Jun Dang

Member, IEEE Stefaan Tavernier
email: stefaan.tavernier@vub.ac.be

Student Member, IEEE Benjamin Frisch

Member, IEEE Paul Lecoq

Member, IEEE Philippe Lasaygues
email: lasaygues@lma.cnrs-mrs.fr

Member IEEE Serge Mensah

Dachun Zhang

Mingxi Wan

Nicolas Felix

Member IEEE Joao Varela

Etiennette Auffray

Development of an anthropomorphic breast phantom

Keywords: Breast, Elastography, PET, Ultrasound

Combining the advantages of different imaging modalities leads to improved clinical results. For example, ultrasound provides good real-time structural information without any radiation and PET provides sensitive functional information. For the ongoing ClearPEM-Sonic project combining ultrasound and PET for breast imaging, we developed a dual-modality PET/Ultrasound (US) phantom. The phantom reproduces the acoustic and elastic properties of human breast tissue and allows labeling the different tissues in the phantom with different concentrations of FDG. The phantom was imaged with a whole-body PET/CT and with the Supersonic Imagine Aixplorer system. This system allows both B-mode US and shear wave elastographic imaging. US elastography is a new imaging method for displaying the tissue elasticity distribution. It was shown to be useful in breast imaging. We also tested the phantom with static elastography. A 6D magnetic positioning system allows fusing the images obtained with the two modalities. ClearPEM-Sonic is a project of the Crystal Clear Collaboration and the European Centre for Research on Medical Imaging (CERIMED).

shown to have excellent performance [START_REF] Varela | Characterization of the Clear-PEM Breast Imaging Scanner Performance[END_REF]. The ClearPEM-Sonic dual-modality PET/US scanner is a combination of a ClearPEM breast imager and an Aixplorer US system produced by Supersonic Imagine, into a dual imaging device allowing to acquire PET and US images during the same patient imaging session. The breast will be immobilized by a plastic breast contention cone tailor made to the shape and volume of the breast. The ultrasound probe is held by an arm and will be manually pressed against the breast, using a window for US probe in the breast contention cone.

Ultrasound phantoms are commercially available, but there are no commercially available phantoms for combined PET/US. The main motivation for the present project was developing a phantom that can be used with the ClearPEM-Sonic dual-modal PET/US scanner. Different parts in the phantom should have acoustic properties that are typical for human breast tissues and, at the same time, should be impregnated with positron emitting tracers at concentrations that are typical of what is present in human breast during PET scans. In addition, the phantom should also reproduce the elasticity of real tissues in the breast so that the phantom can also be used for US elastographic imaging. In the test reported here the phantom was evaluated separately with a standard whole-body PET system and with the Aixplorer elastographic US system. Moreover, we also tested the phantom with static elastography, which is another method for elastography imaging.

II. MATERIALS AND METHODS

The phantom was made of mixed gels of gelatine (GELITA EUROPE, Ballistic 2, photographic grade) and "high gel Agar" (SIGMA-ALDRICH Co.). However, such gels will degrade within a few days after production. We found that adding less than 0.5% of Germall-plus [START_REF] Huber | Multi-Modality phantom development[END_REF] as a conservation agent greatly improved the conservation properties of the gel, and in this way, the samples were stored for several weeks in a fridge. To find gels with the desired properties, we measured the velocity and the attenuation of the acoustic waves and Young's modulus (the modulus of elasticity) of a large number of samples. The acoustic velocity and the attenuation coefficient were measured in a set-up with two parallel single US transducers immersed in a water tank, and with the sample between the transducers [START_REF] Lasaygues | Ultrasonic characterization of orthotropic elastic bovine bones[END_REF]. Young's modulus was measured by static stretching. The stretching force and corresponding deformation were recorded, and the slope gives the sample's Young modulus.

We evaluated samples with gelatine mass percentages varying from 1% to 15% and with agar mass percentages varying from 0.5% to 6%. Mixed gels with more agar than gelatine remained liquid and could not be used for making phantoms. We found that the acoustical velocity of the waves mainly depends on, and increases with, the gelatine fraction, and that the attenuation of the waves mainly depends on, and increases with, the agar fraction. This allowed us to find gel compositions that gave samples with acoustic and elastic signature similar to fat tissue, normal glandular tissue, fibrous/hard tissue, and carcinoma in the breast. Table 1 lists a few of the gels tested and their acoustical properties. Table II shows the acoustic properties and Young's moduli for real breast tissues reported in the literature [START_REF] Hasni | Ultrasound in the assessment of the palpable breast mass[END_REF][START_REF] Krouskop | Elastic Moduli of Breast and Prostate Tissues Under Compression[END_REF]. A comparison of Tables I andII shows that it is possible to find gels that closely mimic the properties of real breast tissues. We chose a gel with 1% gelatine and 0.5% agar to mimic the fat medium in the breast and a gel with 12% gelatine and 4% agar to mimic hard tissue and carcinoma. All phantoms consist of a volume of gel simulating fat tissue with a number of inserts of varying dimensions (diameters from 1 to 3 cm), simulating cancer lesions. For the positron emitting tracer concentration we used 4 kBq/ml in the fat tissue and 16 kBq/ml in the cancer lesions.

The phantom was prepared as follows: 1) To make gels simulating cancer lesions, 12% gelatine and 4% agar were mixed with the correct amount of deionized water, and heated in the microwave oven until the liquid nearly boils. The mixture was stirred moderately and cooled to 45 ⁰C, and FDG was added as needed. Moulds were used to make "tumour volumes" of desired size and shape, and were solidified in a fridge.

2) To make gels mimicking fat tissue, 1% gelatine and 0.5% agar were mixed with the correct amount of deioniszed water, and the beaker with the mixture in a microwave oven was heated until the temperature reaches 70 ⁰C till all the gelatine and agar dissolved. The mixture was cooled down to 40 ⁰C and FDG was added as needed.

3) The "tumour volumes" were removed from the molds and inserted into the phantom while the gel simulating fat tissue was still at 40 ⁰C.

4) The whole phantom was left to cool to room temperature where it solidified.

The time taken for preparing the phantom, including the cooling phase, was about 1 h. The PET scan should be taken immediately after the preparation of the phantom, i.e., before the tracer decays. The US scan can be taken any time after that.

III. RESULTS

A. Description of the equipment

The PET images of the phantoms were obtained with a PHILIPS GEMINI TF 64 scanner. We used a modified DP-9900 scanner (Mindray, P. R. China) for studying static elastography, and a SuperSonic Imagine Aixplorer scanner (SuperSonicImagine, Aix en Provence, France) for obtaining shear wave elastographic images.

The static elastogram obtained with the modified DP-9900 scanner is a strain image and not an image of Young's modulus. The Aixplorer scanner produces real elastograms, i.e., images of Young's modulus. This machine allows obtaining elastograms in a way that is independent of the amount of compression applied by the operator and therefore gives more reproducible results. SuperSonic Imagine uses a proprietary concept to introduce the shear wave into the body: ultrasound beams are focalized successively at different depths in the tissue, creating a shear wave in the medium [START_REF] Tanter | Quantitative assessment of breast lesion viscoelasticity: initial clinical results using SuperSonic shear imaging[END_REF][START_REF] Bercoff | Supersonic shear imaging: a new technique for soft tissue elasticity mapping[END_REF]. Using ultrafast imaging, the velocity of the shear wave at every point in the image is measured, and a quantitative elasticity map can be obtained. The result is a real-time, quantifiable, user-independent, and reproducible ShearWave™ Elastographic image.

B. Tests with a cuboi-shaped phantom

In the first experiment, we studied static elastography with a rectangular cuboid-shaped phantom measuring 134 mm × 88 mm × 70 mm. The phantom contained several globular inclusions representing cancerous lesions measuring between 10 and 40 mm, immersed in a homogeneous volume of fat tissue mimicking medium.

