Enright, Michael A.

Neal, Steven P.

Speckman, Paul L.

Steven P. Neal

Paul L. Speckman

Michael A. Enright

Crossref

Error Analysis and Ultrasonic Scattering Amplitude Estimation Using the Wiener Filter with Limited Prior Information

In ultrasonic nondestructive evaluation (NDE), flaw characterization is inhibited by the effects of the measurement system and by acoustic noise due to non-flaw related scattering of the sound. The Wiener filter can be formulated to optimally remove the effects of the measurement system and suppress the noise; however, prior information must be available about the noise and flaw distributions, respectively. The objective of this research is to develop an approach for the optimal implementation of the Wiener filter given prior noise information but no prior flaw information

Iowa State University Computer Science Technical Reports

Loper, K

HKU Scholars Hub

Title Toward Comprehensive Refugee Legislation in Hong Kong?Reflections on Reform of the 'Torture Screening' ProceduresAuthor(s) Loper, KCitation Hong Kong Law Journal, 2009, v. 39 n. 2, p. 253-259Issued Date 2009URL http://hdl.handle.net/10722/92988Rights Creative Commons: Attribution 3.0 Hong Kong LicenseVol 39 Part 2 Hong Kong refugee torture screening procedures 253CommentToward Comprehensive refugee LegisLaTion in hong Kong? refLeCTions on reform of The “TorTure sCreening” proCedures■Kelley Loper*Introductionthis comment reflects on the Hong Kong government’s announcement that it plans to reform its “torture screening” mechanism and provide a statutory framework for the consideration of claims under Article 3 of the Conven-tion against torture and other Cruel, Inhuman or Degrading treatment or Punishment (torture Convention). I argue that the new mechanism should go beyond the requirements of the torture Convention and also examine claims for protection from non-refoulement (non-return) based on other standards in customary international law and the International Covenant on Civil and Political Rights (ICCPR).BackgroundAs a result of the Court of Final Appeal’s (CFA) judgment in Secretary for Security v Sakthevel Prabakar (Prabakar)1 in 2004, the Hong Kong Government established an administrative screening procedure to pre-vent the refoulement (return or expulsion) of individuals to states where they would be in danger of torture. the Government had articulated its policy to respect its obligation under Article 3 of the torture Convention, which prohibits refoulement,2 in its report to the United nations Com-* Assistant Professor, Director of the LLm in Human Rights Programme, Deputy Director of the Centre for Comparative and Public Law, Faculty of Law, the University of Hong Kong. the author would like to thank Simon nm Young for his helpful comments on an earlier draft of this com-ment.1 [2005] 1 HKLRD 289; [2004] HKeC 678.2 Article 3(1) of the torture Convention provides that “no State Party shall expel, return (‘re-fouler’) or extradite a person to another State where there are substantial grounds for believing that he would be in danger of being subjected to torture”. Article 3(2) states that “For the purpose of determining whether there are such grounds, the competent authorities shall take into account all relevant considerations including, where applicable, the existence in the State concerned of a consistent pattern of gross, flagrant or mass violations of human rights”.04_Comment.indd   253 10/20/09   12:04:09 PM254 Kelly Loper (2009) HKLJmittee against torture in 1999.3 the court cited common law principles of procedural fairness and held that high standards of fairness are required when considering torture claims since “[a] determination under the [non-refoulement] policy was plainly of momentous importance to the individual concerned” and that “[l]ife, limb and [the claimant’s] fundamental right not to be subjected to torture was involved …”.4the CFA, however, concluded that the Hong Kong Government’s sole reliance on determinations of refugee status made by the Hong Kong Sub-office of the United nations High Commissioner for Refugees (UnHCR) was insufficient. the UnHCR’s refugee status determination procedures are designed to prevent the refoulement of refugees as defined by article 1(A)(2) of the Refugee Convention.5 Article 33 of the Convention prohib-its the expulsion or return of refugees to the frontiers of territories where their lives or freedom would be threatened on account of race, religion, nationality, membership of a particular social group or political opinion (the Convention grounds). Although valid claims under the torture Con-vention may also fall within the refugee framework, this is not always the case. While torture would certainly amount to “persecution”, it may not be based on one of the Convention grounds and the torture Convention does not require this nexus. After the