- Email: [email protected]
Empirical modeling of ultrasonic phenomena
Ultrasonics
34 (1996) 451-454
Empirical modeling of ultrasonic phenomena Igor Grabec a*b*,Wolfgang Sachse b aFaculty of Mechanical Engineering, University of Ljubljana. PO Box 394, 61001 Ljubljana, Slouenia b Theoretical and Applied Mechanics, Cornell University, Ithaca, New York 14853-1503,
USA
Abstract An exact treatment of ultrasonic phenomena is based on a quantitative description of measured displacement or velocity vector fields, but the corresponding mathematical relation can be derived from the partial differential equations of elastodynamics only for idealized and rather simple examples. To avoid these limitations, we have investigated empirical methods, which stem from either a non-parametric or a parametric regression and they can be related to simulations of artificial neural networks. The nonparametric approach is convenient for a general, non-linear modeling while the parametric one is more suitable for linear modeling. Here we summarize the associated procedures and describe their applicability with examples that include both active and passive ultrasonic phenomena. Keywords:
Modeling of ultrasonic phenomena; Neural networks
An exact treatment of ultrasonic phenomena is based on a quantitative description of measured displacement or velocity vector fields. It is well known that this field is related to the properties of the source and the specimen. However, the corresponding mathematical relation can be derived from the partial differential equations of elastodynamics only for idealized and rather simple examples. In order to avoid the difficulties inherent in a strict, analytical description of ultrasonic phenomena, yet to provide a quantitative treatment, we have studied a number of empirical methods in the last decade [l-5]. These methods stem from either a non-parametric or a parametric regression and they can be related to simulations of artificial neural networks. The non-parametric approach is convenient for a general, non-linear modeling while the parametric one is more suitable for linear modeling. Our aim in this note is to summarize the associated procedures and to describe their applicability with examples that include both active and passive ultrasonic phenomena. These methods have been applied to obtain a description of discrete events corresponding to the scattering of ultrasound by defects in materials, the generation of acoustic emission signals resulting from the fracture of stressed solids [l] and the con-
* Corresponding author. Fax: + 386-61-21-85-67; e-mail: [email protected]. 0041-624X/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0041-624X(95)00107-7
tinuous acoustic emissions accompanying chaotic manufacturing processes [ 2,3]. Empirical modeling relies on information obtained from previous observations of the same phenomenon. For ultrasonic phenomena, it is analogous to the human recognition of voices [ 11. The ultrasonic measurement system consists of an array of sensors and a computer with a modeling algorithm. This algorithm stems from a statistical treatment of empirical data [ 11. The empirical model is based on learning from examples and it is developed in a computer. During measurements the sensors provide empirical data comprising ultrasonic signals as well as the variables that describe the properties or characteristics of the source or the medium. These data are either stored in the memory as a database or employed to estimate missing ultrasonic waveform data from the characteristics of the source or the medium. Alternatively, they can be used to estimate the characteristics of the source or the medium from the measured waveform data. Because this estimation proceeds autonomously, we call such a system an automatic modeler. We briefly review here the operation of the automatic modeler. An observation of the ultrasonic phenomenon is described by a data vector which comprises M measured components: x = (xi, . . . , xM). These components represent digitized ultrasonic signals as well as
452
I. Grabec, W Sachsel Ultrasonics 34 (1996) 451-454
signals representing source and sample properties [ 11. From repeat experiments, we obtain a set of N samples An empirical model represents the {x1, x2, ‘.. 9 xN}. statistical relations between these components. In the non-parametric approach these relations are estimated from a joint probability distribution of the components. For this purpose, the automatic modeler must be able to determine from the samples the probability distribution of the variable x and to estimate from the measured components of the data vector the missing components in accordance with the stored probability distribution. The density of the joint probability distribution of the variable x is empirically estimated by the expression [l]
P(x)=; f W(X--A n-1
(1)
in which w(x -s) denotes a smooth approximation of the delta function. The Gaussian function w(x - s) = exp[lx - ~1~/2a~], in which the parameter czcorresponds to the average distance between samples, can be used. The probability density p(x) represents an empirical model of the phenomenon which can be applied to estimate the complete vector x from incomplete measurements. For this purpose we assume that the sensors provide partial data described by a truncated vector g=(x,, . . . . x,; #), from which is to be estimated the complementary, or hidden data that is represented by h=(#;x,+i, . ..> xM). Here # denotes the missing components. The vector h is optimally estimated by the conditional average, which is expressed in terms of the prototypes as [l] c(g) = c A,(g)h, where A,(g) = n
w(g -g*) T w(g - g!J.