In this study, a linear US probe with nominal frequency of 7.5 MHz and sampling frequency of 25 MHz was used. A rectangular compressor with a polyethylene plane of size 88 mm × 66 mm was used to enlarge the US probe's contact surface. The US probe was inserted in a slot cut in the compressor plane. US radio frequency (RF) data were acquired with a 32 bit data I/O card. Both pre-and post-compression US RF data were acquired. The images obtained with this setup, the compression plate, and typical signals are shown in Fig. 1.

For this measurement a compression of 0.5 mm was used. The strain image was derived from the difference in the RF data before and after compression. The local tissue displacements were estimated using the 1D cross-correlation technique by comparing the gated pre-and post-compression RF signals [START_REF] Ophir | Elastography: a quantitative method for imaging the elasticity of biological tissues[END_REF]. For all the static elastograms, the length of the correlation window was fixed at 2.5 mm, and an overlap window of 80% was used. To improve the correlation between the pre-and post-compression signals, and therefore to improve the measurement of the displacements, temporal stretching [START_REF] Varghese | Enhancement of echo-signal correlation in elastography using temporal stretching[END_REF] is applied to the post-compression signals with a stretching factor equal to the mean strain. After obtaining the displacement data, the axial strain was computed from the displacement estimates using a third-order optimum low-pass differentiator [START_REF] Luo | Axial strain calculation using a low-pass digital differentiator in ultrasound elastography[END_REF]. The dynamic range displayed in the strain image is 1.5%. The attenuation coefficient of the polyethylene plate had been tested to be very near to 0 dB / mm , so we believe it can be seen transparent to the US RF signals.

Fig. 2 shows the images obtained with different modalities for the cuboid-shaped phantom with the size of 134 mm × 88 mm × 70 mm. The white rectangle in the PET image shows the US scan region. In B-mode imaging, the tumour shows as a brighter region, which means that the tumour reflects the sound waves more strongly. In static elastography, the tumour shows up as a darker region, which means that the strain distribution in the hard tumour parts is much smaller than in the softer fat-simulating material. The phantom was scanned for 2 min in a PHILIPS GEMINI TF 64 scanner, resulting in a total of 3.6 × 10 6 counts. The default energy window of 440-665 keV was used. The PET data were acquired in list mode, and a TOF (time-of-flight) list-mode OSEM algorithm was used for image reconstruction. The OSEM algorithm has 33 subsets and stops after 3 iterations [START_REF] Surti | Performance of Philips Genimi TF PET/CT scanner with special consideration for its time-of-flight imaging capabilities[END_REF][START_REF] Popescu | Iterative image reconstruction using geometrically ordered subsets with list-mode data[END_REF]. Attenuation correction is performed using the CT transmission data, whereas scatter is estimated using a single-scatter simulation algorithm [START_REF] Surti | Performance of Philips Genimi TF PET/CT scanner with special consideration for its time-of-flight imaging capabilities[END_REF]. For clinical images taken with ClearPEM-Sonic, this simple approach will not be possible. In this case we will rely on a 6D magnetic positioning system (3D Guidance trakSTAR™, Ascension Tech. Co.). According to the specifications, the spatial accuracy of the tracking system is 1.4 mm r.m.s for the position and 0.5 degree r.m.s for the orientation angles. In the ClearPEM-Sonic system, the breast will be contained in a plastic holder that is well adapted to the shape and size of the breast. If this breast container has fiducial markers that are visible in the PET image and if the positions of these fiducials can be measured by the magnetic tracking system, it is possible to superimpose both images.

D. Tests with the anthropomorphic phantom

To test this idea we made an anthropomorphic breast phantom by filling one of the breast contention cones with breast-simulating gelatine-agar gels (Fig. 4(a)). For this phantom, as for the orthogonal cuboid-shaped phantom, the US probe scan region

US B-mode image

Superposition of the US B-mode and the whole body PET image B-mode Shear wave elastography

Full body PET image

US transducer scan region

Reflection of polyethylene plate bulk of the breast tissue was taken to be fat, and a number of cancerous lesions of various sizes (diameters 1-2 cm) were inserted in the volume. As fiducial markers, we used three low-activity (few micro-Ci) point sources. The plastic breast container cone, with the three fiducial markers attached to it, is shown in Fig. 4(a). The breast contention cone has a window for placing the US probe. For our test this window was covered with a Mylar foil. The width of the window is the same as the width of the US 3D probe width.

In the present test, we used a whole-body PET/CT scanner for obtaining the PET images. We found that the activity in the fiducial markers was too low for these to be clearly visible in the PET image. Fortunately, with a whole-body PET/CT scanner, the field of view is very large and the 10 metal screws close to the edge of the phantom are in the field of view. We therefore relied on these metal screws instead. The diameter of the screws is 5 mm. These are clearly visible in the CT image, and their position can also be measured with the magnetic positioning tracker. When performing the test, the phantom was also equipped with a wire strung horizontally in the symmetry plane of the breast contention cone. See Fig. 4(b). With the help of the laser beams, we could therefore position the phantom horizontally and exactly in the middle of the PET/CT image with the wire, and therefore the symmetry plane of the phantom, perpendicular to the axis of the scanner.

The US probe has a smooth and rounded shape so that it is comfortable in the operator's hand, but it has no suitable points for attaching the magnetic positioning probes. We therefore made a small plastic piece that can be attached to the US probe and that is equipped with two groves housing two magnetic probes in such a way that the axis of the US probe is exactly in the middle of, and exactly parallel to, the axes of the two magnetic positioning probes (Fig. 5(a)). The distance between the tip of the magnetic probes and the front of the US probe was measured with a ruler and found to be 80 mm.

Using the information provided by the positioning probes, we made sure that while recording the US image, the US probe was held such that its central image plane coincided with the symmetry plane of the breast phantom. During the US scan, we recorded the position and the dip angle of the US probe. In this way, the central plane of the US image coincides with the symmetry plane of the breast phantom, and this symmetry plane coincides with the transaxial plane in the center of the PET image. The metal screw allows us to check the correct axial location of the phantom in the PET scanner. Finally we only need to apply the correct magnification factor and rotate the US image by the measured dip angle to superpose both images, see Fig. 5(b).

Several factors influence the accuracy of the image fusion: First, the US probe deforms the breast when it is making contact. The deformation of the breast is several millimeters at the point of maximum deformation. The uncertainty of the position of the probe is 1.4 mm r.m.s. This has to be multiplied by because both the fiducial and the probe position are measured. The uncertainty on the orientation of the probe translates into an uncertainty of 0.8 mm r.m.s on the position of the elevation of the US probe. Finally the PET image has voxels of 4mm. Considering all these factors, it is very likely that the relative position error of the PET and US is at most on PET voxel.

IV. DISCUSSION AND CONCLUSION

A. Comparison between B-mode US image with static elastography

The results presented here confirm the conclusions of several previous studies [START_REF] Cui | Displacement and strain estimation of breast tissue using multi-compressed ultrasound RF signals[END_REF][START_REF] Madsen | Anthropomorphic breast phantoms for tesing elastography systems[END_REF][START_REF] Zhang | Improvement on the elastic visualization of thermal lesion suing block wavelet shrinkage[END_REF] regarding the good performance of static elastography compared to B-mode US imaging. Young's modulus between tumor and normal tissue mimicking material is 5:1, dynamic range of B-mode images are 65 dB, and dynamic range of strain image is 0-0.015. All the image fields of view are 33 mm × 75 mm. In (a) and (b), the white arrow shows in static elastography that the strain seems to be higher between two hard lesions, but in fact it is caused by mechanical artifact by former studies [START_REF] Cui | Displacement and strain estimation of breast tissue using multi-compressed ultrasound RF signals[END_REF][START_REF] Li | Ultrasound estimation of breast tissue biomechanical properties using a similarity-based non-linear optimization approach[END_REF]. In (c) and (d), the white arrow indicates a small lesion can be easily found by static elastography, while in B-mode image it is hardly to be found. This supports that elastography can be an ideal supplementary tool for conventional B-mode exam, to help clinical biopsy. In (e) and (f), elastography can overtake the shadowing effect compared with B-mode imaging and also it can detect very small-sized lesions, parallel to conventional B-mode imaging. it. This is a well-known artifact of US B-mode imaging, and the effect is absent in the elastogram 7(f).

