CHAPTER 8 ANALYSIS OF AVERAGE SQUARED DIFFERENCE SURFACES

Transcription

1 CAPTER 8 ANALYSS O AVERAGE SQUARED DERENCE SURACES n Chapters 5, 6, and 7, the Spectral it algorithm was used to estimate both scatterer size and total attenuation rom the backscattered waveorms by minimizing the average squared dierence (ASD) given by Equation (5.3). owever, the ASD surace over which the minimization is to be perormed has not yet been analyzed. the surace has a local minimum, then the global minimum could not be ound by the simple minimization routine that was utilized in the previous chapters and a more complicated implementation o the algorithm would be required. n addition, the properties o the ASD surace might provide insight into the puzzling dependence o the precision on the initial ka e values or dierent sources discussed in Section 7.3. ence, in this chapter, an analysis is perormed o the ASD surace. The analysis considers the impact scatterer size, hal-space attenuation, spectral variance resulting rom random scatterer spacing, requency range, ka e range, and initial requency have on the shape o the ASD surace. The analysis is done using the simulated backscattered waveorms rom the earlier chapters as well as through mathematical derivations. The changes o the shape o the ASD surace were then related to the precision o the Spectral it algorithm. 8.1 Properties o ASD Suraces rom Simulated Waveorms nitially, the eatures o the ASD surace were investigated using some o the simulated data generated or the hal-spaces discussed in the previous chapters. The investigation was done by observing qualitative changes in the overall structure o the surace with changes in the properties o the hal-space. An example ASD surace or a hal-space with an attenuation o 0.3 db/cm/mz containing Gaussian scatterers with eective radii o 5 µm at a density o 35/mm 3 is shown in igure 8.1. This surace was generated by solving or ASD using Equation (5.3) or dierent input values o scatterer size and total attenuation. n the igure, the colors correspond 14

2 to the 1/ASD values. Normally, the Spectral it algorithm would yield the scatterer size and total attenuation corresponding to the minimum ASD value. When generating this surace, P scat was ound by averaging in the normal spectral domain the irst 5 waveorms (1 st set o 40 sets) o the 1000 waveorms that had been generated or this hal-space using a source with iltering characteristics given by Equation (7.5). The waveorms had been windowed in the time domain by a hamming window with a length o.5 mm and the requency range selected, over which the mean value in Equation (5.3) was ound, was given by Equation (7.). Although the width o the trough (i.e., light red portion in image) was dierent or the dierent sets, as can be observed in Appendix, the suraces corresponding to the other sets were not qualitatively dierent rom the one shown. owever, the width o the trough did not correlate with the accuracy o the inal estimate (i.e., see 13 th and 14 th sets in Appendix ). igure 8.1: Example ASD surace or a hal-space with an attenuation o 0.3 db/cm/mz containing 5 µm scatterers generated rom 5 waveorms windowed with a.5 mm hamming window rom a source whose iltering unction () was given by Equation (7.5). Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. The minimum or this surace (dark red) occurs at a scatterer size o 6.6 µm. The ASD surace varies smoothly throughout the entire region with only one minimum. ence, our simple minimization routine employed in Matlab was appropriate. Also, there is a range o a e and α values or which the ASD values are relatively small as is seen by the parabolic light red region o the image. t was observed in Appendix that the location o the minimum is always ound along this parabolic trough, even when it does not occur at the correct 15

3 value or scatterer size and total attenuation. The parabolic trough is symmetric about a e = 0 µm because a e is squared in the evaluation o the Gaussian orm actor. O course, a negative value or scatterer size is not physical and hence would only be a mathematical peculiarity. Ater establishing the basic shape o the ASD surace, the inluence o the dierent halspace parameters on this basic shape was investigated. ence, an ASD surace, shown in igure 8., was generated corresponding to the irst 5 waveorms (1 st set o 40 sets) or a hal-space with a hal-space attenuation o 0.5 db/cm/mz that still contained Gaussian scatterers with eective radii o 5 µm at a density o 35/mm 3. The requency range was once again selected by Equation (7.) and a hamming window with a length o.5 mm was still used to gate the time domain signals. Now the parabolic trough intersects the a e = 0 µm axis at ~0.55 db/cm/mz. owever, the same change in scatterer size results in the same change in attenuation along the parabolic trough. ence, the attenuation appears to shit the surace to higher values o attenuation without aecting the overall shape o the surace. igure 8.: Example ASD surace or a hal-space with an attenuation o 0.5 db/cm/mz containing 5 µm scatterers generated rom 5 waveorms windowed with a.5 mm hamming window rom a source whose iltering unction () was given by Equation (7.5). Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. The minimum or this surace (dark red) occurs at a scatterer size o 18.0 µm. Ater assessing the impact o attenuation on the ASD surace, the eect o dierent scatterer sizes was evaluated. ence, ASD suraces, shown in igures 8.3 and 8.4, were 16

