USE OF RICIAN DISTRIBUTIONS TO PREDICT THE DISTRIBUTIONS OF ULTRASONIC FLAW SIGNALS IN THE PRESENCE OF BACKSCATTERED NOISE I. Valda, F. 1. Margetan, and R. B. Thompson Center for NDE Iowa State University 1915 Scholl Road Ames, IA 50011 INTRODUCTION In assessing the probability of detection (POD) of flaws during ultrasonic inspection, one needs to know the distribution of signal strengths that would be produced by flaws of the same nominal size [1]. Several factors contribute to creating such a distribution, including variations in flaw morphology, variations in the position of the flaw with respect to the insonifying beam, modifications of the beam by the microstructure, and the interaction between the flaw signal and noise signal backscattered from microstructural inhomogeneities. This paper deals with the latter mechanism, developing models which predict the distribution of signal in the presence of noise in terms of independent measures of the noise and the noise free flaw response. In its most elementary form, the problem is illustrated by the addition of a noise free signal, So, and a noise signal, Nr/ B Elementary trigonometry shows that, if 9 is a uniformly distributed random variable, and S = So + Nr/ B In agreement with intuition, <S>ISo approaches unity for high signal-to-noise (SNR) ratios and <S> approaches N 2 for low signal-to-noise ratios. A more sophisticated analysis has been developed in the radar and communication communities [2]. Considering a process in which signal and noise add linearly (as in the above example), and assuming, normally distributed noise and envelope detection, it has been shown that the resulting distribution has the following form (1) (2) Review of Progress in Quantitative Nondestructive Evaluation, Vol17 Edited by D.o. Thompson and D.E. Chimenti, Plenum Press, New York, 1998 105
fir) A =0 --~~----~~~~----------~------~~----~r o o Figure 1 A sketch of the Rician distribution for three signal-to-noise ratios. known as the Rician distribution. Here, A is the noise free envelope of the signal, a is the standard deviation of the noise, and 10 is the modified Bessel function of the first kind and zero order. Figure I sketches this function. When A=O, it has the form of the Rayleigh distribution while, when A2/2cf» 1 (high SNR), it approaches a normal distribution centered on A. In this paper, we consider the degree to which the Rician distribution is applicable to an ultrasonic inspection situation in which the signal is broadband, the noise may be neither white nor normally distributed, and gated peak-to-peak detection is used. The latter is selected because of its importance in industry, such as in the inspection of titanium billet material from which the rotating components of aircraft engines are fabricated [3]. DIRECT COMPARISON OF GATED, PEAK-TO-PEAK SIGNALS TO RICIAN DISTRIBUTION To ascertain the degree to which the Rician distribution is applicable to the ultrasonic case, an empirical study was conducted in which a scaled broadband "signal" was added to each of 252 noise waveforms taken from a Ti-17 sample at a selected time. The noise characteristics of this sample (AI) have been discussed elsewhere [4]. The "signal" was taken to be a reflection from a planar surface, scaled in accordance with the defmition where Vpp represents the peak-to-peak voltage within the gate. Insonification was with a 5 MHz, 0.74 in. diameter, 7.85 in. geometrical focal length probe. For various values of X, the scaled reference was added to each of the 252 noise waveforms, the peak-to-peak value of the superimposed waveforms within the gate was recorded, and a histogram of the result was constructed. These histograms were compared to the Rician distribution, where A was taken to be the peak-to-peak signal amplitude and a the rms noise level, as determined previously [4]. For the case of an 0.75 gate selected near the focal plane of the transducer, Figure 2 shows the results for X=O.I, 1, and 10. Note from Eqn. (3) that X is a noise-to-signal ratio, so high values of X correspond to low values of SNR. Examination of Figure 2 shows that agreement between the Rician distribution and the empirical distributions appears reasonable at high SNR (X=O.I), but that there is obviously a problem at low SNR (X=lO). (3) 106