B. Signa-to-Noise ratio

For the PET modality, Fig. 8 shows the signal and background activity of the breast phantom. The signal-to-noise ratio of the background region is 19. The tumour activity peak value is 2.2 times higher (tumour 1) than the background region (Fig. 8(a)). This is less than the ratio of the activities in tumour and fat medium, and the difference is due to the small size of the tumours compared to the resolution of the scanner, namely partial volume effect. The diameter of the tumour 1 is 1.5 cm. The activity ratio between tumour and fat medium is 4:1. According to partial volume effect [START_REF] Soret | Partial-Volume Effect in PET Tumour Imaging[END_REF], the measured activity should be around 50% of the original activity when the PET scanner FWHM spatial resolution is 4 mm. So this explains why we have measured 2.2 of signal to noise ratio for tumour 1. For tumour 2, as the image plane we used did not cross the tumour center but the edge of it, the measured activity is even lower than tumour 1. For the US modality, the signal-to-noise ratio of the background region is 10. The signal reflected by the tumour is only a factor 1.2 larger than the background signal, giving a signal to noise ratio of about 2. This is a normal situation for US images (Fig. 8(b)). C. A phantom for the ClearPEM/Sonic project Our tests show that it is possible to make phantoms that can be used for testing a multimodal PET/US scanner. If the phantom is prepared with inclusions simulating cancerous lesions with dimensions in the range 1-10 mm, it will be very useful for comparing the performance of the ClearPEM with the performance of standard whole-body PET systems. The Supersonic Aixplorer scanner can detect lesions in the 1 mm range. The ClearPEM scanner will allow the observation of hot spots in the same size range [START_REF] Neves | The ClearPEM breast imaging scanner[END_REF]. The phantom will be an ideal training tool for studying volume registration and image fusion, and for system performance evaluation.

Fig. 9 shows the layout of the ClearPEM scanner and the plastic head contention cone. It also shows how the US probe will be held in position in the combined ClearPEM/Sonic scanner.

D. Image registration

Image registration is an essential step in multimodal imaging. We tested both fiducial marker based image registration and "mutual information" software based image registration for phantom images [START_REF] Wells | Multi-modal volume registration by maximization of mutual information[END_REF][START_REF] Josien | Mutual information based registration of medical images: A survey[END_REF]. Although mutual information based image registration has been reported to be robust for multimodality image registration in case of PET/MR and PET/CT [START_REF] Josien | Image registration by maximization of combined mutual information and gradient information[END_REF][START_REF] Jenkinson | Improved Optimization for the Robust and Accurate Linear Registration and Motion Correction of Brain Images[END_REF], we obtained only very disappointing results. This is probably due to the limited field of view of the US image, and therefore the limited overlap region between the two images, and to the low signal-to-noise ratio [START_REF] Maes | Multimodality Image Registration by Maximization of Mutual Information[END_REF], particularly for the US image.

For fiducial marker-based registration, on the other hand, we obtained good results. Unfortunately, in the present test, we could not use the radioactive point sources as fiducials for the image superposition because the activity in the fiducials was too low, and we had to rely on the metal screws visible in the CT image. We are nevertheless confident that it is possible to use radioactive point sources for superposing the images. This method can only be used if the breast retention cone immobilizes the breast sufficiently while both images are recorded. Since the fiducial markers are not visible on the US image, we must rely on the 3D magnetic positioning system. In this method, only rigid linear transformations are applied.

E. Conclusion

We have developed a gelatine-agar-based dual-modality phantom for PET/US breast-imaging research. Gels with acoustic and mechanical properties that closely resemble real tissues such as fat tissue, glandular tissue, fibrous tissue, and carcinoma could be obtained. During the preparation of the phantom, FDG can be added to give the different components activity levels similar to what will be observed during PET scans. Tests with a whole-body PET/CT and US imaging systems confirmed the good properties of the phantom. On the basis of our preliminary tests, we are confident that image registration is possible by using radioactive fiducial markers and a 6D magnetic positioning system.

1 .

 1 Comparison of results obtained with and without polyethylene plate compressor, and the corresponding 1D ultrasound radiofrequency signals. (a) B-mode image of the phantom with the US probe fixed inside the polyethylene compressor plate, (b) B-mode image obtained without the polyethylene compressor plate, (c) US probe with its polyethylene compressor, and (d) comparison of the RF signals with and without the polyethylene compressor plate.

2 .

 2 Corresponding planes in the images obtained with different modalities for the cuboid shaped phantom. (a) B-mode US image (506×239 pixels, field of view 69mm×33mm), (b) static elastogram(506×239 pixels, field of view 69mm×33mm), and (c) PET image(interpolated as 540×660 pixels to have the same voxel size as US images, field of view 90mm×80mm). The white box in the PET image shows the US probe scan region. The dynamic range of B-mode image is 65 dB. For elastogram the strain dynamic range is 0 to 0.015.C. Image superposition Fig.3shows images of another cuboid shaped phantom (134 mm × 88 mm × 60 mm) obtained with the SuperSonic Imagine's Aixplorer shear wave elastographic imager. This figure shows the images obtained with US B-mode imaging, with ShearWave elastograpy, and with a whole-body PET. The superposition is also shown. The edges of the cuboid-shaped phantom are clearly visible on the PET image. Therefore, with this phantom, the relative position of the two images could simply be obtained with a ruler measuring the position of the US probe relative to the edges of the cuboid and by making sure the US probe was held vertically.

Fig. 3 .

 3 Fig. 3. Images of the cuboid-shaped phantom obtained with Supersonic Imagine's Aixplorer scanner and with a whole body PET: (a) B-mode image (749 × 711 pixels, field of view 60 mm × 45 mm), (b) shear wave elastography (749 × 711 pixels, field of view 60 mm × 45 mm), (c) whole-body PET image (15 × 22 pixels, field of view 60 mm × 88 mm), white box stands for the US probe scan region, and (d) superposition of the two images (interpolated as 1050 × 1573 pixels, field of view 60 mm × 88 mm). Young's modulus ratio between tumour and tissue background is 6:1. The dynamic range of B-mode images is 65dB and the elasticity range of elastography information is 300 kPa.

Fig. 4 .Fig. 6 Fig. 6 .

 466 Fig. 4. (a) Breast shaped phantom and in a plastic breast contention cone. The positioning fiducials are the three gray discs, 20 mm in diameter. (b) Positioning of the phantom on the bed of the PET/CT scanner.

Fig. 7

 7 Fig. 7 Comparison of B-mode US image and static elastogram. Young's modulus between tumor and normal tissue mimicking material is 5:1, dynamic range of B-mode images are 65 dB, and dynamic range of strain image is 0-0.015. All the image fields of view are 33 mm × 75 mm. In (a) and (b), the white arrow shows in static elastography that the strain seems to be higher between two hard lesions, but in fact it is caused by mechanical artifact by former studies[START_REF] Cui | Displacement and strain estimation of breast tissue using multi-compressed ultrasound RF signals[END_REF][START_REF] Li | Ultrasound estimation of breast tissue biomechanical properties using a similarity-based non-linear optimization approach[END_REF]. In (c) and (d), the white arrow indicates a small lesion can be easily found by static elastography, while in B-mode image it is hardly to be found. This supports that elastography can be an ideal supplementary tool for conventional B-mode exam, to help clinical biopsy. In (e) and (f), elastography can overtake the shadowing effect compared with B-mode imaging and also it can detect very small-sized lesions, parallel to conventional B-mode imaging.