4 generated corresponding to two hal-spaces containing scatterers with a e o 5 µm and 45 µm, respectively. The attenuation o the hal-spaces was maintained at 0.3 db/cm/mz while the source and windowing parameters were the same as the previous two surace plots. Once again, the dierent scatterer sizes appear to primarily shit the ASD suraces along the α axis with the smaller scatterer sizes shiting the surace to smaller α values and the larger scatterer size shiting the surace to large α values. owever, the area colored light red in the igures relative to the true scatterer size appears to be much smaller as the scatterer size is increased. ence, the minimum is more pronounced or the larger sized scatterers. Once the impact o attenuation and scatterer size on the ASD surace was determined, the inluence o window length was investigated because the earlier results indicated that the precision o the estimates should be improved by increasing the length o the window used to gate the time domain signals. igure 8.5 shows the ASD surace or the same waveorms used to generate the ASD surace in igure 8., only with a hamming window o length o 8 mm instead o.5 mm. The ASD surace in igure 8.5 is indistinguishable rom the ASD surace in igure 8.. ence, although the larger window lengths improve the precision o the Spectral it algorithm, the improvement does not appear to result rom any changes to the overall structure o the ASD surace. owever, it may be that the changes are not visible in this simple qualitative comparison o the suraces. Ater exploring the impact o window length, the eect o dierent requency ranges on the ASD surace, while maintaining the same ka e range, was investigated. igure 8.6 shows the ASD surace corresponding to the irst 5 waveorms (1 st set o 40 sets) o the 1000 waveorms that had been generated or a hal-space with an attenuation o 0 db/cm/mz containing Gaussian scatterers with eective radii o 5 µm. The source used to generate the waveorms had iltering characteristics given by Equation (7.4), and the waveorms had been windowed in the time domain by a hamming window with a length o 3 mm. requencies corresponding to ka e values o 0.5 to 1.5 were used to ind the ASD. Likewise, igure 8.7 shows the ASD surace corresponding to the irst 5 waveorms (1 st set o 40 sets) o the 1000 waveorms that had been generated or a hal-space with an attenuation o 0 db/cm/mz containing scatterers with eective radii o 50 µm using the same source and window length. The requency range or igure 8.7 was given by the requencies corresponding to ka e values o 1 to (same initial requency and ka e range but dierent requency range). 17

5 igure 8.3: Example ASD surace or a hal-space with an attenuation o 0.3 db/cm/mz containing 5 µm scatterers generated rom 5 waveorms windowed with a.5 mm hamming window rom a source whose iltering unction () was given by Equation (7.5). Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. The minimum or this surace (dark red) occurs at a scatterer size o 13. µm. igure 8.4: Example ASD surace or a hal-space with an attenuation o 0.3 db/cm/mz containing 45 µm scatterers generated rom 5 waveorms windowed with a.5 mm hamming window rom a source whose iltering unction () was given by Equation (7.5). Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. The minimum or this surace (dark red) occurs at a scatterer size o 43.0 µm. 18

6 igure 8.5: Example ASD surace or a hal-space with an attenuation o 0.5 db/cm/mz containing 5 µm scatterers generated rom 5 waveorms windowed with a 8 mm hamming window rom a source whose iltering unction () was given by Equation (7.5). Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. The minimum or this surace (dark red) occurs at a scatterer size o 15.8 µm. igure 8.6: Example ASD surace or a hal-space with an attenuation o 0 db/cm/mz containing 5 µm scatterers generated rom 5 waveorms windowed with a 3 mm hamming window rom a source whose iltering unction () was given by Equation (7.4) using requencies in the ka e range o 0.5 to 1.5. Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. The minimum or this surace (dark red) occurs at a scatterer size o 18.4 µm. 19

7 igure 8.7: Example ASD surace or a hal-space with an attenuation o 0 db/cm/mz containing 50 µm scatterers generated rom 5 waveorms windowed with a 3 mm hamming window rom a source whose iltering unction () was given by Equation (7.4) using requencies in the ka e range o 1 to. Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. The minimum or this surace (dark red) occurs at a scatterer size o.0196 nm. By comparing igures 8.6 and 8.7, one can observe that the ASD surace has once again been shited to larger α values or the larger scatterer size. urthermore, the parabola describing the parabolic trough has been broadened when a smaller requency range was used (igure 8.7). ence, a change in a e produces a smaller change in α along the trough in igure 8.7 as compared to igure 8.6. Also, the minimum is more pronounced or the smaller sized scatterers when the larger requency range was selected. When compared to the results in igure 8.4, this suggests that the more pronounced minimum observed earlier or the larger scatterer sizes was probably due to the increased ka e range just as the more pronounced minimum in this case is probably due to the increased requency range. Another eect observed in Chapter 7 was the dependence o the precision on the initial requency used in the minimization routine (i.e., Section 7.3). This dependence was also shown to be related to the peak requency o the ideal backscattered spectrum. ence, the next goal was to determine i the dependence on the initial requency and the corresponding relationship to the peak o the backscattered spectrum could be observed by analyzing the appropriate ASD suraces. 130

