORRELATEDNOISE SOURCES

3.1 Classical Spectral Estimation

In this chapter we will apply the popular spectral estimation method based on Fourier analysis:the periodogram. The definition of the periodogram spectral estimator relies on an alternativedefinition of the PSD and is repeated in appendix 1.To obtain the periodogram from figure 3.1, 50 adjacent records of 256 points were used. Thesample frequency was fixed at 44.1kHz. It can clearly be seen that there is more energy in thelower frequencies than in the higher.With the classical estimators such as the periodogram, it is possible to estimate the frequenciesof periodic components in the PSD. Unfortunately, it remains very difficult to overtake theorigin of all periodic components. Furthermore, if one is interested in the modelling of thePSD, modern spectral estimators are more appropriate methods than the periodogram describedabove.

3.2 Modern Spectral Estimation

These spectral estimators are based on a parametric approach of the estimation problem. Thediscrete random process is approximated by a time series or transfer function model

p q

x[n] =- La[k]x[n-k] + Lb[k]u[n-k]k=l k=O

(1)

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Fig. 3.2.1.c : Burg method ( 50 realisations)

Although, the amplitude is biased, one can state that the Burg method (Eu) produces anaccurate spectral estimate.Not only the selection of the estimator is a critical point in applying spectral estimator methods,but also the selection of the order of the model. The best choice for the order of the model isnot known a priori. Therefore, several orders should be tried out. However, criteria exist tochoose the best model order. The most common method is based on the defmition of whitenOIse.Filtering the sequence x[n], of which we try to estimate the PSD, with the inverse AR, MA orARMA fIlter obtained with the estimated parameters, should yield white noise since one hadchosen white noise as the driving sequence. Due to that property, the selection of the order isreduced to determining whether the filtered sequence is white noise or not. To solve thatproblem, several methods can be found [4],[5].The importance of the model order will be shown in the figure 3.2.1.d: the Autocorrelationwas used for an order of p =5 respectively p = 8.It is seen that in both cases an accurateestimate has been produced. Clearly, one should take a model with as few parameters aspossible and therefore the model with order p =5 is a better candidate.In this chapter, several AR spectral estimator methods have been applied to the describedproblem. The model order selection methods [4] imply that an estimation has been done beforeone can determine whether the correct order was chosen.If the modelling of the PSD has to beimplemented in an automated device, it is obvious that some expert knowledge will be requiredto determine the model orders of interest [6].

3.2.2 AutoRegressive Movin~ Average (ARMA) methods

In practice, one does not know a priori which model, AR, MA or ARMA, to choose. It is verylikely that in other environments the PSD will differas well. More spectral peaks, valleys and agreater roll-off can be present. Yet, one wants to estimate the PSD in all encounteredenvironments in the acoustic measurement procedures. From that point of view it is also wiseto look at other then AR-estimators. Therefore two ARMA estimators, the 'Least SquaresModified Yule Walker algorithm' (LSMYWE) and the ' Akaike algorithm', have beenimplemented.

---

Burg0

-10

- -20cc"C-

,Q -30(/')

0-

-40

-50

-600 5 10 15 20 2S

f (kHz)

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Fig. 3.2.1.d. : order selectionThe 'Least Squares Modified Yule Walker algorithm' is based on the Yule Walker equations[4]. As depicted in the chapter concerning the AR estimators, the solving of these equationsyields the AR-parameters. The datasequence x[n] is then filtered with A(z), where A(z) is thepolynomial constructed with the estimated AR parameters.The output sequence of that filter isthen a pure MA process, so that the MA parameters can be found by applying an MA estimatorto that sequence.It can be shown that the approximate MLE ( Maximum Likelihood Estimation) of theparameters of a real ARMA process can be determined as the minimum of a highly nonlinearfunction [4]. The 'Akaike algorithm' uses a Newton-Raphson iteration to minimize thisfunction. This approach is iterative and therefore not guaranteed to converge. If convergencedoes occur, the minimum found may not be the global minimum. It is important to start theiteration with initial values of the ARMA parameters that are as close as possible to the realvalues. The initial values can be found using anotherARMA estimator first, such as the 'LeastSquares Modified Yule Walker algorithm'.

Modified Yule Walker

Ls (p=8,q=6)

5 10 15

f ( IcHz)20 25

Fig. 3.2.2.a: Least Squares Modified Yule Walker

Order Selection

-10

-20

-40CQ

"Q'-'101

O(f)Cl.

-<i0

O

0 5 10 15 20 25

f (IcHz)

-20

40

-CQ O"Q'-'101(f) -<i0Cl.