Fig. 7

 7 Fig. 7 illustrated the differences. This figure shows corresponding slices through the US B-mode image and the elastographic images obtained with the cuboid-shaped phantom. The top arrow in Fig. 7(b) points to a bright, high-strain region between two hard tumours. This seems to indicate the presence of very soft material, while in fact it is a mechanical artifact [26, 29] when compared with Fig. 7(a). In Fig. 7(c) and 7(d), the white arrow points at a small lesion that is hardly visible in the US B-mode image and shows clearly on the elastogram. The small lesion indicated with an arrow in the upper part of Fig. 7(e) and 7(f) is also much more visible in the elastogram. In Fig. 7(e), the outlined arrow points at a dark vertical band. This is due to the shadow of the hard tissue above

  Fig. 8 (a) Signals of phantom tumour part and background part from PET image. (b) signals of phantom tumour part and background part from SuperSonic B-mode image.

Fig. 9

 9 Fig. 9 ClearPEM/Sonic combination. (a) ClearPEM patient scanner with the patient scanned by a prone position; (b) US probe arm fixed with ClearPEM detector heads; (c) plastic breast contention cone; and (d) US scanning window on the breast contention cone

Table I .

 I Acoustical properties and Young's moduli for mixed gelatine-agar samples with a small amount of Germall-plus.

	Composition	Acoustical	Acoustical	Young's
	Gelatine%	Agar%	wave velocity [m/s]	wave attenuation	modulus [KPa]
				[db/Hz/mm]
	1%	0.5%	1503±8	0.05±0.02	22±2
	5%	2%	1510±4	0.10±0.02	33±1
	12%	4%	1543±11	0.13±0.02	73±9
	15%	6%	1550±7	0.25±0.02	117±8

Table II .

 II Acoustical properties and Youngs's moduli for real breast tissues.

	Real tissue type	Acoustical	Acoustical	Young's
		wave	attenuation	modulus
		velocity[m/s]	[dB/Hz/mm]	[kPa]
	Fat	1479±32	0.05	19±7
	Glandular /Fibrous	1553±35	0.14	33±11
	tissue
	Carcinoma	1550±35	0.3	99±33

ACKNOWLEDGMENT

We thank colleagues from LIP (Laboratory for Instrumentation and Experimental Particle Physics, Lisbon), especially Catarina M. F. Ortigã o and Andreia M. A. Trindade Rodrigues, and the Instituto Portuguê s de Oncologia, Porto, for precious help in several phases in this work


Keywords: Ultrasonic, crack detection, non-destructive testing, acoustical signature discrimination, spectral analysis, wavelet decomposition PACS numbers: 43.35.Zc, 43.60.Ac, 43.60.Hj Ultrasonic discrimination and modeling for crack-tip echoes, P. Lasaygues Ultrasonic crack discrimination and modeling Ultrasonic discrimination and modeling for crack-tip echoes, P. Lasaygues

The problem of discriminating between the acoustical signatures of an open crack tip and an interface is approached by modeling the various ultrasonic signatures, and by a signal-processing method. In the modeling study, the Kirchhoff approximation (the physical optics approximation) is used to describe the far field scattered by an arbitrary object with non-negligible dimensions in comparison with the wavelength, and to express this field depending on the incident field. It is established that, under this condition, the impulse response of an open crack tip is proportional to the first-order derivative of the impulse response of the transmitter. In the signal processing study, spectral and wavelet analysis was applied to the discrimination problem. The basic idea was to discriminate signatures simultaneously in time and in frequency (scale) domain.

This method was found to be an effective means of testing models for the interactions between waves and flaws. Both aspects (modeling and signal processing) were studied numerically and experimentally, and the validity of the results was tested on an industrial sample.

Introduction

Ultrasonic crack detection is a signal discrimination problem: one has to differentiate between the signatures of flawed regions of the material and those of intact regions. In the field of ultrasonic inspection methods, echoes of several kinds are known to occur, since they can be generated by interfaces, by the structure, by the body or tips of cracks, or by inclusions. Those echoes result from various acoustic processes [START_REF] Morse | Theoretical acoustics[END_REF], ranging, under approximations, between two extremes: pure reflection where the echoes are "copies" of the emitted signal, with some distortions due to the propagation, and pure scattering where the echoes are "copies" of the derivative of the emitted signal, again with some distortions [START_REF] Vadder | Caractérisation ultrasonore des bords de fissure par traitement numérique du signal[END_REF]. The spectral signatures of these various processes differ significantly, and the idea naturally arose of attempting to model crack/wave interactions and to use a wavelet analysis to discriminate different ultrasonic waves phenomena.

Ultimately, our aim is to solve an inverse scattering problem (the discrimination of defaults), and first the corresponding forward problem must be solved. Different methods have been applied to the forward problem, including equation integral, hybrid finite-elements methods and the geometrical theory of diffraction. The latter method provides asymptotic approximations for diffracted fields, valid for high frequencies and large distances from the diffracting body [START_REF] Borovikov | Geometrical Theory of Diffraction[END_REF]4,[START_REF] Keith | High-wavenumber acoustic radiation from a thin-walled scarfed cylinder[END_REF]. Several strategies are possible allowing simple, efficient, and precise numerical modeling of the forward and the copyright / 5 inverse problem. With the Kirchhoff approximation, also known as the physical optics approximation, which applies to non-penetrable targets and when the wavelength is smaller than the characteristic size of the diffracting body, the far-field back-scattered impulse response of the object can be seen to be proportional to the second derivative of the cross-sectional area function of the object in the transmitted-receiver direction with the time-space variable change. This identity is known as the Bojarski identity [START_REF] Bojarski | A survey of the Physical Optics Inverse Scattering Identity[END_REF] in the context of electromagnetic scattering. Its inversion leads to the POFFIS (Physical Optics Far Field Inverse Scattering) identity, which is used in radar and underwater technology for identifying the shape of targets [START_REF] Li | Fredholm integral equation for shape reconstruction of an underwater object using backscattered ramp response signatures[END_REF].

In ultrasonic non-destructive testing, since the same transducer is used for both the transmission and the reception of the waves, the signal recorded is the convolution between the back-scattered impulse response of the medium and the impulse response of the whole electro-acoustic system, which is defined as the back-scattered impulse response of a perfect normal flat reflector.

If the crack, which is assumed to be sufficiently open, is modeled in the form of a nonthin parallelepiped and if we consider only the terminal portion of the crack, the crosssection area function of this diffracting object is a Heaviside function and its second derivative is the first derivative of a Dirac distribution. The echographic response of the terminal portion of the crack will therefore be the first derivative of the impulse copyright / 6 response of the electro-acoustic system, i.e. the first derivative of the response to a normal flat reflector.

The results obtained with this model show good agreement with our experimental data.

It can be used to discriminate between the echoes back scattered by a crack and by an interface by performing spectral analysis: the transfer function between the two echoes must be a ramp function [START_REF] Varadan | Image reconstruction of flaws using ramp response signatures[END_REF][START_REF] Varadan | Image reconstruction of flaws using ramp response signatures[END_REF].

When the crack-tip and the interface coincide, however, the spatial (temporal) resolution, lost in the case of classical Fourier analysis, has to be preserved. It is therefore necessary to perform time-scale analysis using wavelet analysis methods, for example.

I. Far-field back-scattered impulse response of a scatterer

Let us take a convex obstacle V, which gives rise at the observation point N to a scattered field 1 ψ . The scattered field (displacement) can be modeled by the following integral in terms of the density µ ~ and the 3-D Green function g , which is the solution of the Dalembert homogenous equation, on the surface S (the boundary of V):

∫∫ µ - = ψ ' t S 1 dP ' dt ) ' t , P ( ~ ) ' t , P ; t , N ( g ) t , N ( (1) Since ( ) c r t r 4 1 ) t , r ( g   - δ π =
where NP r = and c is the compressional wave velocity (acoustical assumptions only), we obtain the following expression for the total field ψ copyright / 7

∫ - µ π - ψ = ψ S 0 dP ) c r t , P ( ~ r 1 4 1 ) t , N ( ) t , N (  (2)
where

) t , N ( 0 ψ
is the incident field at the point source A.