8 irst, only the dependence on the initial requency was considered by using the same waveorms associated with the ASD surace shown in igure 8.6 in order to generate igure 8.8. This time, however, the requency range used in the ASD calculation was given by the requencies corresponding to ka e values o 1.5 to.5 instead o ka e values o 0.5 to 1.5. All other hal-space and processing parameters were the same. ence, the only dierence between the two suraces would be the initial requency selected or the calculation. An initial ka e value o 1.5 corresponds to an initial requency ater the deviation peak (good precision or this source) while an initial ka e value o 0.5 corresponds to an initial requency three-quarters o the way up the deviation peak (poor precision or this source), as can be observed in igure 7.8. The ASD surace in igure 8.8 has been shited to a slightly higher α value, and the parabola describing the parabolic trough has been signiicantly narrowed (change in a e produces a larger change in α along trough). Also, the minimum has become much more pronounced or the surace corresponding to the larger initial requency. igure 8.8: Example ASD surace or a hal-space with an attenuation o 0 db/cm/mz containing 5 µm scatterers generated rom 5 waveorms windowed with a 3 mm hamming window rom a source whose iltering unction () was given by Equation (7.4) using requencies in the ka e range o 1.5 to.5. Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. The minimum or this surace (dark red) occurs at a scatterer size o 1.9 µm. Now the impact o the iltering characteristics o the source can be considered by generating the example ASD surace shown in igure 8.9 or a source with iltering 131

9 characteristics given by Equation (7.1) with a σ R o 18 Mz and a o o 8 Mz. All o the halspace and processing parameters will be the same as the surace shown in igure 8.8, including the selection o requencies corresponding to ka e values o 1.5 to.5. An initial ka e value o 1.5 corresponds to an initial requency close to the top o the deviation peak (poor precision) or this source as can be observed in igure 7.8. The parabola describing the parabolic trough has basically the same width and position in igure 8.9 as in igure 8.8. Also, the minimum is much less pronounced or the ASD surace shown in igure 8.9 as compared to igure 8.8 and is about as pronounced as the minimum in igure 8.6. owever, the reason or the peak being more pronounced in igure 8.8 cannot be deduced based on a qualitative comparison o the dierent ASD suraces. igure 8.9: Example ASD surace or a hal-space with an attenuation o 0 db/cm/mz containing 5 µm scatterers generated rom 5 waveorms windowed with a 3 mm hamming window rom a source whose iltering unction () was given by Equation (7.1) with a σ R o 18 Mz and a o o 8 Mz using requencies in the ka e range o 1.5 to.5. Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. The minimum or this surace (dark red) occurs at a scatterer size o 5.6 µm. 8. Mathematical Derivation and Analysis o deal ASD Suraces n the previous section, the impacts o attenuation, scatterer size, window length, requency range, initial requency, and source iltering unction were investigated by looking at example ASD suraces or some o the previously simulated waveorms. To better understand these example suraces, a mathematical description o the surace is derived in this section. The 13

10 derivation can then be used to provide additional insight into the minimization o the ASD surace Derivation o ASD surace To begin the derivation o the surace, assume the expression or X(,a e,α o ) rom Equation (5.4) can be written as. ka d i b g 0 87 b g d i 4 c h G ka CPreb g sb ge e 1 b g = ln ln maxcpscatb G gh J max Prebg e G = G 087 π c sbg h J d real e i 4 Tbαreal αog + e α ozt P P e e scat re X, ae, α o = ln ln max Pscat. kae α ozt G b g J max 0 87 G Pre b ge d i 4 e b g d i real real T e o T α z 087. ka 4α z Pre e e ln. a a z C, c d J e o T 087. ka 4α z where C 1 and C are constants independent o requency related to the acoustic concentration and transmission coeicient or the intervening tissue layers, a real and α real are the real values o scatterer size and attenuation respectively, and s () are the spectral luctuations related to the random scatterer spacing. ence, X becomes and yielding d e i bg Xa b π, αo = mean ln s. areal ae mean G 0 87 c J b 4z α α mean + C T real o c h d i g c bgh c bgh G J d i g π X X = ln s mean ln s 087. areal ae mean c 4z α α mean T real o i e J (8.1) (8.) (8.3) 133

11 c h c bgh c bgh b g 4 π + G areal ae c J 087. d i mean b gd i X X = s s zt real o + ln mean ln 16 α α mean + zt G π. real o areal ae c J α α mean mean c sbg h c sbg h ln mean ln 087. π G c J da real ae i mean lnc 4z sbgh mean lncsbgh Tbαreal αog mean. Thereore, the mean o c X Xh is given by ASD mean ln mean ln = L N M O s s QP G d c bghi c bgh J 4 π + G areal ae c J d i mean mean G J + zt real o 16 bα α g mean mean G J π + zt G. real o areal ae c J bα α gd i mean mean mean G 087 π J. dareal ae i mean lncsbg h mean mean lncsbg h c 8z α α mean ln mean mean ln Tb real og c sb gh c sb gh. (8.4) (8.5) Equation (8.5) could be simpliied i ln( s ()) did not have any consistent dependence on cbg s h c sbg requency allowing m ean ln = mean mean ln and mean ln = mean mean ln. n order to determine i it is reasonable to cbg s h c sbg h assume that ln( s ()) is independent o requency, the simulated backscattered waveorms generated or a hal-space with an attenuation o 0 db/cm/mz containing 5 µm scatterers at a density o 35/mm 3 were considered. The source had iltering characteristics given by Equation (7.4). The waveorms were windowed in the time domain with a hamming window with a length o 3 mm. Once again, the 1000 waveorms were grouped into 40 sets o 5 waveorms, and the h 134