O

-70

0

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The estimation shown in figure 3.2.2.a reveals interesting features. First a totally differentshape of the PSD due to a change of measurement location is observed. The measurements forthis estimation were carried out in a harbor dock, in very silent conditions and in stagnantwater. The level of the PSD is remarkably lower (some 20 dB). This PSD has a lot of valleys,peaks and a great roll-off so that an ARMA estimator was most suitable. With the chosen order(AR order 8 and MA order 6), the 'Least Squares Modified Yule Walker' estimator yields verygood results, as well in estimating the frequencies as the noise level.The 'Akaike' algorithm has proved not to be able to estimate the PSD of the obtainedmeasurements, and therefore the results have been rejected. The reason is that the estimationproduces a non minimum phase fIlter, which is considered as a stop criterium by the algorithm.It is difficult to compare spectral estimators due to a lack of a commonly accepted definition ofthe optimality of estimators. Nevertheless, cenain criteria are proposed to describe theestimators perfonnance. The resolution is a prime example. Other criteria are the bias, thevariance, etc. Still the comparison remains difficult because spectral estimation is a statisticaloperation, so that the perfonnance can only be described statistically. In this chapter it wasshown that it is possible to estimate the Power Spectral Density of the noise corrupting theacoustic measurements in a marine environment. Doing so, one can incorporate the effects ofthe noise sources, when processing the acoustic measurements.

4. MODEL VALIDAnON

In order to simulate the behaviour of the marine system, a suitable model has to be chosenwhich includes the characteristics of the marine deposits, shape and duration of the acousticinput signal, possible present correlated noise sources, etc. Therefore, it is convenient tosuggest the following convolutional model:

r(t) =het) * set) + net) (1)

with: r(t) : reflectogram in the time domain

het) : impulse response of the system

set) : acoustical input signal

net) : additive colored noise source

* : the convolution integral

A brief overview of the modelling approach of the acoustic wave propagation phenomena isgiven, since the main purpose here is to discuss the influence of the noise sources representedby net).The model is based on a layered structurerepresentation of the marine system, characterized bythe thickness of the layers, frequency dependent absorption and velocity of sound, and thedensity of the sediment layers [7,2]. This is in accordance with the assumption that the seabedis composed of linear elastic, or inelastic materials when absorption and dispersion effects arepresent. Furthermore, it is assumed that the nonnal incident acoustical waves propagate only inone dimension.This system is completely characterized by the impulse response het). In the literature, a lot ofattention has been paid to calculate the response of a layered structure, when a spike is used asthe input signal. When attenuation is incorporated in the model, the calculations are elaboratedin the frequency domain and usually the absorption-dispersion theory of Futtennan is used[2,8]. Without absorption often the z-domain representation is utilized [9]. Since the modeltakes absorption and dispersion into account, the frequency domain implementation is chosen

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here.Starting from the estimated parameters for the PSD of the colored noise, the time domainrepresentation n(t) is calculated. It then becomes possible to validate the complete model,adding the calculated time signal n(t) to the syntheticechogram. For this purpose an acceptableestimation of the PSD is obtained with the LSMYWE algorithm, choosing the order of the ARbranch 6 and of the MA branch 4 (ARMA [6-4]). In fig. 4.a comparison is made between theestimated PSD with ARMA modellisationand with the periodogram.

Overlay periodogram and ARMA [6 4]

2000 4000 6000

frequency (Hz)8000 10000

fig. 4.a : Estimated PSD with ARMA modellisation and with the periodogram.

The fundamental components, appearing in the estimated PSD with the ARMA representationas well as with the periodogram, are located at the same frequency values, but their amplitudesare different.

The synthetic echogram itself includes already the effect of the TVG (Time Varying Gain)power amplifier [10]. In fig. 4.b the measured reflectogram is presented, in order to comparethis with the synthetic echogram of fig. 4.c, including the effects of the measurementenvironment Although only a qualitative correspondence is observed, this modelling approachbrings forward an excellent tool to refine the proposed model.This simulation is carried out with the following numericalvalues:

10

0-I:Q."

-10......

.":I....

-20....

E<

-30

-400

layerdepth density quality velocity

(m) (kglm1 factor (m/s)

water 10 1030 40 1500

mud 0.8 1200 20 1600

sand 00 1970 0.2 1800

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Measured reflectogram

-1

-20.00 0.01 0.02 0.03 0.04

time ( s )0.05 0.06

Fig. 4.b : Measured reflectogram.

Mode1 ya1idation

400

300

200

- 100....'-'... 0

-100

-200

-JOO .

0.00 0.01 0.02 0.03 0.04

time ( s )0.05 0.06

Fig. 4.c : Synthetic echogram includingthe effect of the TVG amplifier and the colored noisesource n(t).

5. CONCLUSION

This paper has treated the modelling of the power spectral density of correlated noise sourcesin acoustic measurements in a marine environment. Classical estimators, such as theperiodogram as well as modern spectral estimators based on an ARMA approach bave .beendiscussed. It is shown that low order parametric models are adequate to dcscribethe .and measured correlated noise sources. These models have made itt\mbc=-

3

2

1-....-...0

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validate the synthetic echograms, which are required for the development of algorithms solvingthe inverse problem.