establishment of the government’s torture screening procedures, the UnHCR’s refugee status determination process has continued to operate separately and in parallel. many claimants have accessed both mechanisms, either simultaneously or sequentially, and this overlap has created delays and duplication of resources, and has been blamed for encouraging disingenuous claims.In December 2008, the Court of First Instance, in a strongly worded judgment, held that the Immigration Department’s torture screening procedures did not conform to the fairness requirements set out in the Pra-bakar judgment.6 Saunders J ruled that a number of procedural limitations, including the lack of publicly funded legal assistance, were unlawful. In re-sponse, the government halted further screening of claims and announced in February 2009 that it would consider developing a legislative framework governing the processing of torture claims.7 In July 2009, the Security 3 See China’s third periodic report under the torture Convention, Un doc. CAt/C/39/Add.2, 4 may 1999, para 122, submitted pursuant to Art 19 of the torture Convention. the Committee against torture is the body which monitors the implementation of states’ obligations under the Convention. 4 See n 1 above.5 According to Art 1(A)(2) of the Refugee Convention, a “refugee” must have a well-founded fear of persecution based on one of five grounds (race, religion, religion, nationality, membership of a particular social group or political opinion). 6 FB v Director of Immigration [2008] HKeC 2072.7 Security Bureau, Hong Kong SAR Government, Paper for discussion by the Legislative Council Panel on Security on 3 February 2009, LC Paper no CB(2)737/08-09(03), January 2009, Annex B.04_Comment.indd   254 10/20/09   12:04:10 PMVol 39 Part 2 Hong Kong refugee torture screening procedures 255Bureau further indicated that it would implement “enhanced screening procedures” by September or october 2009 and would brief the Legislative Council’s panel on security on its plans to introduce legislation by the end of 2009.8 At the same time, the government reiterated its position that it is not bound by the Refugee Convention and has no obligations to handle refugee status determination.9 this position has been challenged, however, in C v Director of immigration.10 In that case, the applicants argued that Hong Kong has an obligation under customary international law to refrain from the refoulement of refugees and must therefore establish refugee status determination procedures. Although the applicants lost at first instance, their appeal was considered.Hong Kong’s Non-Refoulement ObligationsAlthough Hong Kong has consistently resisted application of the Refugee Convention to its territory, it nevertheless has international and domes-tic human rights obligations which provide considerable protection from refoulement for claimants. Indeed, Hong Kong’s legal obligations extend beyond Article 3 of the torture Convention and are also based on broader standards reflected in the ICCPR and customary international law. the new statutory screening procedures in Hong Kong should incorporate these more comprehensive standards. they should be designed to prevent the return of individuals to territories where they could face a range of human rights vio-lations, including torture, arbitrary deprivation of life, cruel, inhuman, and degrading treatment or punishment, and other forms of persecution.The ICCPR, the Basic Law and the Bill of Rightsthe Human Rights Committee, the body which monitors states’ imple-mentation of their obligations under the ICCPR, has interpreted the Covenant to include an implicit right to non-refoulement when “there is a real risk that [an individual’s] rights under the Covenant will be violated in another jurisdiction”.11 Articles 6 and 7 are particularly relevant. Article 6 provides for a right to be free from arbitrary deprivation of life and Article 7 prohibits torture or cruel, inhuman, or degrading treatment or punish-8 the Security Bureau provided a further update on the development of new screening procedures on 29 September 2009. See Security Bureau, Hong Kong SAR Government, “torture Claim Screening mechanism: Latest Progress,” 29 September 2009, LC Paper no. CB(2)2514/08-09(01).9 Ibid.10 C v Director of Immigration [2008] HKCU 256.11 Human Rights Committee, A.R.J v Australia, Communication no 692/1996, Un doc CCPR/C/60/D/692/1996, 11 August 1997, para 6.8.04_Comment.indd   255 10/20/09   12:04:10 PM256 Kelly Loper (2009) HKLJment. the Committee’s jurisprudence clarifies that both of these provisions contain an implied right to non-refoulement. For example, in one of its General Comments, the Committee indicates that “States parties must not expose individuals to the danger of torture or cruel, inhuman or degrading treatment or punishment upon return to another country by way of