To introduce the Kirchhoff approximation for the field on S, we divide the surface S into the "lit" and "dark" side depending on the straight rays emitted by the point source A, which are incident on S + (Figure 1). We assume that the exact boundary condition stipulates that the total field must be zero on S + . Under this condition, the density µ ~ is given by:

' n ) t , P ( 2 ) t , P ( ~0 ∂ ψ ∂ = µ (3)
and the diffracted field by:

∫ + - - ψ ∂ ∂ - π - = ψ S 2 0 d x d ) c ' x x t , ' x ( ' n ' x x 1 4 2 ) t , x (       (4)
where '

x  is the distance from the origin O to point P on S + , and x  is the distance to point N.

We now assume that the receiver and the transmitter are located far from the scatterer.

Taking the origin O to be inside the obstacle, we then use the following asymptotic approximation:

x . n x ' x x      - ≅ - (5) copyright / 8 where x x n    =
is the unit normal vector in the observed direction. The following result is obtained for the diffracted field:

c x t ' t S 2 0 ' x d ) c ' x . n ' t , ' x ( ' n x 4 2 ) t , x ( d       - = ∞ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ψ ∂ ∂ π - = ψ ∫ + (6)
The value of the incident field 0 ψ is the free-space solution of the wave equation:

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - δ - π - = ψ c x x t x x 4 ) M ( f ) t , x ( 0 0 0      (7)
where ) M ( f is a function describing the directivity of the source (typically a transducer with a directivity which is restricted to an axial cone, and no side lobes are take into account).

Introduce the asymptotic behavior:

x n x x x r r r r r . 0 0 0 + = - (8)
where 0 n  is the unit normal vector in the incident direction.

From these assumptions, we can also conclude that the diffracted field can be written:

( ) c x x t ' t S 2 0 0 0 d ' x d ) c ' x ). n n ( ' t ( ' n x x 4 ) M ( f 2 ) t , x (          + - = ∞ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - - δ ∂ ∂ π - = ψ ∫ + (9) copyright / 9
If a single transducer is used, we have

) 0 , 0 , 1 ( n n 0 = - =  
, and the measured field is the back-scattered field:

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - = ψ ∞ ∞ c x 2 t , x h x ) M ( f ) t , x ( r 2 r     (10)
where

( ) ∫ + - δ ∂ ∂ π = ∞ S 2 i 2 r ' x d ) c ' x 2 t ( ' n 8 1 t , x h   (11)
is the infinite back-scattered impulse response of the obstacle:

( ) ∫ + - δ π = ∞ S 2 ' i 2 r ' x d ) 2 t c x ( ' 8 c t , x h   (12)
We then decompose the volume of the obstacle into a "lit" part with a surface S + consisting of the wave front and the last section before the dark part, and we define the characteristic function of the "lit" part as follows:

not if 0 V x if 1 ) x ( = ∈ = = Γ +   (13)
and the cross-sectional area function of the scatterer (Figure 2):

' 3 ' 2 x V dx x d ) x ( ) x ( A 1 ∫ ⊥ + ∩ Γ =  (14)
We then obtain: copyright / 10 ( )

2 ct x V 2 2 2 r ) x ( A x 8 c t , x h = ∞ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ π = +  (15)
The far-field back-scattered impulse response of an obstacle is proportional to the second derivative of the cross-sectional area function, in the transmission/reception direction, after making the appropriate change in the time-space domain.

II. Diffraction of a crack tip

If we now introduce a non-thin crack modeled in the form of a stiff rectangular parallelepiped (Figure 3) with no internal propagation, the area function can be written:

) x ( Y L L ) x ( A 1 z y 1 V * = (16)
where ) x ( Y 1 is the Heaviside function, y L is the thickness of the crack, which is much smaller than the wavelength λ in the material, and z L is the depth of the crack, which is much larger than the thickness. Since it has been established that:

x ) x ( x ) x ( Y 2 2 ∂ δ ∂ = ∂ ∂ (17)
The far field back-scattered impulse response of the open tip of a non-thin crack is proportional to the first derivative of a Dirac distribution:

( ) ( ) 2 ct x L L 8 c t , x h z y 2 r δ ∂ ∂ π = ∞  (18) copyright / 11
In the frequency domain, the transfer function is:

( ) z y r L L 2 ik k H π - = ∞ ( 19
)
where k is the wave number of the compressional wave.

III. Ultrasonic modeling

The data required here obviously has to be collected over a broad range of arbitrary directions. In our application, the scatterer is immersed in a water bath and the backscattered echoes are measured at various incident angles. The ultrasonic mechanical device is therefore composed of a main symmetric arm, and since the transducer can be exactly positioned and oriented, both linear and sectorial scanning can be performed.

Six stepping motors sequentially driven by a programmable translator-indexer device fitted with a power multiplexer are used to generate all the movements. The translatorindexer device and the power multiplexer are integrated into a control rack that also includes other remote control devices. The increments are multiples of 0.75 10 -2 millimeter for translations and of 10 -2 degree for rotations. The transducers are Panametric ® transducers with a nominal frequency of 5 MHz. The Panametric ® multiplexer excites the transducers and organizes the ultrasonic Radio-Frequency (RF-) signals received, which were composed of 256 samples and were digitized (8 bits, 20 MHz).

copyright / 12

The back-scattered impulse response of the scatterer measured at angle φ can be written (Figure 4):

(

)

) t ( s h h ) t ( s i m t ⊗ ⊗ = (20) where ) t ( h t
is the impulse response of the transducer including the whole electroacoustic apparatus, which is assumed to be linear, ) t ( h m is the impulse response of the scatterer and ) t ( s i is the source, which is assumed to be a perfect electronic pulse δ(t).

⊗ denotes the operation of "convolution".

If the scatterer is a perfect flat reflector, the back-scattered impulse response can be assumed to be a Dirac function (distribution):

) t ( ) t ( h m δ = (21)
we obtain:

( )

) t ( h ) t ( h ) t ( s t t = δ ⊗ δ ⊗ = (22)
First result:

In the case of a perfect flat reflector, which in the present case was a rectified brass plate (Figure 5 and Figure 6), the RF-signal recorded in the back-scattered mode will be a "copy" of the impulse response of the transducer: copyright / 13 ( )

2 ct h ) t ( s t = (23)
If the scatterer is a crack tip, however, ) (t h m is given by Eq. (18), and s(t):

( )( ) ( ) ( ) 2 ct t h L L 4 c t h h ) t ( s t z y r t δ ∂ ∂ ⊗ π = ⊗ = ∞ (24)
Second result:

The signal diffracted by a non-thin crack tip is a "copy" of the first derivative (Figure 7) of the impulse response of the transducer:

( ) 2 ct h t C ) t ( s t ∂ ∂ = (25)
where C is a constant.

IV. Results

The present study focuses on an industrial problem, namely the non-destructive testing of thick cracked stainless steel/steel coated plates measuring 10 cm in width and thickness, and 30 cm in length. The stainless steel coating was 1 cm in width and thickness, and 30 cm in length. The compressional wave velocity measured was 5866 m/s in the steel, and was 6330 m/s in the stainless steel. Figure 8 shows the enhancement procedure used, which was adapted for use with this ultrasonic method of assessment. A transducer is moved linearly along the interface between the water and the steel target (on the steel side), working from the intact part to the damaged part. In copyright / 14 the case of protocol "A", the transducer forms an incident angle of 0° with respect to the normal vector of the water/steel interface. In the case of protocol "B", the transducer is tilted so that the incident beam reaching the interface between the two steels is directed at an angle of 60°, in keeping with Snell-Descartes laws. Since this angle is lower than the critical angle (68°) of the compressional waves, no shear waves propagation is taken into account in this case.