12 5 waveorms in each set were averaged in the normal spectral domain to obtain 40 estimates or the P scat versus requency curves or requencies corresponding to ka e values between 0.1 and kareal 4α real zt Each P scat curve was then divided by ma x P P e e and evaluated as a bg c scat h re bg b g natural log in order to obtain 40 examples o ln(c1 s ()). These 40 examples were then averaged together and the resulting estimate o lncsb gh mean ln sb g versus requency was plotted in igure Clearly, on average ln( s ()) does not exhibit any consistent requency dependence. As a result, Equation (8.5) can be simpliied as ASD mean ln sb g O mean ln sb g G = L N M π c + G 087. J b d a QP i G 4 a mean mean e real 16z α α mean mean + T real o G d c 4 g b hi gd c π eal o areal a e 3 + zt G αr α mean m c J ean mean. i h J J c h J (8.6) ln(s())-mean(ln(s())) igure 8.10: lncsbg h mean lncsbg h requency (Mz) or waveorms rom a hal space with an attenuation o 0 db/cm/mz containing 5 µm scatterers at a density o 35/mm 3. The source had iltering characteristics given by Equation (7.4). 135

13 8.. Analysis o ideal suraces Ater deriving Equation (8.6), the equation was used to calculate some ideal ASD suraces. nitially, the parameters were selected to duplicate the results obtained using the simulated waveorms in order to validate our derivations. Then, the impact o each term in the summation o Equation (8.6) was evaluated. or all the calculated ideal ASD suraces, z T and c were the same as were used in the simulations or the /4 source discussed previously. The ideal ASD suraces corresponding to hal-spaces with 5 µm scatterers and attenuations o 0 db/cm/mz and 0.3 db/cm/mz are shown in igures 8.11 and 8.1, respectively. Likewise, the ideal ASD surace or a hal-space with 50 µm scatterers and an attenuation o 0 db/cm/mz is shown in igure The requency range used when generating all three ideal suraces was rom 4.88 Mz to 14.6 Mz, corresponding to ka e values o 0.5 to 1.5 or the 5 µm scatterers and 1 to 3 or the 50 µm scatterers. Also, G mean ln mean ln L NM b g O QP d c s hi c sb g h J was set to a value o 0.03 that is comparable to the values ound rom analyzing some o the earlier simulation data (values varied rom ~0.0 to ~0.08). Just as in the earlier suraces rom the simulation data, an increase in scatterer size and an increase in attenuation both shit the ASD surace to higher values o α. Also, the width o the parabola describing the parabolic trough is independent o both scatterer size and attenuation (change in a e produces the same change in α or all three suraces). n addition, the minimum is much more pronounced or the larger scatterer size shown in igure 8.13 as was also observed in the suraces rom the simulation data (igures 8.3 and 8.4). The impact o the total requency range on the ideal ASD surace was also considered. igure 8.14 shows an ideal surace calculated or a hal-space with 50 µm scatterers and an G attenuation o 0 db/cm/mz with mean ln mean ln L NM O QP d c sbghi c sbg h J still set to a value o This time the requency range extended rom 4.88 Mz to 9.75 Mz corresponding to ka e values o 1 to. Once again, a decrease in the requency range resulted in a less pronounced minimum, and the parabola describing the parabolic trough was broadened (change in a e produces smaller change in α along the trough). 136

14 igure 8.11: deal ASD surace or a hal-space with an attenuation o 0 db/cm/mz containing 5 µm scatterers using requencies in the ka e range o 0.5 to 1.5 or a G L NM b g O QP d c s hi c sb g h mean ln mean ln value o Dark red corresponds to a small ASD J value and dark blue corresponds to a large ASD value. G igure 8.1: deal ASD surace or a hal-space with an attenuation o 0.3 db/cm/mz containing 5 µm scatterers using requencies in the ka e range o 0.5 to 1.5 or a L NM b g O QP d c s hi c sb g h mean ln mean ln value o Dark red corresponds to a small ASD J value and dark blue corresponds to a large ASD value. 137

15 igure 8.13: deal ASD surace or a hal-space with an attenuation o 0 db/cm/mz containing 50 µm scatterers using requencies in the ka e range o 1 to 3 or a G L NM b g O QP d c s hi c sb g h mean ln mean ln value o Dark red corresponds to a small ASD J value and dark blue corresponds to a large ASD value. igure 8.14: deal ASD surace or a hal-space with an attenuation o 0 db/cm/mz containing 50 µm scatterers using requencies in the ka e range o 1 to or a G L NM b g O QP d c s hi c sb g h mean ln mean ln value o Dark red corresponds to a small ASD J value and dark blue corresponds to a large ASD value. 138