their extradition, expulsion or refoulement” and that they “should indicate in their reports what measures they have adopted to that end”.12Articles 6 and 7 have been incorporated into Hong Kong law by Article 39 of the Basic Law and are duplicated in Articles 2 and 3 of the Hong Kong Bill of Rights ordinance. Article 28 of the Basic Law contains similar articulations of fundamental human rights and also arguably provides for implicit protection for individuals fearing serious human rights violations upon return to their countries.13 In Ubamaka Edward Wilson v Secretary for Security, Reyes J recognised that the implicit right to non-refoulement in Hong Kong law includes a right not to be returned to inhuman treatment of a severity proscribed by the [Bill of Rights], the ICCPR and [the torture Convention]”.14Customary International Lawthere is general consensus that a principle of non-refoulement exists in customary international law.15 While it reflects, in part, the obligation ex-pressed in Article 33 of the Refugee Convention its content goes beyond this more narrow formulation. According to a comprehensive analysis by Lauterpacht and Bethlehem, this rule obligates states to ensure that “no person shall be rejected, returned, or expelled in any manner whatever where this would compel him or her to remain in or return to a territory where substantial grounds can be shown for believing that he or she would face a real risk of being subjected to torture or cruel, inhuman or degrading treatment or punishment...”16 In addition, “no person seeking asylum may be rejected, returned, or expelled in any manner whatever where this would compel him or her to remain in or to return to a territory where he or she may face a threat of persecution or a threat to life, physical integrity, or liberty”.1712 Human Rights Committee, General Comment 20, 1992, para 9.13 Article 28 provides the right to inviolability of the freedom of the person and prohibits arbitrary or unlawful arrest, detention or imprisonment, arbitrary or unlawful search of the body, deprivation or restriction of the freedom of the person, torture, and arbitrary or unlawful deprivation of life.14 [2009] HKeC 710 at [111].15 Although a minority of scholars have argued otherwise.16 Sir elihu Lauterpacht and Daniel Bethlehem, “the scope and content of the principle of non-re-foulement: opinion”, para 253(a). this analysis was cited favourably by Hartmann J in C v Director of Immigration (see n 10 above), at [103].17 See Lauterpacht and Bethlehem (cited above, n 16), para 253(b).04_Comment.indd   256 10/20/09   12:04:10 PMVol 39 Part 2 Hong Kong refugee torture screening procedures 257Hong Kong’s Court of First Instance held that rules of customary inter-national law which do not conflict with domestic law are automatically incorporated into Hong Kong law.18 Although the court concluded that Hong Kong law is inconsistent with a rule of non-refoulement specific to the Refugee Convention, it did not consider the implications of the broader obligations discussed above which reflect human rights standards clearly ap-plicable to Hong Kong.Overlap Between Refugees and “Human Rights” ClaimantsAlthough Hong Kong is not bound by the Refugee Convention, the ex-plicit and implicit rights to non-refoulement in the torture Convention and the ICCPR overlap considerably with the protection offered by Article 33 of the Refugee Convention. the Convention does not define the term “persecution” but recent state practice indicates that it amounts to serious human rights violations. James Hathaway’s suggestion that persecution is “the sustained or systemic violation of basic human rights demonstrative of a failure of state protection” has been accepted by a number of domes-tic courts.19 the UnHCR’s Handbook on Procedures and Criteria for Determining Refugee Status states that serious violations of human rights constitute persecution.20 As a result, claims made under the Refugee Con-vention are often determined with reference to international human rights standards.International and Comparative ResourcesIn addition to the scope of the non-refoulement obligations implemented by the new screening system, a number of other substantive and procedural is-sues need to be resolved during the legislative process. A few of these issues include: (1) the evidence necessary to establish the claim; (2) methods of assessing a claimant’s credibility; (3) the role of legal representatives and the training and qualifications necessary to ensure that the representation 18 C v Director of Immigration (cited at n 10 above) at [80] and [82]. the court relied on Trendtex Trad-ing Corporation v Central Bank of Nigeria [1977] 1 QB 529.19 James C. Hathaway, The Law of Refugee Status (Butterworths: 1991), pp 104–105.20 UnHCR, “Handbook on Procedures and Criteria for Determining Refugee Status under the 1951 Convention and the 1967 Protocol relating to the Status of Refugees”, HCR/IP/4/eng/ReV.1, 1979, re-edited January 1992. 