The interface between the two steels is plane and weakly contrasted. Any cracks occurring will develop near the interface, approximately perpendicularly to it, so that the crack tip echo and the specular echo generated by the interface will completely overlap when normal ultrasonic inspection procedures are used (protocol "A"). The problem is therefore not how to discriminate between crack and interface, but more subtly, how to discriminate between a cracked interface and a healthy interface. The ultrasonic power of the back-scattered echoes generated by the interface between the two steels and by the crack tip is weak enough to be able to assume that the interference introduced by the wave propagation is linear, and that the corresponding signal result from the sum of two echoes generated by the interface and the crack tip. The modeling problem can then be solved with a weak scattering solution, such as the physical optics approximation. Each signal is modeled separately. When dealing with the crack, the reference echo obtained on a brass-rectified plate was used for the derivation, and when copyright / 15 dealing with the interface, we focused on the back-scattered echo originating from the interface between the two steels.

Figure 9 shows the experimental signal obtained from a back-scattered echo at the interface between the two steels in the intact region (protocol "A"). Figure 10 shows the experimental signal obtained from the back-scattered echoes originating from the crack tip (protocol "B"). In this case, there are no specular echoes because the incident beam is at an angle of 60°, and the signal recorded corresponds to the echo generated by the tip of the crack alone. Figure 11 shows the normalized comparison between the two signals.

Figure 12 shows the modulus of the spectra of the echoes generated by the interface between the two steels (solid line) and the crack tip (plus line). The division was made in the bandwidth of the transducer (point line) at -6 dB. Figure 13 gives the spectral law (solid line) and the polynomial approximation (point line), which takes the form of a ramp function with a slope equal to 0.98. It can be concluded that the spectral division varies proportionally with the wave number k. This is in agreement with Eqs. ( 19) and (25). The impulse response of a crack tip is a first-order derivative of the impulse response of the transmitter depending on the incident field.

In Figure 14 and Figure 15, the experimental back-scattered echo generated by the crack tip can be compared with the model (i.e. the first-order derivative of the impulse response of the transducer, Figure 7). The two signals are in good agreement, which copyright / 16 confirms the theoretical assumption adopted and the modeling procedure presented here.

Lastly the echo back scattered by the cracked interface is compared with the model.

Figure 16 shows the echo obtained at the cracked interface using protocol "A" in the damaged region. This signal results in theory from the separate contributions of the echoes generated by the interface and by the crack.

Figure 17 shows the back-scattered impulse response of a cracked interface predicted by the model. This prediction is the numerical sum of the impulse response of the transducer and its first order approximate derivative. Figure 18 shows the normalized comparison between the two time graphs.

To check the validity of our modeling procedure, we used a time-scale approach to the signals, namely the wavelet analysis method (see appendix VI). An analysis was performed between scales j = -4 (0.83 MHz) and 0 (13.3 MHz), and 2 5 -1 intermediate scales (γ) were used between each entire scale. The experimental signal (Figure 19) and the model (Figure 20) for the interface in the intact region (signals on the left) and for the cracked interface in the damaged region (signals on the right) were analyzed. The maximum coefficient was obtained for the model on the scale γ = -2.06 and for the experimental signal on the scale γ = -2.15. The behavior of the material on a scale corresponding to low frequencies, as well as the timing of events was very similar in both cases.

copyright / 17

In both analyses, we noted the halving of the decompositions on scales γ > -1.5, 4.71 MHz, near the nominal frequency of the transducer, 4.3 MHz, which extended a little less towards scale 0 in the case of the model, however.

Two conclusions can be drawn from the results of the present study. First, the comparisons between the wavelet analysis of the experimental and simulated signals confirmed that there was good agreement between the predictions of the model and the experimental data, as regards the acoustical signature of the damaged/cracked interface and that of the intact interface. The modeling procedure presented here therefore provides a useful tool, since it is a simple method giving accurate results.

Secondly, the wavelet method can be used to discriminate between an echo resulting from an intact region and one resulting from a damaged region. The halving of the upper part of the wavelet decomposition showed the existence of two contributions, corresponding to the echoes originating from the interface and from the crack.

V. Conclusion

Here we address the problem of detecting and characterizing cracks in the field of nondestructive testing of materials, taking a signal processing approach. In the frequency domain, the ratio between the signal back scattered by an intact interface and that back

VI. Appendix

For the wavelet algorithm, a redundant algorithm in scales, which was developed specifically for ultrasonic signal processing, was used [START_REF] Lasaygues | Deconvolution and Wavelet Analysis on Ultrasonic Reflexion Tomography[END_REF]. It quickly calculates the wavelet coefficients of the signal on the time and (frequency) scale bands.

  scattered by the tip of an open crack was found to be proportional among the waves number k. Based on the physical optics approximation, it is established that the impulse copyright / 18 response of scatterers of this kind is associated with the first-order derivative of a Dirac distribution. Adapting this response to the electro-acoustical context, the impulse response of a non-thin open crack can be said to be associated with the first-order derivative of the impulse response of the transducer, i.e. the incident wave. This result, which has often been assumed to be true in the literature, is proved here explicitly, adopting the hypotheses and the experimental conditions involved in real-life tests. The predictions of the model developed here and the results obtained by performing wavelet analysis on real and simulated signals show this theoretical law to be in particularly good agreement with the experimental data.The present method based on a combination of modeling and wavelet analysis provides a useful time-frequency tool for processing the signals and analyzing the physical processes involved in the interactions between ultrasonic waves and materials. It could serve as the basis for developing industrial methods of non-destructive in situ crack testing.

copyright / 19

The time-frequency and time-scale analyses are often used when processing biological signals [START_REF] Akay | Time frequency and wavelets in biomedical signal processing[END_REF]. Wavelet decomposition of the signals is a transformation that depends on discrete parameters [START_REF] Meyer | Orthonormal wavelets[END_REF].

We associated the series of coefficients with a signal S as follows:

where (j, k) belongs to Z 2 . When j < 0, we have dilation and when j > 0, we have compression. C j,k is the coefficient associated with S on scale j and with sample k.

The redundant algorithm in scales is an extension of the definitions published in studies on non-integer scales. In particular, we extended the definition of C j,k , j∈Z to γ ∈ R.

The wavelet Ψ was previously analyzed by S. Jaffard [START_REF] Jaffard | Décompositions en ondelettes "Developments of Mathematics 1950-2000[END_REF][START_REF] Jaffard | Wavelets: Tools for Science and Technology[END_REF]. Its spectral modulus reaches a maximum at 2/3 F e , where F e is the frequency rate of the signal (20 MHz in this study). It has a bandwidth equal to F e and a -3dB bandwidth equal to ½ F e . In the time domain, it is real and lower than 10 -3 except for an interval 13U t (U t = 1/Fe) in length centered on ½ U t , which is symmetric with respect to ½ U t .


Dominique Guégan
email: dominique.guegan@univ-paris1.fr

Wayne Tarrant
email: w.tarrant@wingate.edu

Viewing Risk Measures as Information

Keywords:

Regulation and risk management in banks depend on underlying risk measures. In general this is the only purpose that is seen for risk measures. In this paper we suggest that the reporting of risk measures can be used to determine the loss distribution function for a financial entity. We demonstrate that a lack of sufficient information can lead to ambiguous risk situations.

We give examples, showing the need for the reporting of multiple risk measures in order to determine a bank's loss distribution. We conclude by suggesting a regulatory requirement of multiple risk measures being reported by banks, giving specific recommendations.

Introduction

In every instance that we can find, risk measures are used to compute an amount of capital that an institution should hold in order to remain financially solvent, whether this calculation is performed for a bank's own internal risk department or for the requirements of an external entity. For instance regulatory bodies like the Basel Committee on Banking Supervision require the reporting of the Value at Risk (VaR) in order to legislate the amount of cash reserves a bank must have. But should this be the only function of a risk measure?