16 Lastly, the impact o dierent initial requencies on the ideal ASD surace was investigated. igure 8.15 shows an ideal surace calculated or a hal-space with 5 µm scatterers and an attenuation o 0 db/cm/mz with mean ln mean ln G L NM b g O QP d c s hi c sb g h J still set to a value o 0.03 and using requencies rom 14.6 Mz to 4.4 Mz (ka e values o 1.5 to.5). The parabola describing the parabolic trough has been narrowed (change in a e produces smaller change in α along the trough) or the larger initial requency, just as had been observed in the analysis o the suraces rom the simulation data. igure 8.15: deal ASD surace or a hal-space with an attenuation o 0 db/cm/mz containing 0 µm scatterers using requencies in the ka e range o 1.5 to.5 or a G L NM O QP d c sbghi c sbg h mean ln mean ln value o Dark red corresponds to a small ASD J value and dark blue corresponds to a large ASD value. Upon urther comparison o igure 8.15 to igure 8.11, however, it is observed that peak has not been enhanced by using the larger initial requency. Recall that in the analysis o the ASD suraces rom the simulation data, the peak was enhanced in igure 8.8 (() by Equation (7.4), ka e o 1.5 to.5) where a larger initial requency was used compared to the peak in igure 8.6 (() by Equation (7.4), ka e o 0.5 to 1.5). owever, the peak in igure 8.9 (() by Equation (7.1), ka e o 1.5 to.5), which also had a larger initial requency, was comparable to the peak in igure 8.6. Thereore, there is some eature o the waveorms used to generate 139

17 igure 8.8 that has not been captured by Equation (8.6) and that might also have been responsible or the improved precision at these requencies or the source whose iltering unction is given by Equation (7.4). Even though the ideal ASD surace given by Equation (8.6) does not capture the dependence o the precision on the initial requency on the high requency end o the deviation peak, it still is able to capture all o the other dependencies observed in our analysis o the simulated waveorms. Thereore, the impact o each term in the summation o Equation (8.6) will be evaluated in order to provide urther insights into the minimization surace o the Spectral it algorithm. The analysis was done by multiplying each term in the summation given by Term mean ln mean ln Term Term 16z α α mean mean Term = L s s N Md c hi c bg a real e e O QP G bg h J 4 π = G a a c J d i mean mean G J b g G J = G J b gd i α = T real o π 3 8z 087. α α a a mean mean mean c α, a T real o real e e by a actor independent o the other terms and then recalculating the ideal ASD surace (i.e., ASD = Term Term + Termα + Termα, a ). The calculations were or a hal-space with 50 a e e µm scatterers and an attenuation o 0 db/cm/mz. The requency range used when generating the suraces was rom 4.88 Mz to 14.6 Mz, corresponding to ka e values o 1 to 3. Also, Term was set to a value o 0.03 except when evaluating its impact when it was given a value o (8.7) irst, consider the impact o Term by comparing the new ASD surace corresponding to Term = 008. shown in igure 8.16 to that shown in igure 8.13, where Term = The width o any given cross section o the parabolic trough is larger and the minimum is not as pronounced or the larger value o Term. owever, the parabola describing the parabolic trough occurs at the same location along the α axis and has the same width (change in ae produces the same change in α along the trough) regardless o the amount o spectrum noise. 140

18 igure 8.16: deal ASD surace to evaluate impact o Term using a hal-space with an attenuation o 0 db/cm/mz containing 50 µm scatterers using requencies in the kae range o 1 to 3 or Term = Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. n Section 8.1, some observations were made regarding the relative importance o the minimum as hal-space parameters, such as the scatterer size, and selected requencies were varied or the simulated waveorms. owever, the amount o spectral variation (i.e., Term ) can vary or dierent waveorm sets and requencies or the same hal-space depending upon the random arrangement o the scatterers. Since Term can inluence the relative importance o the minimum, the previous observations need to be validated by estimating Term or each o the ASD plots rom the simulated waveorms. The estimates or each o the suraces were ound by. ka z dividing each P scat curve by ma x P P 087 real 4α real T e e to obtain an estimate o bg bg c scat h b g re ln(c1()) rom which Term was then calculated. The Term values or all o the previous ASD curves or the simulated waveorms are given in Table 8.1. The Term values or the simulated cases are all approximately the same except or igure 8.7 that is signiicantly less. When we compared igure 8.7 to 8.6 previously, the minimum o the ASD surace was less pronounced or the larger sized scatterers using the smaller requency range (igure 8.7). owever, the smaller Term value o igure 8.7 should have resulted in a more pronounced minimum in the absence o 141

19 other eects. ence, the previous observations regarding the prominence o the minimum in relation to the hal-space parameters and selected requencies are still valid. Table 8.1: Estimates o Term values or the ASD curves or the simulated waveorms. igure Term Term ae Now consider the impact o Term ae and Term α by comparing igure 8.17, where was multiplied by a actor o 1.01, and igure 8.18, where Term was multiplied by a actor o 1.01, to igure igures 8.17 and 8.18 are indistinguishable, with the minimum being much more pronounced than the minimum in igure owever, the parabola describing the parabolic trough occurs at the same location and has the same width (change in a produces the same change in α along the trough) or all three igures. α e igure 8.17: deal ASD surace to evaluate impact o 101. Term ae using a hal-space with an attenuation o 0 db/cm/mz containing 50 µm scatterers using requencies in the ka e range o 1 to 3 or Term = Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. 14