04_Comment.indd   257 10/20/09   12:04:10 PM258 Kelly Loper (2009) HKLJmeets the required high standards of procedural fairness;21 (4) the necessary burden and standard of proof; and (5) determining the necessary degree of risk facing the individual.the jurisprudence of the Un Committee against torture, the Human Rights Committee, and domestic courts, as well as procedural guidelines produced by the UnHCR could usefully inform Hong Kong’s approach to these and other issues. For example, the Committee against torture, the monitoring body for states’ implementation of the torture Convention, has published views on communications it has received by individuals alleging state violations of Article 3. In several cases, the Committee has provided guidance on the evidence necessary to establish a torture claim. It has clarified that “[t]he existence of a consistent pattern of gross, flagrant or mass violations of hu-man rights in a country does not as such constitute sufficient grounds for determining whether the particular person would be in danger of being sub-jected to torture upon his return to that country; additional grounds must be adduced to show that the individual concerned would be personally at risk”. At the same time, however, “the absence of a consistent pattern of gross violations of human rights does not mean that a person cannot be considered to be in danger of being subjected to torture in his or her specific circumstances”.22the Committee’s jurisprudence also assists when determining how to approach the assessment of a claimant’s credibility and indicates that deci-sion makers must bear in mind the difficulties and trauma faced by torture victims. Although there may be contradictions and inconsistencies in the claimant’s story “complete accuracy is seldom to be expected by victims of torture and that such inconsistencies as may exist in the author’s presen-tation of facts are not material and do not raise doubts about the general veracity of the author’s claims”.23Conclusion: Toward a Comprehensive FrameworkWhen developing the new statutory determination mechanism, the Hong Kong government should implement its broader obligations of non-refoulement in addition to the explicit obligation in Article 3 of the torture 21 this issue will be the subject of a comparative study of legal representation for asylum seekers in other jurisdictions conducted by the author and co-investigator Simon n.m. Young. the study is funded by a Public Policy Research Grant awarded by the Research Grants Council in 2009.22 See, for example, Committee against torture, Sadiq Shek Elmi v Australia, Communication no 120/1998, may 1999, para 6.4 and R. K. et al v Sweden, Communication no 309/2006, para 8.2.23 CAt Communication no. 21/1995, para 11.3.04_Comment.indd   258 10/20/09   12:04:10 PMVol 39 Part 2 Hong Kong refugee torture screening procedures 259Convention. Specifically, the new legislation should provide protection for claimants based on the full range of applicable standards found in custom-ary international law (including the non-refoulement of refugees), Article 3 of the torture Convention, Articles 6 and 7 of the ICCPR, Articles 28 and 39 of the Basic Law, and Articles 2 and 3 of the Bill of Rights. the upcoming legislative process provides an opportunity for Hong Kong to develop a fair, efficient screening procedure to ensure the protec-tion of individuals who risk a range of serious human rights violations in their countries of origin. to achieve these objectives, the Hong Kong government should conduct regular consultation with – and facilitate par-ticipation by – non-governmental organizations and legal representatives working with asylum seekers and torture claimants. It also needs to establish a single mechanism which combines all of its non-refoulement obliga-tions.24 the continuing existence of separate refugee and torture screening procedures implemented by different decision-making bodies wastes resources and generally impedes the interests of justice and procedural fair-ness. Incorporating broader non-refoulement standards into the screening procedures at the start could also avoid potential judicial review challenges grounded in constitutional rights and the ICCPR. It would also prevent inequality of treatment since those relying upon ICCPR grounds currently need to apply for judicial review, which can be costly and time-consuming, while those making a torture claim can access a process of administrative remedies before going to the court as a last resort.24 there has been a trend toward combining refugee and human rights standards in one status deter-mination procedure in a number of jurisdictions including new Zealand and Canada.04_Comment.indd   259 10/20/09   12:04:10 PM