taining four horizontal, side-drilled holes of different diameters and whose acoustic properties were varied by filling them with various fluids. The ultrasonic pulses were launched and detected by a transducer placed directly above each hole. The signals from the ultrasonic transducer were recorded on a waveform digitizing system with the start of each record synchronized with the start of the ultrasonic excitation pulse. The original digitized waveform was reduced to a 128-component data vector. The reflector radius and the inclusion material were encoded by two descriptors into the field of the last 22 components of the data vector. The acquired database is shown in Fig. 1, while the test ultrasonic signals and corresponding descriptors are shown in Figs. 2 and 3. The initial portions of the records correspond to the scattered ultrasonic signal. The position of the first and the second peak of the descriptors is the encoded hole diameter and the inclusion material, respectively. In this example we extract from the scattered signals an estimate of the size of the scatterer and the wavespeed of the inclusion material. This example is just one in a general class of
(7-J
The estimator b(g) is determined by a linear superposition of truncated prototype vectors h,. The coefficients A,(g) are highly nonlinear functions which represent a measure of similarity between the input vector g and the training samples g,. Eq. (2) corresponds to a non-parametric regression and coincides with the mapping relation of a radial basis function neural network which, in turn, can be shown to be equivalent to a three-layer perceptron. When the number of samples increases without limit, a saturation of the system memory occurs. This can be prevented by adapting a fixed number of prototype data vectors. The adaptation is driven by the sensory signal x and it is modified by a self-organized interaction between the stored prototype data, which can be interpreted as the contents of memory cells [4]. The operation of the modeler has been demonstrated in processing the ultrasonic signals scattered by discontinuities and inhomogenities in materials cl]. The experiment was performed on an aluminum block con-
Fig. 1. Database
of the scattering
experiment.
Fig. 2. Partial ultrasonic scattered signals used in testing the empirical modeling. Only the signals scattered by the inclusion were input as given data in the recall of scatterer descriptors.
I. Grabec, W. Sachse / Ultrasonics 34 ( 1996) 451-454
Fig. 3. Test descriptors
of the scattering
experiment.
inverse problems. Its solution is found by using the modeler to process the information in the scattered pulses. When test signals are to be processed to estimate the source descriptors, the completed signals shown in Fig. 4 are obtained. Good agreement between the true and estimated descriptors indicates the applicability of the empirical approach to model ultrasonic phenomena and to solve the inverse problem. The same modeler has also been successfully applied to model discrete acoustic emission phenomena and to obtain the solution of corresponding direct and inverse problems [ 11. In the foregoing example, the initial echo as well as the scatterer characteristics were recalled from the other scattered signals. This demonstrates that the modeler can also be applied to estimate the properties of one portion of a signal from another portion. Because of this property, the modeler is capable of predicting even a chaotic time series from the partial input and stored signals in the database [3]. For this purpose, the data vector is represented by a sequence of successive digitized values of the ultrasonic signal x(t) = (x,, x, _ 1, . . . , x,_ M). By recording such vectors at different times, a database is generated. Using the conditional average, this database
can be used to forecast the ultrasonic signal at some given time to. To do this, the record of the ultrasonic signal in the time interval prior to t,: from t, - 1 to t, - M, must be provided as a condition. Using the estimated signal value x(to), we can create a condition at time t, to forecast the signal value at to + 1 etc. The forecasting method was tested on a chaotic acoustic emission signal that is generated during drilling. It was found experimentally that the discrepancy between the forecast and the observed signals increases on average approximately exponentially with the number of forecasting steps. Fig. 5 shows an example of the actual and the forecast acoutic emission signal generated during drilling [ 31. The characteristic increment of the forecasting RMS error is approximately equal to the Lyapunov coefficient of the chaotic signal. Because of the increasing error, the forecasting is, on average, acceptable only in a time interval whose length is approximately determined by the correlation time of the measured chaotic signal 1I31In addition to forecasting the acoustic emission signals, which are generated in chaotic manufacturing processes, a modification of the modeler that relies on the detected AE signals has been applied for the on-line estimation of machining parameters, such as tool sharpness [2]. In many cases of practical interest, the source of ultrasound is described by a force at the source position that is linearly related to the displacement field at a receiver position. The corresponding relation can be modeled parametrically using a linear multivariate regression [S]. For this purpose, we first describe the data representing the displacement and the force by two vectors: u and f, respectively and express the relation between them as u=Gf.