MPENDIX 1

The defInition of the periodogram is :

N-I

PperCf)= ~ I L,x[nl exp(-j21tf) 12n=O

(1)

To improve the statistical properties, i.e the bias and the variance, of the periodogram we canuse K independent data records of the same random process [4]. Then, the averagedperiodogram estimator can be applied as

K-I

P AvPer(f) = L,P~~~(f)m=O

(2)

The data blocks can be adjacent or may overlap, usually by 50% or 75 %. Also data windowscan be used to reduce leakage by reducing the sidelobelevels of the spectral estimates.

ApPENDIX 2

The ARMA model is represented as shown in figure 1:

The system function H(z) between the input u[n] and the output x[n] for the ARMA process isthe rational function

H( ) = B(z)z A(z)

where A(Z)=Z{ ~a[k]z-k}

(1)

andB(z) =z{ ~b[k]Z -k}

A(z) is the AR-branch and p the AR order, while B(z) represents the MA-branch and q the MAorder, or the number of Moving Average parameters used in the model. Assuming the zeros ofA(z) are situated in the unit circle, the filter H(z) becomes stable and causal. Theautocorrelation function (ACF) of the output signal can then be written as

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urn]

x[n]

1

L

Fig. 1 : Representation of an ARMA process.

*P xx(z) = H(z) H (z) Puu(z) (2)

ApPENDIX 2

An AR-process is a process where all the b[k] coefficients except b[O]=l are zero in the ARMAmodel. Thus,

p

x[n] = - La[kJx[n-k] + urn]k=l

(1)

It is also called an all-pole model and the PSD is given by

20-

PAR(0 = 1 A(f) 12(2)

Because of the all-pole character an AR model can be used to estimate peaky spectra.The autocorrelation method [4] is based on solving the Yule-Walker equations given by

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A

rxx[O]

A

. rxx[-(p-l)]A

a[1]

A

r xx[1]

- (3)

A

r xx[p-l]

A

r xx[O]

A

a[p]

A

r xx[p]

These equations express the relationship between the model parameters and an estimate of theautocorrelation function. In fact, the AR parameters are estimated by minimizing an estimate ofthe so called prediction error power

1 pp = N L I x[n] + La[k]x[n-k] /2

n=-oo k=l(4)

The only difference between the covariance method [4] and the autocorrelation method is therange of summation in the prediction error power

N-l P

P = ~L /x[n] + La[k]x[n-k] 12n=p k=1

(5)

For an AR process the optimal forward predictor is

p

~[n] =- La[k]x[n-k]k=l

(6)

while the optimal backward predictor is

p

~[n] = - La *[k]x[n+k]k=l

(7)

The modified covariancemethod [4] estimates the AR parameters by minimizing the average ofthe estimated forward and backward prediction error powers or

A 1 A fA

P =l( P + pb) (8)

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In contrast to the autocon-elation, covariance, and modified covariance methods, whichestimates the AR parameters directly, the Burg method [4] estimates the reflection coefficients.Those are obtained by minimizing the prediction en-orpower for different order predictors in arecursive manner. The AR parameters are then derived from the estimates of the reflectioncoefficients [4].

REFERENCES

[1] Fredric J. Hams, "On the use of windows for harmonic analysis with the discreteFeurier transform", proceedings of the IEEE, Vol. 66, No. 1, 1978.[2] L. Van Biesen, L. Peirlinckx, S. Masyn and S. Wartel, "Modelling of Multi-LayerMarine Geological Deposit Systems and the Computer Simulation of the Acoustic PropagationPhenomena by Synthetic Generation of Echograms", Proceedings of the European SimulationCongress, p.87-93, Edinburgh, 5-8 September, 1989.[3] R. Pintelon, Y. Rolain, M. Vanden Bossche and J. Schoukens, "Towards an IdealData Acquisition Channe1.", IEEE Trans. Instrum. Measurm., Vol IM-39, n° 1, February1990.[4] Steven M. Kay, "Modern Spectral Estimation", Prentice-Hall, 1988.[5] J.P. NOTton,"An introduction to identification", Harcourt Brace Jovanovich, 1986.[6] James Kulubi, Leo Van Biesen, "A framework for knowledge based spectrumanalysis", accepted for publication in the proceedings of the IEEE-IMTC-90 conference.[7] D.C. Ganley, "A method for calculating synthetic seismograms which include theeffects of absorption and dispersion.", Geophysics, Vol.46. No.8; Aug. 1981.[8] WaIter 1. Futterman, "Dispersive Body Waves.", J. of Geophysical Research,V01.67,No. 13,Dec.1962.[9] P.R. Gutowski and S.Treitel, "The generalized one-dimensional syntheticseismogram", Geophysics, Vo1.52,No.5, May 1987[l0] O.R.E., "Operating and Maintenance Manual", Model 140, October, 1975.