Toward Comprehensive Refugee Legislation in Hong Kong? Reflections on Reform of the 'Torture Screening' Procedures

In ultrasonic nondestructive evaluation (NDE), flaw characterization is inhibited by the effects of the measurement system and by acoustic noise due to non-flaw related scattering of the sound. The Wiener filter can be formulated to optimally remove the effects of the measurement system and suppress the noise; however, prior information must be available about the noise and flaw distributions, respectively. The objective of this research is to develop an approach for the optimal implementation of the Wiener filter given prior noise information but no prior flaw information.</p

Neal, Steven

Speckman, Paul

Enright, Michael

Digital Repository @ Iowa State University (ISU)

ERROR ANALYSIS AND ULTRASONIC SCATTERING AMPLITUDE ESTIMATION USING THE WIENER FILTER WITH LIMITED PRIOR INFORMATION INTRODUCTION Steven P. Neal Department of Mechanical and Aerospace Engineering University of Missouri-Columbia Columbia, MO 65211 Paul L. Speckman Department of Statistics University of Missouri-Columbia Columbia, MO 65211 Michael A. Enright Computer Sciences Corporation Herdon, VA 22071 In ultrasonic nondestructive evaluation (NDE), flaw characterization is inhibited by the effects of the measurement system and by acoustic noise due to non-flaw related scattering of the sound. The Wiener filter can be formulated to optimally remove the effects of the measurement system and suppress the noise; however, prior information must be available abeut the neise and flaw distributions, respectively. The objective of this research is to develop an approach for the optimal implementation of the wiener filter given prior noise information but no prior flaw information. This paper beg ins with a background section in which necessary models are formulated, previous work is discussed, and the objective of the research is restated. The estimation error is then stated and analyzed. Finally, three techniques for the optimal application of the Wiener filter with limited prior information are formulated and tested. BACKGROUND A measured flaw signal can be modeled in the time domain as the convolution of the measurement system impulse response function, H(t), with the flaw's impulse response function, R(t), plus noise, n(t). The model can be written as F(t) = H(t)*R(t) + n(t) (1) Review of Progress in Quantitative Nondestructive Evaluation, Vol. 11 Edited by D.O. Thompson and D.E. Chimenti, Plenum Press, New York, 1992 991 The measured signal, F(t), is a blurred and noise-corrupted estimate of the flaw's true signature, R(t). The model can be stated in the frequency domain as F(oo) = H(oo)A(oo) + n(oo) (2) Here, F(oo) is a band-limited, distorted, and noise-corrupted estimate of the flaw's true frequency domain signature, A(oo), known as the flaw's far-field scattering amplitude. The problem is to estimate A(oo). A common approach is via the frequency domain application of a filter of the Wiener filter variety [1-9]. The Wiener filter can be formulated based on the model stated in Eq. (2) with A(oo) and n(oo) taken to be uncorrelated, complex Gaussian random variables with known power spectral density functions and known mean [2,5,6,10,11]. Using ~(oo) to denote an estimate of A(oo), the Wiener filter as formulated by Neal and Thompson [5,6] specifically for the scattering amplitude estimation problem can be stated as F (00) H* (00) Q2 (00) E [A (00) ] ~(oo) + (3) IH (00) 1 2 + Q2 (00) 1 H (00) 12 + Q2 (00) where E[A(oo)] is the expected value or mean of the scattering amplitude distribution, and Q2(00), known as the desensitization term, desensitizes the deconvolution to division by near-zero values in IH(OO) 12 . For the filter to be applied optimally, Q2(00) must be equal to Sn (oo)/SA(oo) where these are the. power spectral density functions of the noise and scattering amplitude, respectively. Since the noise has zero mean, Sn (00) is the average power spectrum of the noise. The Wiener filter is typically applied with E[A(oo)] equal to zero at all frequencies and Q2(00) is taken to be a non-optimally chosen, frequency independent constant [1-6]. The most commonly used value for Q2 has been 0.01 times the maximum value of IH(oo) 12 [3-6]. This non-optimal form of the Wiener filter can be written as F (00) H* (00) ~(oo) where Q2 (4) The Wiener filter with Q2=0.01(IH(00) 12 )max works weIl in removing the effects of the measurement system, but it is independent of noise and is thus unacceptable