Sla--_..,.,. S9*+-HoIM dialrtluuon. TI-17 A1 : PO, 1 T\.17 A1:,1 _0 -. -~,...- - ~,.»0r------------------------. ~.r---------------------~ (a) (b) ~.NoIM dilatrlbuuona fl.17.1:.-100.. ~0r-------~------------, 0.. (c) Figure 2 Comparison of Rician distribution to empirically determined distributions of signal and noise in Ti-l7. Gate width = 0.75 J.lsec. (a) x = O.l ; (b) x = 1.0; (c) x = to.o. 450 300 0 Sig.al+Noise distributions in ESaW gate T~17 A1 :.=10 0 0 Histogram 0 0 Riclan Fit 0 H(r') 150 00 +---------+--o<~~----- 000 005 r' 010 Figure 3 Comparison of Rician distribution to empirically determined distributions of signal and noise in Ti-17. Gate width = 0.185 J.lsec and x = to.o. 107
In assessing the possible causes of this difficulty, it was postulated that the length of the gate was playing an important factor. For sufficiently low SNR, one is essentially looking at the noise only. Margetan has shown that the distribution of gated peak-to-peak noise can be modeled by taking twice the maximum value of N independent samples of the distribution which governs the noise envelope observed at an instant of time [5]. N is determined by the ratio of the gate width and the equivalent square wave (ESQ) duration of the input pulse and can be thought of as the number of times that the noise assumes independent values within the gate. For the case under consideration, the ESQW duration of the input pulse was 0.185 flsec. Figure 3 shows the improved agreement for X=lO when the signal-plus-noise was sampled in such a gate. EXTENDED RICIAN MODEL FOR GATED PEAK-TO-PEAK DETECTION Based on these results, an extended Rician model has been developed. As illustrated in Figure 4, the total time gate is divided into a set of intervals whose length is the duration of the ESQW. In constructing the gated peak-to-peak noise model discussed above, one constructed the distribution by independently sampling the instantaneous noise envelope distributions in each of the intervals. The gated peak-to-peak noise distribution was then determined by a histogram of the maximum of these sampled distributions. For the signal-plus-noise case, the same basic idea is used. However, in one of the time intervals, the distribution of signal-plus-noise replaces the noise distribution. In our first implementation of this idea, it was assumed that the underlying statistics governing the instantaneous noise was normal. Figure 4 independent. I time gate I I I I + + ESQW I Gate divided into intervals in which signal and noise can be considered to be Then the noise envelope governed by the Rayleigh distribution, was independently sampled in all but one interval. In the remaining interval, the Rician distribution was sampled, based on the procedures described in the previous section. Again, the maximum value for all intervals was selected. Note that, for low SNR, this reduces to the procedure previously employed for noise only. For high SNR, this reduces to a simple Rician distribution, since the maximum will nearly always fall in the interval selected to represent the signal. Figures 5-7 represent the fit of this modified Rician distribution to data on 3 samples, obtained by the procedures discussed previously with a gate width of 0.75Ilsec. Figure 5 contains the data previously discussed to Ti-17 sample AI. Figure 6 contains results for Ti-64 sample KI. Figure 7 contains results for a stainless steel sample. Although no statistical measures of degree of fit have been employed, it is evident that the fit is much better for this extended Rician than for the simple Rician, particularly for low SNR. 108
lo. SJgnal+Noise dd:tributlom 11-17 A1 : :1""0.'\.. ~------,-~-."." l ". "" Slg "'1I_1oM 1\.\7 ",: ~,.O "'- - """'-." 010 003 SigIYl+ NoIse dii!.tl.bulloo!i Ti-t1 M : r1gg o _ -. Ho.. (a) (b) (c) Figure 5 Comparison of extended Rician distribution to empirically determined distributions of signal and noise in Ti-17. Gate width = 0.75I-1sec. (a) x = 0.1; (b) x = 1.0; (c) x = 10.0. " Slgf\l.l+Nocw dlstribu'\ic)ns 'JI.64 Kt: 11:"'0.1,., StgnaHNOIIH da1uoo\.ons r.. 6of Kl ~ :1""1.0 O)a..,,.,,.. =_- -1. 001) Sigl\ll.~ dl'slflbulions n.64 K1 : x.'0.0.. L--""''-- ~ _.. - Q.-.cIA.c..M1 Cit! (a) (b) (c) Figure 6 Comparison of extended Rician distribution to empirically determined distributions of signal and noise in Ti-6AI-4V. Gate width = 0.75I-1sec. (a) x = 0.1; (b) x = 1.0; (c) x = 10.0. 109