(3)
Here G can be interpreted as a response matrix of the system. The basic problem of modeling is then reduced to the estimation of this matrix from the set of experimental joint records {u,, f,; . . . ; uN, fN}. This generally represents an ill-posed problem, which can be regularized using the statistical approach. For this purpose we treat
I Fig. 4. Partial ultrasonic signals completed descriptors and the front surface reflected
with the recovered echoes.
scatterer
453
t-
Fig. 5. AE signal from a drilling process (lower record) and the forecast signal (upper record) as functions of time (horizontal axis). The correlation time of the chaotic signal is, in this case, approximately equal to four typical AE signal periods.
454
I. Grabec, W. Sachse / Ultrasonics 34 (1996) 451-454
the linear relation u = Gf as a definition of the estimator of the ultrasonic displacement. During learning, we require that the optimal response matrix on average yields a minimum of the discrepancy between the observed and the estimated displacement vector. This occurs when the response matrix satisfies the matrix equation: GR,, = R,,,
(4)
in which R, and R,, denote the auto- and crosscorrelation matrices of the force and displacement variables repectively. In order to avoid solving this matrix equation, one often proceeds with sequential modeling. In this case, it is assumed that each new presented experimental sample only slightly modifies the correlation matrices, which, in turn, is reflected in small modifications of the response matrix. In a first approximation, the change of the response matrix is then expresses as AG = @V)(u - Gf )f?
(5)
Here T denotes the transposed vector and cl(N) is the adaptation rate parameter whose value should be inversely proportional to the product of the number of samples N and the mean square value of the force signal f. Experiments have shown that such an approach is applicable to the solution of forward and inverse problems related to the description of simple acoustic emission phenomena, such as impacts on a structure [ 51.
References
Cl1 I. Grabec and W. Sachse, J. Appl. Phys. 69 (1991) 6233. PI E. Govekar and I. Grabec, J. Engineering for Industry 116 (1994) 233. c31 I. Grabec, Conf. Proc. Ultrasonics International ‘93 Vienna (Butterworth, London, 1993) 771. c41 I. Grabec, Biological Cybernetics 63 (1990) 403. CSI I. Grabec and W. Sachse, J. Acoust. Sot. Am. 85 (1989) 1226.
34 (1996) 451-454
Empirical modeling of ultrasonic phenomena Igor Grabec a*b*,Wolfgang Sachse b aFaculty of Mechanical Engineering, University of Ljubljana. PO Box 394, 61001 Ljubljana, Slouenia b Theoretical and Applied Mechanics, Cornell University, Ithaca, New York 14853-1503,
USA
Abstract An exact treatment of ultrasonic phenomena is based on a quantitative description of measured displacement or velocity vector fields, but the corresponding mathematical relation can be derived from the partial differential equations of elastodynamics only for idealized and rather simple examples. To avoid these limitations, we have investigated empirical methods, which stem from either a non-parametric or a parametric regression and they can be related to simulations of artificial neural networks. The nonparametric approach is convenient for a general, non-linear modeling while the parametric one is more suitable for linear modeling. Here we summarize the associated procedures and describe their applicability with examples that include both active and passive ultrasonic phenomena. Keywords:
Modeling of ultrasonic phenomena; Neural networks
An exact treatment of ultrasonic phenomena is based on a quantitative description of measured displacement or velocity vector fields. It is well known that this field is related to the properties of the source and the specimen. However, the corresponding mathematical relation can be derived from the partial differential equations of elastodynamics only for idealized and rather simple examples. In order to avoid the difficulties inherent in a strict, analytical description of ultrasonic phenomena, yet to provide a quantitative treatment, we have studied a number of empirical methods in the last decade [l-5]. These methods stem from either a non-parametric or a parametric regression and they can be related to simulations of artificial neural networks. The non-parametric approach is convenient for a general, non-linear modeling while the parametric one is more suitable for linear modeling. Our aim in this note is to summarize the associated procedures and to describe their applicability with examples that include both active and passive ultrasonic phenomena. These methods have been applied to obtain a description of discrete events corresponding to the scattering of ultrasound by defects in materials, the generation of acoustic emission signals resulting from the fracture of stressed solids [l] and the con-
* Corresponding author. Fax: + 386-61-21-85-67; e-mail: [email protected]. 0041-624X/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0041-624X(95)00107-7