for noise-corrupted data [5,6]. Neal and Thompson [5,6] have shown that the optimal wiener filter works weIl even for noise-corrupted data, but it requires SA(oo) and E[A(oo)] as input and thus has limited applicability. The objective of this research is to investigate the application of the Wiener filter with Q2 (00) =ASn (00) . In this form, the frequency independent constant, A, replaces the unknown l/SA(oo), and E[A(oo)] is set equal to zero since it is assumed that its actual value is unknown. This non-optimal form of the Wiener filter can be stated as F (00) H* (00) ~(oo) (5) IH (00) 1 2 + ASn (00) The goal is to choose A in order to minimize the scattering amplitude estimation error for this form of the wiener filter. This goal will be 992 pursued in two major tasks: 1) analysis of the estimation error and 2) formulation and application of techniques for optimally choosing A. ERROR ANALYSIS Since the task is to choose that A at which the error between the actual scattering amplitude for a given flaw, A(m), and the estimate, ~(m), is minimum, a plot of estimation error versus A should show a clear and unique minimum. The error used here was determined by summing IA(m)-~(m) 1 2 over the bandwidth of the system response for a 15 MHz planar transducer [6]. Errors were determined by summing from approximately 2-18 MHz. Using ~i to denote the i th flaw, the sum of squared errors for the i th flaw can be stated as lerrorl 2 = I. IA(m,~il-~(m'~i) 12 m (6) The empirical or sampie mean square error (SMSE) determined by averaging the sum of squared errors over N flaws can be stated as MSE 1 N I. (I. JA(m'~i)-~(m,U J2) N i=l m (7) The theoretical MSE can be calculated by evaluating E[lerrorI 2]. It is assumed that each flaw to be considered is out of a flaw distribution with SAlm) and E[A(m)] unknown. In general, E[A(m)] is non-zero. Therefore, by forcing the estimate to zero when Q2(m) is large relative to IH(m) 12, the filters given in Equations (4) and (5) are introducing a bias to the estimate. This bias is represented by E[A(m)]2 in Eq. (8). From Equations (3) and (6), a fairly straightforward calculation shows that the theoretical MSE is given by Sn (m) 1 H (m) 12 (SA (m) + E [A (m) ]2) Q4 (m) MSE ------------------- + ------------------------ (8) (IH(m) 1 2 + Q2(m»2 The theoretical MSE for the most commonly applied form of the Wiener filter with Q2(m)=0.01(IH(m) 12 )max can be evaluated by substituting this value for Q2(m) into Eq. (8). The theoretical MSE for the form of interest with Q2(m)=AS n (m) can be similarly evaluated. Having stated the theoretical error, the next steps are to 1) evaluate the behavior of the error as a function of A to determine if it has a clear and unique minimum, and 2) compare the theoretical error with empirically determined errors from actual noise-corrupted flaw signals. Noise-corrupted flaw signals were generated using measured acoustic noise from a stainless steel block [11], the measurement system response for an actual ultrasonic system, and computer genera ted flaw signatures [5,6]. The noise-corrupted flaw signal for the i th flaw was created as F(t'~i)=H(t'~i)*R(t'~i)+Bn(t'~i) where B is a scale factor used to scale the noise in order to achieve the desired S/N. The acoustic noise power spectrum, Sn(m), was estimated from measured noise signals [11] and was scaled by B2 when the noise was scaled by B. Flaw sizes were chosen at random from an assumed lognormal distribution of spherical voids in stainless steel with a mean radius of 250 ~ and a standard deviation of 25 ~ [5,6,10]. Three different signal-to-noise ratios (10/1, 2/1, and 1/1) were considered. Representative noise-corrupted flaw signals are shown in Fig. 1 for each S/N. 993 .g S/N=2/1 .3 <.) ] S/N=1O/1 <.) ~ :.::l 0. ~ .U :.::l 0. ~ 13 ~ 8 ö. ~ ] Ci! § 1 ~ +---------------~ z+-______________ ~ Z +-______ ~ ______ __; Oe+O Time (s) 2e-6 Oe+O Time ( ) 2c-6 Oc+O Time (s) 2e-6 Fig. 1. Noise-corrupted flaw signals for three signal-to-noise ratios. The graphs in Fig. 2 show the theoretica1 average error versus A for Q2=0.01(IH(OO) 12 )max and Q2(OO)=ASn (OO), respectively, for the 2/1 and 1/1 S/N cases . Also shown is the empirical average error for a sampIe of 50 noise-corrupted flaw signals. Note three points: 1) the graphs of error versus A show clear and unique minima; 2) the theoretical and empirical errors agree weIl; and 3) as the S/N decreases, the potential advantage of incorporating Sn(OO) into the Wiener filter becomes clear. SCATTERING AMPLITUDE ESTlMATION The problem is to choose A such that the error with Q2 (OO)=ASn (OO) is near the minimum error for this form of the filter. While the plots in Fig . 