Signal+No[se distributions Stainless St~t!t x~o _1 Signal+Noise distributions Stainms Steel: x:1.0 200r-.----------------------------, 300r-------------------------------, b 150 " " " ~ 100 '00.0 00 O' D. o. 00 07 000 005 0,0 0.. 300 S;gnal+NoIse distributions Stainless Steel: ""10.0 "., " _ GM"'~ ~==1 200 C :r '.0 00 000 005 0'0 0', Figure 7 Comparison of extended Rician distribution to empirically determined distributions of signal and noise in stainless steel Gate width = 0.75 flsec. (a) x = 0.1; (b) x = 1.0; (c) x = 10.0. 110
18.,.. Signllt+Noise distributions lor SHA'5 Slgnal+Nolso dis'ribu'lons lor SHA *3 \-~ Eo I dit.arom "C M~ _Vpo ". ". c r I. 0 s b 00 DO 02 O. O..0 o. 02 0'.' Signal+_ distributions Slgnal+NoIse dislributions lor SHA for SHAI2 '8.,.. aiii.. 1ibIn I W_uredVpp 120 l- a...-/ o ~uredv~ 12. C r,. 10.0 '0 D. 02 00., c DO d 0, 0' Figure 8 Comparison of extended Gaussian distribution to experimentally observed signals of synthetic hard-alpha inclusions of various sizes (diameter=height). (a) #2; (b) #3; (c) #4; (d) #5. Our final initial test of this approach made use of experimental waveforms obtained on a sample containing 8, nominally identical synthetic hard-alpha inclusions of each of the sizes #2, #3, #4, and #5 [6]. In Figure 8, the fitted Rician is plotted as the solid line. The individual peak-topeak values are indicated by the square symbols plotted along the abscissa. In the development of the Rician distribution, cr was deduced from experimental measurements of the noise distribution and A was predicted theoretically [6]. The agreement is not quite as good as for the previously presented results, in part because of some significant outliers. It is possible that these might be a consequence of fabrication errors. In any case, we consider the results to be quite encouraging for a first attempt with no adjustable parameters. CONCLUSIONS An extended Rician model for the distribution governing gated peak-to-peak detection of signal in the presence of noise has been developed and compared to experimental data. The initial comparisons are quite favorable. Future work involves considerably more experimental validation, including the effects of greater gate widths, with model modifications perhaps required as a result of those comparisons. ACKNOWLEDGEMENT This work was supported by the Engine Titanium Consortium under the Federal Aviation Administration Grant No. 94-G-048. III
REFERENCES 1. W. D. Rummel, G. L. Hardy, and T. D. Cooper, "Applicability ofnde Reliability to Systems", in Metals Handbook, Ed. 9, Vol. 17, Nondestructive Evaluation and Quality Control (ASM, Metals Park, QH, 1989), pp. 674-684. 2. M. Schwartz, Information Transmission, Modulation and Noise (McGraw-Hill, NY, 1980), pp.377-382. 3. E. 1. Nieters, R. S. Gilmore, R. C. Trzaskos, J. D. Young, D. C. Copley, P. 1. Howard, M. E. Keller, and W. 1. Leach, "A Multizone Technique for Billet Inspection", in Review of Progress in Quantitative Nondestructive Evaluation, Vol. 14, D. Q. Thompson and D. E. Chimenti, eds. (Plenum Press, NY, 1995), pp. 2137-2144. 4. 1. Yalda, F. J. Margetan, K. Y. Han, and R. B. Thompson, "Survey of Ultrasonic Grain Noise Characteristics in Jet Engine Alloys", in Review of Progress in Ouantitative Nondestructive Evaluation, Vol. 15, D. Q. Thompson and D. E. Chimenti, eds. (Plenum Press, NY, 1996), pp. 1487-1494. 5. F. 1. Margetan, I. Yalda, and R. B. Thompson, "Predicting Gated-Peak Grain Noise Distributions for Ultrasonic Inspections of Metals", in Review of Progress in Quantitative Nondestructive Evaluation, Vol. 15, D. Q. Thompson and D. E. Chimenti, eds. (Plenum Press, NY, 1996), pp. 1509-1516. 6. C.-P. Chiou, F. J. Margetan, R. B. Thompson, "Modeling of Ultrasonic Signals from Weak Inclusions", in Review of Progress in Quantitative Nondestructive Evaluation, Vol. 15, D. O. Thompson and D. E. Chimenti, eds. (Plenum Press, NY, 1996), pp. 49-55. 112