20 igure 8.18: deal ASD surace to evaluate impact o 101. Term α using a hal-space with an attenuation o 0 db/cm/mz containing 50 µm scatterers using requencies in the ka e range o 1 to 3 or Term = Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. The impact o Term can now be analyzed by comparing igure 8.13 to igures 8.19 α,a e and 8.0 where Term has been multiplied by actors o and 1.008, respectively. The α,a e parabola describing the parabolic trough occurs at the same location and has the same width (change in ae produces the same change in α along the trough). owever, the minimum is less pronounced as the actor multiplying Term is increased rom 1 (igure 8.13), to (igure 8.19), and then to (igure 8.0). Also, the degradation o the minimum is more severe or larger Term than or smaller Term or Term α or the same percent change. α,a e ence, increasing Te or Term may not enhance the minimum i Term is also increased rm ae α,a e at the same time. Also, the existence o the parabolic trough can be attributed to Term because the minimum is less pronounced as this term increases. α a e α, a e α,a e 143

21 igure 8.19: deal ASD surace to evaluate impact o Termα, ae using a hal-space with an attenuation o 0 db/cm/mz containing 50 µm scatterers using requencies in the ka e range o 1 to 3 or Term = Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. igure 8.0: deal ASD surace to evaluate impact o Termα, ae using a hal-space with an attenuation o 0 db/cm/mz containing 50 µm scatterers using requencies in the ka e range o 1 to 3 or Term = Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. 144

22 8.3 Relating Properties o the ASD Suraces to Precision o Spectral it Algorithm n the previous sections, changes to the ASD surace were observed or dierent scatterer sizes, hal-space attenuations, spectral variance (i.e., Term ), requency ranges, and initial requencies. t was observed that the minimum o the ASD surace became more pronounced or larger ka e ranges or the same requency range (igures 8., 8.3, 8.4, 8.11, and 8.13), or larger ka e ranges or the same scatterer size (igures 8.13 and 8.14), or the same ka e ranges or larger requency ranges (igures 8.6, 8.7, 8.11, and 8.14), and or smaller spectral variances (igures 8.13 and 8.16). Also, the parabola describing the parabolic trough o the ASD surace was narrowed or the same ka e range with larger requency ranges (igures 8.6, 8.7, 8.11, and 8.14) and or the same ka e range and requency range with larger initial requency (igures 8.6, 8.9, 8.15, and 8.11). owever, these changes o the ASD suraces have not been related to the precision o the Spectral it algorithm. n addition, the ideal ASD suraces in Section 8. always had a minimum at the correct location even as the overall shape o the ASD surace was altered by the dierent hal-space parameters and requency ranges. ence, the irst step is to determine the cause o inaccuracies in the estimate. The impact o the causes can then be addressed as the ASD suraces are changed. The most likely cause o inaccuracies in the estimate is ln( s ()) having some requency dependence or some set o waveorms. This would not contradict the results presented in igure 8.10 because on average ln( s ()) would have no requency dependence yielding accurate average estimates as was observed in the earlier chapters. n order to explore this possibility, values or σ, = mean ln mean mean ln σ = mean ln mean mean ln, c sbgh c sbgh c sb gh c sb gh were estimated or some o the simulated waveorms along with the corresponding Term. These values were then input into Equation (8.5) with the same hal-space parameters and requency range that had been used to generate and obtain estimates or the simulated waveorms in order to generate a corresponding ASD surace. When generating the suraces, a e was varied rom 0 µm to 50 µm in steps o 0.0 µm while α was varied rom db/cm/mz to db/cm/mz in steps o db/cm/mz. The true scatterer size used in the simulation was 5 µm and the true hal-space attenuation was 0 db/cm/mz. A scatterer size was then (8.7) 145

23 determined by inding the minimum o the generated ASD surace. The scatterer size rom the generated surace was then compared to the scatterer size given by the Spectral it algorithm or the same set o waveorms. The comparison o the two scatterer sizes along with the estimated values o Term, σ,, and σ are given in Table 8.. The irst three rows correspond to the data sets used to generate igures 8.6, 8.8, and 8.9, respectively. The minimum value o the generated ASD surace has also been provided., The values or the scatterer size rom the generated ASD surace are in good agreement with the scatterer sizes rom the Spectral it algorithm or the same data set or all o the cases except or the data set corresponding to igure 8.8 (i.e., 1.9 µm rom the Spectral it algorithm compared to 0 µm rom the ASD surace). The discrepancy or the igure 8.8 data set is another indication that the current theory cannot account or the deviation peak discussed in Section 7.3. The agreement or the other data sets would probably be urther enhanced i smaller step sizes were used to generate the ASD surace. Because when larger step sizes o 0.5 µm and db/cm/mz were used to generate an ASD or the irst case in Table 8., the scatterer size estimate was 19.5 µm instead o 19.1 µm. ence, the errors in the estimates rom the Spectral it algorithm are due to the inite values o σ and σ,, due to the good agreement in ae rom the Spectral it algorithm and the generated ASD surace. n addition, the inclusion o σ, and σ, in Equation (8.5) reduces the ASD value below the ideal minimum o Term with a greater reduction or the poorer estimates. t may be possible to exploit this eect in a new minimization algorithm i an accurate estimate o Term could be obtained or any given data set. Table 8.: Comparison between the scatterer size rom the generated surace and the scatterer size given by the Spectral it algorithm or the same data set along with the minimum value o the generated ASD surace as well as the estimated values o Term, σ, and σ,,. a e rom a e rom Minimum o Spectral it Generated Term Generated σ σ,, Algorithm ASD Surace ASD Surace 18.4 µm 19.1 µm µm 0 µm µm 5.7 µm nm 0 µm µm 8.7 µm µm 9.0 µm µm 36.5 µm