tinuous acoustic emissions accompanying chaotic manufacturing processes [ 2,3]. Empirical modeling relies on information obtained from previous observations of the same phenomenon. For ultrasonic phenomena, it is analogous to the human recognition of voices [ 11. The ultrasonic measurement system consists of an array of sensors and a computer with a modeling algorithm. This algorithm stems from a statistical treatment of empirical data [ 11. The empirical model is based on learning from examples and it is developed in a computer. During measurements the sensors provide empirical data comprising ultrasonic signals as well as the variables that describe the properties or characteristics of the source or the medium. These data are either stored in the memory as a database or employed to estimate missing ultrasonic waveform data from the characteristics of the source or the medium. Alternatively, they can be used to estimate the characteristics of the source or the medium from the measured waveform data. Because this estimation proceeds autonomously, we call such a system an automatic modeler. We briefly review here the operation of the automatic modeler. An observation of the ultrasonic phenomenon is described by a data vector which comprises M measured components: x = (xi, . . . , xM). These components represent digitized ultrasonic signals as well as
452
I. Grabec, W Sachsel Ultrasonics 34 (1996) 451-454
signals representing source and sample properties [ 11. From repeat experiments, we obtain a set of N samples An empirical model represents the {x1, x2, ‘.. 9 xN}. statistical relations between these components. In the non-parametric approach these relations are estimated from a joint probability distribution of the components. For this purpose, the automatic modeler must be able to determine from the samples the probability distribution of the variable x and to estimate from the measured components of the data vector the missing components in accordance with the stored probability distribution. The density of the joint probability distribution of the variable x is empirically estimated by the expression [l]
P(x)=; f W(X--A n-1
(1)
in which w(x -s) denotes a smooth approximation of the delta function. The Gaussian function w(x - s) = exp[lx - ~1~/2a~], in which the parameter czcorresponds to the average distance between samples, can be used. The probability density p(x) represents an empirical model of the phenomenon which can be applied to estimate the complete vector x from incomplete measurements. For this purpose we assume that the sensors provide partial data described by a truncated vector g=(x,, . . . . x,; #), from which is to be estimated the complementary, or hidden data that is represented by h=(#;x,+i, . ..> xM). Here # denotes the missing components. The vector h is optimally estimated by the conditional average, which is expressed in terms of the prototypes as [l] c(g) = c A,(g)h, where A,(g) = n
w(g -g*) T w(g - g!J.
taining four horizontal, side-drilled holes of different diameters and whose acoustic properties were varied by filling them with various fluids. The ultrasonic pulses were launched and detected by a transducer placed directly above each hole. The signals from the ultrasonic transducer were recorded on a waveform digitizing system with the start of each record synchronized with the start of the ultrasonic excitation pulse. The original digitized waveform was reduced to a 128-component data vector. The reflector radius and the inclusion material were encoded by two descriptors into the field of the last 22 components of the data vector. The acquired database is shown in Fig. 1, while the test ultrasonic signals and corresponding descriptors are shown in Figs. 2 and 3. The initial portions of the records correspond to the scattered ultrasonic signal. The position of the first and the second peak of the descriptors is the encoded hole diameter and the inclusion material, respectively. In this example we extract from the scattered signals an estimate of the size of the scatterer and the wavespeed of the inclusion material. This example is just one in a general class of
(7-J
The estimator b(g) is determined by a linear superposition of truncated prototype vectors h,. The coefficients A,(g) are highly nonlinear functions which represent a measure of similarity between the input vector g and the training samples g,. Eq. (2) corresponds to a non-parametric regression and coincides with the mapping relation of a radial basis function neural network which, in turn, can be shown to be equivalent to a three-layer perceptron. When the number of samples increases without limit, a saturation of the system memory occurs. This can be prevented by adapting a fixed number of prototype data vectors. The adaptation is driven by the sensory signal x and it is modified by a self-organized interaction between the stored prototype data, which can be interpreted as the contents of memory cells [4]. The operation of the modeler has been demonstrated in processing the ultrasonic signals scattered by discontinuities and inhomogenities in materials cl]. The experiment was performed on an aluminum block con-
Fig. 1. Database
of the scattering
experiment.
Fig. 2. Partial ultrasonic scattered signals used in testing the empirical modeling. Only the signals scattered by the inclusion were input as given data in the recall of scatterer descriptors.