2 demonstrate that an optimum value of A exists, for the realistic ca se of scattering amplitude estimation for an unknown flaw out of an unknown distribution, neither the theoretical nor the empirical error can be calculated. The theoretical error cannot be calculated since SA(OO) and E[A(OO)] are not known, and the empirical error obviously cannot be calculated since the quantity to be estimated, A(OO), is unknown. Therefore, the task is to find some other function which has a minimum near the value of A at which the minimum in the actual error curve occurs. This function must be such that it can be calculated based on the known quantities F(W), H(W), and Sn(W) . 0.012 0.010 .... ~ lI.l 0.008 <l.l ~ 0.006 ~ 0.004 ..q:: 0.002 0.000 --;' ~ Fig . 2 . 994 0.012 "T"""--------- - ------, S/N=2/l .... 0 Ji <l.l C>() ~ ~ ..q:: omo 0.008 0.006 0.004 0.002 0.000 S/N=l/l 0 '" ,... ... .,.., .., .... 00 --;' 0 '" ,... ... .,.., .., ..... 00 ~ ~ ~ 0 0 ~ 0 0 ~ - ).. - ~ ~ ~ o <;) ~ ~ ~ ~ - ).. -Comparison of theoretical (solid) and empirical (dotted) MSE. The data for the filter with Q2=O.01(IH(OO) 12)max are given by the frequency independent lines. ~ 0.005 -r---------__, 0.004 ~ 0.003 E ~ 0.002 0.001 Actual Error 0.006 .,..-----------, -< 0.005 0.004 ~ 0.003 ::;;, 0.002 0.001 12~---------, Unbiased Risk for F 10 8 6 4+----1000 -r-----------, 800 600 400 200 o Likelihood Function -200 +--r--r--r-"""T"--r"-T~---I 10°101102103104105106107108 A. Fig. 3. Actual error and estimation function value versus L Three approaches were considered. For each approach, it is assumed that H(oo) is known and fixed (i.e., not a random variable), F(oo) has been measured, and n(oo) is an uncorrelated, complex normal random variable with zero mean and variance given by Sn (00) . The assumption relative to A(oo) is stated below for each technique. Due to space limitations, only the resultant function is stated for each approach. The summations in each function are over the frequency range previously indicated. Unbiased Risk for A(W) (UBRA) For this approach, it is assumed that A(oo) is unknown but fixed. The function to be minimized, denoted UBRA(A), is given by where <1>2 = (I H (00) 12 +ASn (00) ) 2 and 1"(00) =H (00) ßI (00) is an estimate of H(oo)A(oo). This estimate is motivated by Bowman [12] for the related density estimation problem, and it can be shown that E[UBRA(A)]=MSE. Unbiased Risk for F(W) (UBRF) For this approach, A(oo) is again assumed unknown but fixed. The resultant function, denoted UBRF(A), is where <I> = IH (00) 1 2+ASn (00). UBRF is similar to UBRA except that the target function is the theoretical MSE for H(oo)A(oo) rather than A(oo) . 995 ... 0.008 ~ S/N=l/l ~ 50 rt .8 S/N=l/l \\) 0.007 t I:>() ~ ~ 0.006 ~ 40 ~ ~ ~ ~ 0.005 ~ - ~ :~ 30 0.004 .--~ ~ 0.003 :.:5 20 10 2 10 3 A 10 4 10 5 10 2 10 3 A 104 10 5 ... 60 ~ 0.0025 ~ S/N=2/1 .8 S/N=2/l '-tl ... ~ I.J ~ 40 ~ ~ ~ 0.0020 ~ ~ ] ~ 20 .!:: ]1 ~ 0.0015 :.:5 0 10 2 A 104 10 5 10 2 10 3 A 10 4 10 5 ... ~ ~ 90 ~ 1.098e-4 S/N=1O/1 ~ S/N=lO/l . .:: 70 \\) I.J ~ ~ I.;; 50 ~ ~ ~ 1.088e-4 ~ 30 -8 .t:: - 10 ! ~ 1.078e-4 .....;j -10 10 2 10 3 A 104 10 5 10 2 10 3 A 10 4 10 5 Fig. 4. Empirical average error and empirical average likelihood function versus A. Maximum Likelihood Estimation for A (MLE) A(ffi) is assumed to be an uncorrelated, complex normal random variable with zero mean and frequency independent variance given by A-l. The resultant function is (11) Each of the two summation terms has been multiplied by (-1) so that the function is to be minimized rather than maximized as imp1ied by MLE. The use of maximum likelihood here is related to the method of generalized maximum 1ikelihood (GML) given by Wahba [13]. The graphs in Fig. 3 are resu1ts for a single flaw and for S/N=2/1. The actua1 errar and the function for each approach are plotted versus A. The function va1ue is irrelevant since only the value of A at the minimum of the function is important. As shown, the function for each approach is well-behaved and shows a clear minimum near the value of A at which the minimum in the actual error curve occurred. Figure 4 shows empirical average results for each of the three signal-to-noise ratios considered. The vertical scale has been set differently for each plot to clearly show the minimum. The average as a 996 0.010 Optimal --e-- OSI. -+- OpLimal 0.003 UBRF .... ] MI.E e 0.008 ----- UBRF UBRA ~ -+- MI.E ~ ----- UBRA '" 0() 0() ~ 0.006 ~ ~ ~ ~ ~ O. 2 "8 0.004 ~ '1::: . !:::; ! 