24 and σ Once the causes o the errors in the estimate were known to be the inite values o σ,, the eect o these terms on the overall ASD surace was investigated. The investigation was done using the ideal ASD surace corresponding to a hal-space with 50 µm scatterers and an attenuation o 0 db/cm/mz using requencies in the kae range o 1 to 3 (i.e. igure 8.13) with a Term o The ideal surace was generated our times corresponding to σ = 1 with σ,, = 0, σ = 1 with σ,, = 0, σ = 0 with σ,, = 005,. and σ = 0 with, σ = 0 05, shown in igures , respectively. The impact o each case was then,. assessed by comparing the generated igures to igure 8.13 where both σ, σ and,, were zero. igure 8.1: deal ASD surace or a hal-space with an attenuation o 0 db/cm/mz containing 50 µm scatterers using requencies in the ka e range o 1 to 3 or a Term o 0.03 with σ = 1, and σ, = 0. The new minimum corresponds to a scatterer size o 34.6 µm. Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. 147

25 igure 8.: deal ASD surace or a hal-space with an attenuation o 0 db/cm/mz containing 50 µm scatterers using requencies in the ka e range o 1 to 3 or a Term o 0.03 with σ = 1, and σ, = 0. The new minimum corresponds to a scatterer size o µm. Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. igure 8.3: deal ASD surace or a hal-space with an attenuation o 0 db/cm/mz containing 50 µm scatterers using requencies in the ka e range o 1 to 3 or a Term o 0.03 with σ = 0, and σ, = The new minimum corresponds to a scatterer size o µm. Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. 148

26 igure 8.4: deal ASD surace or a hal-space with an attenuation o 0 db/cm/mz containing 50 µm scatterers using requencies in the ka e range o 1 to 3 or a Term o 0.03 with σ = 0, and σ, = The new minimum corresponds to a scatterer size o µm. Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. n all our igures, the parabola describing the parabolic trough occurs at the same location and has the same width as the parabola describing the parabolic trough in igure owever, the minimum occurs at dierent locations along the trough or the dierent values o σ and σ,. A, σ o 1 with a σ,, o 0 (igure 8.1) and a σ o 0 with a σ,, o 0.05 (igure 8.4) shit the minimum to a lower scatterer size and larger attenuation, while a σ 1 with a σ, o 0 (igure 8.) and a σ o 0 with a σ,, o 0.05 (igure 8.3) shit the minimum to a larger scatterer size and smaller attenuation. ence, or the same sign, σ, and σ, shit the minimum in opposite directions. Because σ and σ,, typically have the same sign, as is indicated by the results in Table 8., the two requency dependencies will counterbalance each other. n addition, or the same magnitude o σ and σ,,, the shit to smaller scatterer sizes appears to result in a larger percentage error in the new location o the minimum. Upon examining igure 8.13, it is clear that the ASD curve increases slightly aster as the parabolic trough moves to larger scatterer sizes away rom the minimum as opposed to, o 149

27 moving to smaller scatterer sizes. ence, the error introduced by the requency dependence o ln( s ()) appears to be limited by the rate o change o the original ASD surace. A more detailed comparison reveals that the minimum is slightly more pronounced or igures 8.1 and 8.4, which are comparable, than or igure 8.13, and the minimum is even more pronounced or igures 8.3 and 8., which are also comparable. ence, the enhancement o the minimum appears to be related to the rate o change o the original ASD surace beore the requency dependence o ln( s ()) has been included as well. n order to clariy the eect o the shit on the prominence o the minimum, consider the ideal surace shown in igure 8.5 where σ = with σ,, = n igure 8.5, the minimum is 0 comparable to the minimum in igure 8.13 and has not been shited as ar away rom the true scatterer size and attenuation as the minimum in igure 8.4. ence, the enhancement o the minimum also appears to be related to the amount the minimum has been shited away rom its true value. igure 8.5: deal ASD surace or a hal-space with an attenuation o 0 db/cm/mz containing 50 µm scatterers using requencies in the ka e range o 1 to 3 or a Term o 0.03 with σ = 0, and σ, = The new minimum corresponds to a scatterer size o 43. µm. Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. 150