I. Grabec, W. Sachse / Ultrasonics 34 ( 1996) 451-454
Fig. 3. Test descriptors
of the scattering
experiment.
inverse problems. Its solution is found by using the modeler to process the information in the scattered pulses. When test signals are to be processed to estimate the source descriptors, the completed signals shown in Fig. 4 are obtained. Good agreement between the true and estimated descriptors indicates the applicability of the empirical approach to model ultrasonic phenomena and to solve the inverse problem. The same modeler has also been successfully applied to model discrete acoustic emission phenomena and to obtain the solution of corresponding direct and inverse problems [ 11. In the foregoing example, the initial echo as well as the scatterer characteristics were recalled from the other scattered signals. This demonstrates that the modeler can also be applied to estimate the properties of one portion of a signal from another portion. Because of this property, the modeler is capable of predicting even a chaotic time series from the partial input and stored signals in the database [3]. For this purpose, the data vector is represented by a sequence of successive digitized values of the ultrasonic signal x(t) = (x,, x, _ 1, . . . , x,_ M). By recording such vectors at different times, a database is generated. Using the conditional average, this database
can be used to forecast the ultrasonic signal at some given time to. To do this, the record of the ultrasonic signal in the time interval prior to t,: from t, - 1 to t, - M, must be provided as a condition. Using the estimated signal value x(to), we can create a condition at time t, to forecast the signal value at to + 1 etc. The forecasting method was tested on a chaotic acoustic emission signal that is generated during drilling. It was found experimentally that the discrepancy between the forecast and the observed signals increases on average approximately exponentially with the number of forecasting steps. Fig. 5 shows an example of the actual and the forecast acoutic emission signal generated during drilling [ 31. The characteristic increment of the forecasting RMS error is approximately equal to the Lyapunov coefficient of the chaotic signal. Because of the increasing error, the forecasting is, on average, acceptable only in a time interval whose length is approximately determined by the correlation time of the measured chaotic signal 1I31In addition to forecasting the acoustic emission signals, which are generated in chaotic manufacturing processes, a modification of the modeler that relies on the detected AE signals has been applied for the on-line estimation of machining parameters, such as tool sharpness [2]. In many cases of practical interest, the source of ultrasound is described by a force at the source position that is linearly related to the displacement field at a receiver position. The corresponding relation can be modeled parametrically using a linear multivariate regression [S]. For this purpose, we first describe the data representing the displacement and the force by two vectors: u and f, respectively and express the relation between them as u=Gf.
(3)
Here G can be interpreted as a response matrix of the system. The basic problem of modeling is then reduced to the estimation of this matrix from the set of experimental joint records {u,, f,; . . . ; uN, fN}. This generally represents an ill-posed problem, which can be regularized using the statistical approach. For this purpose we treat
I Fig. 4. Partial ultrasonic signals completed descriptors and the front surface reflected
with the recovered echoes.
scatterer
453
t-
Fig. 5. AE signal from a drilling process (lower record) and the forecast signal (upper record) as functions of time (horizontal axis). The correlation time of the chaotic signal is, in this case, approximately equal to four typical AE signal periods.
454
I. Grabec, W. Sachse / Ultrasonics 34 (1996) 451-454
the linear relation u = Gf as a definition of the estimator of the ultrasonic displacement. During learning, we require that the optimal response matrix on average yields a minimum of the discrepancy between the observed and the estimated displacement vector. This occurs when the response matrix satisfies the matrix equation: GR,, = R,,,
(4)
in which R, and R,, denote the auto- and crosscorrelation matrices of the force and displacement variables repectively. In order to avoid solving this matrix equation, one often proceeds with sequential modeling. In this case, it is assumed that each new presented experimental sample only slightly modifies the correlation matrices, which, in turn, is reflected in small modifications of the response matrix. In a first approximation, the change of the response matrix is then expresses as AG = @V)(u - Gf )f?
(5)
Here T denotes the transposed vector and cl(N) is the adaptation rate parameter whose value should be inversely proportional to the product of the number of samples N and the mean square value of the force signal f. Experiments have shown that such an approach is applicable to the solution of forward and inverse problems related to the description of simple acoustic emission phenomena, such as impacts on a structure [ 51.
References
Cl1 I. Grabec and W. Sachse, J. Appl. Phys. 69 (1991) 6233. PI E. Govekar and I. Grabec, J. Engineering for Industry 116 (1994) 233. c31 I. Grabec, Conf. Proc. Ultrasonics International ‘93 Vienna (Butterworth, London, 1993) 771. c41 I. Grabec, Biological Cybernetics 63 (1990) 403. CSI I. Grabec and W. Sachse, J. Acoust. Sot. Am. 85 (1989) 1226.