In this paper we present the thesis that the reporting of a risk measure is the revelation of a piece of information about the financial health of an institution. We show that there are multiple possibilities for loss distributions when only a VaR is reported. We then introduce two other measures, the Expected Shortfall and the Maximum Loss. We show that reporting any one of these measures can lead to multiple possible loss distributions.

Following the old adage that more information is always better, we consider the situation of reporting any two risk measures. Again, we show that there is ambiguity about the loss distribution, no matter which two measures we choose. We show the same result for three measures and for five measures. We then conclude with some suggestions for regulators and with some suggestions for further research.

How can we measure risk?

There is little wonder that people are confused by what the term "risk" ought to mean. For instance, the website businessdictionary.com defines risk in seventeen different "general categories". Now, each of the seventeen definitions are relevant to the situations of banks and securities firms. The unifying theme for each of the definitions is that risk requires both uncertainty and exposure. If a company already knows that a loan will default, there is no uncertainty and thus no risk. And if the bank decides not to loan to a business that is considered likely to default, there is also no risk for that bank as the bank has no exposure to the possibility of loss.

If we desire to measure risk in some way, one goal could be to have a single risk measure that will account for all the risk that a bank or securities firm might encounter. Some have objected to the risk measure being a single number, but there is some support for this idea. Investing is always a binary decision-either one invests or one chooses not to invest. Thus the argument is that, given a single number, one should have enough information to decide whether to invest or not. There have been some general agreements about the kinds of properties that such a risk measure ought to possess.

Virtually everyone would agree that if the payout for an investment is always positive (after accounting for the risk-free interest rate), then there is truly no risk of loss in the investment. Many people would say that there is ten times as much risk in investing $10,000 as there is in investing $1000 simply because there is ten times as much money at stake. And most would acknowledge that having cash on hand makes them feel safer about making an investment. It was exactly these thoughts that were, respectively, codified into the definition of risk measure.

Let X be a random variable. Then ρ is a risk measure if it satisfies the following properties:

• (Monotonicity) if X ≥ 0, then ρ(X) ≤ 0 • (Positive Homogeneity) ρ(αX) = αρ(X) ∀ α ≥ 0 • (Translation Invariance) ρ(X + a) = ρ(X) -a ∀ a ∈ R.
The idea behind this definition is that a positive number implies that one is at risk for losing capital and should have that positive number of a cash balance on hand to offset this potential loss. A negative number would say that the company has enough capital to take on more risk or to return some of its cash to other operations or to its shareholders.

The risk measure that is most used is the Value at Risk (VaR). Essentially the α-Value at Risk is that number L so that we can expect the losses to be worse than L exactly 1 -α of the time. For instance if the 95%-Value at Risk of our position is $100, we would expect to lose more than $100 only 5% of the time. A more formal definition can be stated as follows:

The α-Value at Risk of a position X, VaR α (X) = -in f {x : P(X > x) ≤ 1 -α} Since the 1951 paper of Markowitz, many have also favored the usefulness of diversification. VaR does not account for this preference, and a simple example shows this. Consider the case where a bank has made two $1 million loans and one $2 million loan, each with a 0.04 probability of default and all pairwise independent. Then the 95%-VaR for each loan is $0. Thus, if we construct a portfolio consisting solely of the $2 million loan, we must have a 95%-VaR of $0. If we instead choose diversification and make our portfolio out of the two independent $1 million loans, something paradoxical happens. The probability of both loans defaulting is 0.0016, but the probability of exactly one loan defaulting is 0.0768. This implies that the 95%-VaR of our diversified portfolio is $1 million. Thus, VaR does not favor diversification. This leads Artzner, Delbaen, Eber, and Heath (1997) to term VaR as incoherent. They define a coherent risk measure as a risk measure that also favors diversification in the following way:

if X 1 , X 2 are random variables, then ρ(X 1 + X 2 ) ≤ ρ(X 1 ) + ρ(X 2 )
The most commonly used coherent risk measure is the Expected Shortfall. The α-Expected Shortfall is the amount one would expect to lose on average, given that one is in a case where the loss is greater than the α-VaR. Specifically, the α-Expected Shortfall of a position X, ES α (X), is

ES α (X) = -E[x]|x < -VaR α (X).
Finally, most risk managers want to account for the worst case scenario. Indeed it is only prudent to understand what could be at stake before agreeing to take a position. This leads to the introduction of one final risk measure, called the Maximum Loss. Given a position X, the Maximum Loss of a position X, ML(X), is the worst loss one could experience with nonzero probability, i.e.

ML(X) = -in f {(X)|p(X) > 0}.

Ambiguity left by the reporting of risk measures

We note that in the rest of this paper we consider a single timeframe for all the risk measures under consideration. That is to say that all calculations are assumed to be over the identical historical timeframe. For example we might consider a 100-day 95% VaR and a 100-day 95% ES, but in this section we will not consider a 100-day 95% VaR and a 50-day 95% ES. Of course the Maximum Loss could be an historical one, though this is often computed from a model of expected returns or losses.

Insufficiency of a single measure

Here, we discuss the weakness of any single measure of risk for risk management strategy.

In the following we are concerned with tails of probability distribution functions, i.e. if f (x) is the pdf of a loss distribution, then we are concerned with the region [c, d] with the property that ∫ d c f (x)dx = 0.05 and that f (x) = 0 for all x > d. In Figure 1,[c,d] is the region of the graph that we have shown. This is often called the tail of the loss distribution. In Figure 2 and following, we will only see the region [c, d]. Under our notation, the 95%-VaR(X) = c, the ML = d, and the 95%-ES(X) = [START_REF] Artzner | Thinking Coherently[END_REF] .05

∫ d c x f (x)dx.
We also want to mention that the probability density function of the loss is just the negative of the probability density function for the returns.

In this and the following sections we will also adopt the notation that the first graph in each figure represents a loss distribution X 1 , the second represents a loss distribution X 2 , the third a loss dis-Figure 1: The region of interest [c,d] in a loss pdf tribution X 3 , and so on. There is no significance to this. It is purely for ease of exposition.

Although Value at Risk is used by the Basel Committee for determining the capitalization that a bank needs, it turns out that VaR is insufficient for distinguishing between some kinds of different risk characteristics. Assume that a bank reports that it has a 95%-VaR of 0 for its present balance sheet.

The bank is able to have any of the tails in Figure 2 as the tail of its loss distribution. In each case, the 95%-Value at Risk is 0. This is a great misnomer, as there are probabilities of a loss in each case. We see that the loss distribution X 1 gives a uniform probability of loss from the VaR of 0 out to 5. The loss distribution X 2 is a very standard tail of a probability distribution, with the probability of losses decreasing as the size of losses increase. The position X 3 is something that might look strange at first glance. However, such loss distributions may occur when risks are highly concentrated. For instance we have seen increasing probability of larger and larger losses in the real estate market in some American cities. As one house is foreclosed upon and sits empty, it is a blight to a neighborhood, causing other houses to fall in value and become more likely to enter foreclosure. Since community banks often focus their loans in one particular locality, there is a potential for such a distribution to occur.

It is notable that our other two measures can help us distinguish among the three loss distributions. We can calculate that ES 95% (X 1 ) = 5 2 , ES 95% (X 2 ) = 20 3 = ES 95% (X 3 ), so we are unable to tell the difference between the second and third loss distributions. However, we have that ML(X 1 ) = 5, ML(X 2 ) = 20, and ML(X 3 ) = 10, which distinguishes each of these three loss distributions from the other two.

Whether one argues that certain loss distributions are likely or unlikely, there is a stubborn fact that remains. The Value at Risk cannot distinguish among uniform, rising, and falling probabilities. Since most bank regulations only require the reporting of VaR, this should be a very troubling fact.

Since VaR turned out to be insufficient for our risk measuring purposes, maybe one of the two other measures is a better choice. We will start with the Expected Shortfall at the 95% level.