28 Now that the error in the estimation scheme has been related to the shit in the minimum o the ASD surace away rom the true value, the dependence o the shit on the properties o the ideal ASD surace or any given data set can be explored. Consider the ASD suraces shown in igures 8.6 and 8.7. Both suraces were generated or a hal-space with 50 µm scatterers and an attenuation o 0 db/cm/mz with Term o 0.03, σ, o -1, and σ o 0. owever, in igure 8.6 the minimum was enhanced by multiplying Term by 1.01, and the calculation involved requencies in the kae range o 1 to 3. Likewise, in igure 8.7 the parabola describing the parabolic trough was broadened by using requencies in the ka e range o 0.01 to.01. Upon comparing igure 8.6 to igure 8., it is clear that the minimum has been shited by a smaller amount when the minimum o the original surace (prior to inclusion o σ α, ) is more pronounced. ence, the precision o the estimates should be improved when the minimum o the ASD surace is better deined. The minimum was better deined when the kae or requency range was increased. Thereore, the increased precision with increased requency range and ka e range observed in Chapter 7 agrees with our analysis o the ASD surace., igure 8.6: deal ASD surace or a hal-space with an attenuation o 0 db/cm/mz containing 50 µm scatterers using requencies in the ka e range o 1 to 3 or a Term o 0.03 with σ, = and σ = 0 while enhancing the minimum by multiplying Term by The new minimum, 1 corresponds to a scatterer size o µm. Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. α 151

29 igure 8.7: deal ASD surace or a hal-space with an attenuation o 0 db/cm/mz containing 50 µm scatterers using requencies in the ka e range o 0.01 to.01 or a Term o 0.03 with σ = and σ,, = 0. The new minimum corresponds to a scatterer size o µm. Dark red corresponds to a small ASD value and dark blue corresponds to a large ASD value. 1 Now compare igure 8.7 to igure 8.. The error in the scatterer size is the same or both suraces. owever, because the parabola describing the parabolic trough is broader in igure 8.7, the corresponding error in the hal-space attenuation is less ( db/cm/mz as compared to db/cm/mz). Thereore, broadening the parabola describing the parabolic trough should improve the precision o the attenuation estimate while not aecting the scatterer size estimate. ence, the precision o the attenuation estimate should be reduced as we increase the initial requency without changing the precision o the size estimate, as was also observed or the initial requencies below the deviation peak in Chapter 7. The requencies above the deviation peak were shown to have a more pronounced minimum in igure 8.8 that cannot be accounted or using the current ASD theory. Beore concluding, one case needs to be considered in greater detail. Recall that in Chapter 7, an increase in the requency range or the same ka e range improved the precision o the attenuation estimate while maintaining the same precision or the scatterer size estimate. owever, in this chapter, the parabola describing the parabolic trough o the ASD surace was noted to be narrower or the larger requency range (igures 8.6, 8.7, 8.11, and 8.14). ence, or the same error in scatterer size (in µm), the narrower parabola would have resulted in a larger 15

30 error in the attenuation estimate. This apparent discrepancy can be reconciled by recalling that it was the same percent error in scatterer size that resulted in the improved precision or the attenuation estimate or the larger requency ranges. The same percent change in scatterer size in igure 8.14 and igure 8.7 still produces a larger change in the attenuation along the parabolic trough than the corresponding percent change in igure 8.11 and igure 8.6. ence, the observations regarding the ASD surace or this case agree with the results presented in Chapter Chapter Summary n this chapter, an analysis was perormed o the ASD surace over which the Spectral it algorithm inds the minimum when estimating the scatterer size and total attenuation. The analysis was done using both the simulated waveorms rom the earlier chapters as well as ideal suraces corresponding to a derived expression or the ASD surace. The surace was ound to have a single minimum that always occurred along a parabolic trough. The existence o the parabolic trough could be attributed to a cross term between the attenuation and scatterer size minimizations, Term. α,a e Also, it was observed that the minimum o the ASD surace became more pronounced or larger ka e ranges or the same requency range (igures 8., 8.3, 8.4, 8.11, and 8.13), or larger ka e ranges or the same scatterer size (igures 8.13 and 8.14), or the same ka e ranges or larger requency ranges (igures 8.6, 8.7, 8.11, and 8.14), and or smaller spectral variances (igures 8.13 and 8.16). Also, the parabola describing the parabolic trough o the ASD surace was narrowed or the same ka e range with larger requency ranges (igures 8.6, 8.7, 8.11, and 8.14) and or the same ka e range and requency range with larger initial requency (igures 8.6, 8.9, 8.15, and 8.11). The shit o the minimum along the parabolic trough away rom its true value was then related to the requency dependence o the spectral variations resulting rom the random scatterer spacing, ln( s ()). or the same requency dependence o ln( s ()), the shit o the minimum away rom the true value was ound to be reduced (improvement in precision) when the minimum o the surace was more pronounced. n addition, the error in the hal-space attenuation due to the shit in the minimum is less when the parabola describing the parabolic trough is broader (all other terms remaining the same). ence, the analysis o the ASD surace is in agreement with the results presented in Chapter 7 regarding 153

31 ka e range, requency range, and initial requencies below the deviation peak. The results or initial requencies greater than the deviation peak were ound to not be consistent with ASD analysis, once again indicating a limitation in the current theory or perhaps a simulation artiact. 154