0.002 ~ 0.000 0.001 0 2 3 4 5 6 7 8 9 10 0 2 3 SIN SfN Fig. 5. Empirical average error versus signal-to-noise ratio. function of A was determined by averaging over the 50 noise-corrupted flaw signals. Results are shown for the maximum likelihood approach only. Similar results were obtained for the other two approaches. As the S/N changes, the function minimum for each technique identified approximately the optimum A remained virtually stationary as did the minimum of the empirical average error. Figure 5 shows empirical mean square errors versus S/N based on the sample of 50 flaws. For each signal, five wiener filter variations were applied: 1. Q2=0.01(IH(Ol)1 2 )max 2. Q2(Ol)=AS n (Ol); A chosen by UBRA 3. Q2 (Ol)=AS n (Ol); A chosen by UBRF 4. Q2(Ol)=AS n (Ol); A chosen by MLE of A 5. Q2 (Ol)=AS n (Ol); A chosen to minimize the actual (unobservable) error ... this is the optimal A The left graph in Fig. 5 shows results for all five filter variations at each of the three signal-to-noise ratios considered. The right graph shows on1y the three estimation techniques involving Q2(Ol)=AS n (Ol) and the minimum error for the 1 / 1 and 2/1 S/N cases. For the high S/N case, any of the techniques is acceptable. For the low S/N ratio cases, estimates with Q2(Ol)=AS n (Ol) and A determined by any of the estimation techniques is significantly better than estimates for the most commonly used form of the filter with Q2=0.01(IH(Ol) 12 ) max. On the average, the maximum likelihood estimation approach appears to be slightly superior to the other two estimation approaches. SUMMARY AND DISCUSSION The significance of this research is that it provides a method of incorporating prior noise information, in the form of an estimate of the average noise power spectra, into the Wiener filter, and, thus, make the filter sensitive to increased noise-corruption. 997 Future research will focus in a number of directions: 1) further studies of the three data-driven methods for choosing Ä, 2) formulation of the data driven methods for the two-dimensional case where spacial rather than temporal frequency is of interest, 3) investigation of the application of the Wiener filter with no prior information where Q2=Ä and the choice of Ä is data-driven, 4) modeling of the noise as an auto-regressive process in order to improve the characterization of Sn(oo), and 5) investigation of data-driven adaptive models for SA(oo) and IA(OO) 12 in order to allow Q2(00) to more closely approximate the frequency dependence of Q2(00) for the optimal wiener filter. REFERENCES 1. Murakami, Y., B. T. Khuri-Yakub, G. D. Kino, J. M. Richardson, and S. G. Evans. 1978. Appl. Phys. Lett. 33(8) :685-687. 2. Elsley, R. K., J. M. Richardson, and R. C. Addison. 1980. p. 916-921. In B. R. McAvoy (ed.) Ultrasonic symposium proceedings. Vol. 2. IEEE, New' York. 3. Addison, R. C., R. K. Elsley, and J. F. Martin. 1982. p. 251-261. In D. o. Thompson and D. E. Chimenti (ed.) Review of progress in quantitative NDE. Vol. 1. Plenum Press, New York. 4. Thompson, R. B., and T. A. Gray. 1982. p. 233-249. In D. o. Thompson and D. E. Chimenti (ed.) Review of progress in quantitative NDE. Vol. 1. Plenum Press, New York. 5. Neal, S. P., and D. o. Thompson. 1988. p. 857-864. In D. o. Thompson and D. E. Chimenti (ed.) Review of progress in quantitative NDE. Vol. 7A. Plenum Press, New York. 6. Neal, S. P. 1988. A prior knowledge based optimal wiener filtering approach to ultrasonic scattering amplitude estimation. Doctoral Dissertation. Iowa State University, Ames, IA. 7. Chen, C. H. and S. K. Sin. 1990. J. Acoust. Soc. Am. 87(3) :976-987. 8. Frock, B. G., P. Karpur, and M. J. Ruddell. 1991. p. 781-788. In D. o. Thompson and D. E. Chimenti (ed.) Review of progress in quantitative NDE. Vol. 10A. Plenum Press, New York. 9. Mitche11, K. W. and R. S. Gilmore. 1992. A true Wiener filter implementation for improving signal to noise ratio and resolution in acoustic images. This proceedings. 10. Neal, S. P., and D. o. Thompson. 1990. p. 633-640. In D. o. Thompson and D. E. Chimenti (ed.) Review of progress in quan-titative NDE. Vol. 9A. Plenum Press, New York. 11. Neal, S. P., and D. O. Thompson. 1990. p. 625-632. In D. o. Thompson and D. E. Chimenti (ed.) Review of progress in quan-titative NDE. Vol. 9A. Plenum Press, New York. 12. Bowman, A. 1984. Biometrika 71:353-360. 13. Wahba, G. 1985. Ann. Statist. 13:518-521. 998 

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