In Figure 3 we see that the reporting of the 95%-Expected Shortfall leaves the same issue as before. There are uniform, rising, and falling probabilities, all of which have the same 95%-Expected Shortfall of 10. In this case, the Maximum Loss once again identifies the three situations because ML(X 1 ) = 16, ML(X 2 ) = 15, and ML(X 3 ) = 30. Here we also have that VaR(X 1 ) = 4, while VaR(X 2 ) = 0 = VaR(X 3 ), where each of the VaR calculations are at the 95% level. Because the ES has not been able to distinguish among very different risk situations, we turn to the other measure we have introduced -the Maximum Loss. We will again look at the last 5% of the probability distribution so that we might be able to consider the usefulness of our other two measures. As we see in Figure 4, the risk models have the same Maximum Loss as all the distributions have no probability occurring after the value of 5. Again, there are very different situations of risk here though. The first loss distribution is such that VaR(X 1 ) = -5. For the second and third loss distributions we have that VaR(X 2 ) = 0 = VaR(X 3 ). And this time the Expected Shortfall sorts out the different distributions because ES(X 1 ) = -5 3 , ES(X 2 ) = 10 3 , and ES(X 3 ) = 5 2 .

We hope that the geometry of our figures has pointed out some facts about the different risk measures. In order for us to obtain two distributions with the same 95%-Value at Risk, we only have to begin our loss distribution tails at the same value, making sure that we have 5% of the probability after that common point. For two distributions to have the same 95%-Expected Shortfall, we must have two geometric figures that have the same weighted average. And for two figures to have the same Maximum Loss, we would require a common point as the maximum value for which both loss distributions have a nonzero probability.

Insufficiency of any two risk measures

Since individual measures failed to give a thorough picture of the risk characteristics of a given loss model, maybe the situation will be better if we try using two different risk measures. We will attempt to use measures in pairs to see if they give us a full description of the risk characteristics of a loss model.

We will start by using both the 95%-VaR and the 95%-ES. Again we are left with a problem.

In Figure 5 we see that the loss distribution X 1 is uniform, while X 2 has rising probabilities as losses grow, and X 3 is the most standard looking probability tail. It is discouraging that the two most frequently used measures do not distinguish among these three situations. Each of the distributions has a 95%-VaR of 0 and each of them has an 95%-ES of 10. With such different loss characteristics sharing the same VaR and ES, it is astounding that these are the only two measures mentioned in the Basel 3 document. We are encouraged, though, because here the Maximum Loss will distinguish among the loss distributions as ML(X 1 ) = 20, ML(X 2 ) = 15, and ML(X 3 ) = 30. Now we try to use the pair of measures that are the 95%-VaR and the Maximum Loss. As shown in Figure 6, we can still have some ambiguity, as in previous examples. In this case we have all three loss distributions with 95%-VaRs of 0 and Maximum Losses of 5. Here the 95%-Expected Loss will distinguish the loss distributions as ES(X 1 ) = 5 3 , ES(X 2 ) = 5 2 , and ES(X 3 ) = 10 3 .

So now we will consider the final pair, the circumstance of using the 95%-ES and the ML as our risk measure pair.

Insufficiency of all three measures

With the failure of two measures to definitively distinguish among different risk situations, we move to the next logical step. Here we look at the 95%-VaR, the 95%-ES, and the Maximum Loss.

In Figure 8 all four of our examples have a 95%-VaR of 0, a 95%-ES of 5, and a ML of 10. The most compelling question here is whether these loss distributions are even possible in actual situations. We see uniform distributions with great regularity, so it should be clear that such a distribution is possible in a financial setting. The second loss distribution seems strange, but we argue that it is possible with properly chosen correlation. If we go back to the example of community banks making loans, we would expect a distribution that is very nearly normal. However, if this bank were to make loans in several communities, it is possible to see ramping probabilities in one neighborhood due to contagion. Then there may be other neighborhoods where most of the individuals have a large percentage of equity in their house. Such a situation could lead to the second loss distribution. And similar arguments can apply for the third distribution. Once that final neighborhood sees the blight of an entire region due to contagion of foreclosures, we could see that final neighborhood also experience increasing numbers of foreclosures.

The fourth loss distribution is nearly impossible for us to justify. However, as an academic exercise, it does give rise to a potentially infinite class of examples that share the same VaR, ES, and ML. In order for our examples to have equivalence of all three measures, we have to meet three criteria. We need the same "starting points" of the worst 5% of returns in order to make the VaRs equal. We require identical "ending points" for the MLs to be equal. Since the uniform distribution has symmetry, it will have a 95%-ES of 5. Thus, any distribution with an equivalent 95%-ES (and the same starting and ending points) must also be symmetric in the greatest 5% So our three measures do not distinguish among the potential scenarios that we might face. Obviously, something more is needed.

Extending to different probability levels

We have chosen to use the 95% level for our risk measures, but there is nothing special about the 95% level. Our examples would be just as valid if we substituted 90% or 99% or any other number in all of our measures and then scaled identically in all of the corresponding diagrams. However, this concept of considering different risk levels might also lead us to enough risk measures so that we can determine the universal risk profile from information about the different risk measures.

So now we will try using a vector of measures consisting of the 95%-VaR, the 99%-VaR, the 95%-ES, 99%-ES, and the ML. The different risk profiles demonstrated in Figure 8 are quite oddlooking, but they all have identical 95%-VaRs, 99%-VaRs, 95%-ESs, 99%-ESs, and MLs. The three distributions all have the necessary geometric properties for equality of all five measures. If we wanted to form a larger family of examples with the same values for all five measures, we would need our new distributions to have the same "ending points" in order to match up the MLs. We would need them to have the same "starting points" for both the greatest 5% of losses and the greatest 1% of losses in order to have equal 95%-VaRs and equal 99%-VaRs, respectively. And we would need symmetry in the greatest 5% of losses and in the greatest 1% of losses in order to have 95%-ESs and 99%-ESs, respectively, equal to the 95%-ES and the 99%-ES of the uniform distribution. Any figure meeting all these criteria would also be indistinguishable from the first five when viewed purely by looking at the values in our vector of five measures. Clearly several different profiles can share all five measures.

In spite of our effort in the academic exercise of drawing such figures, we do not claim that such loss distributions will occur naturally, with the obvious exception of the uniform distribution. Further, the loss distributions X 2 and X 3 do not differ by a great deal. Therefore, we are prepared to say that five risk measures should distinguish among differing loss distributions in almost all natural circumstances.

Conclusion and Future Directions

We have shown ambiguity among situations of rising and falling probability in loss distributions when one, two, and three risk measures are reported. Thus, we suggest that banks be required to submit five risk measures to regulatory bodies, specifically a 95%-VaR, 99%-VaR, 95%-ES, and 99%-ES, and a Maximum Loss, where this is possible. We have demonstrated that the reporting of five risk measures can theoretically lead to an infinite family of possibilities for the actual loss distribution, but we have also conceded that such loss distributions are unlikely to occur. Thus, we feel that banking regulations would be much safer if such regulations require the reporting of five different risk measures.

This is an interesting theoretical result that must be pursued on actual loss data. A paper that looks at the ability to differentiate between different types of common loss distributions is being written with my student, Cole Arendt. I am presently considering the question of how many risk measures are needed if we vary the timeframes on the historical data along with the level and the specific measures. Finally, another paper that looks at the predictive ability of historical risk measures is also in the works.

Figure 2 :

 2 Figure 2: Loss distributions with 95%-VaRs equal to 0

Figure 3 :

 3 Figure 3: Loss distributions with 95%-ESs of 10

Figure 4 :

 4 Figure 4: Loss distributions with MLs of 5

Figure 5 :Figure 6 :Figure 7 :

 567 Figure 5: Loss distributions with equal 95%-VaRs and equal 95%-ESs

Figure 8 :

 8 Figure 8: Four loss distributions with equal 95%-VaRs, 95%-ESs and MLs

Figure 9 :

 9 Figure 9: Using five risk measures