Noise limitations on the recovery of average values of velocity profiles in pipelines by simple imaging systems

Transcription

1 Noise limitations on the recovery of average values of velocity profiles in pipelines by simple imaging systems Massimo Brignone Gian Luigi Gragnani Matteo Pastorino Mirco Raffetto Andrea Randazzo April, 26 Abstract This letter deals with a recently proposed two-step algorithm for the reconstruction of dielectric and velocity profiles of axially moving cylinders. In particular, the aim of the present work is to investigate the limitations and capabilities in reconstructing the velocity profile. By considering a best-case scenario, it is shown that, although the algorithm should be able to work even in the presence of flows with very small axial velocity values, reliable results can only be obtained when the axial speed is larger than a lower bound. This bound is shown to be an intrinsic limitation of the algorithm which is only related to the cross-polarization isolation of the measurement system. Other simulations, based on worstcase scenarios, confirm that such a bound is independent of the type of flow considered, of the configuration of the measurement system, and of the errors coming from the step devoted to the reconstruction of the dielectric profile. Finally, the same simulations show that the algorithm is reliable in a wide range of speed values, provided sensors with large cross-polarization isolations are exploited. Keywords: time-harmonic electromagnetic scattering; bianisotropic media; moving media; inverse scattering procedures; reconstruction of velocity profiles. Introduction An important and yet not entirely explored area of electromagnetic applications regards the interaction of electromagnetic fields with moving media. Indeed, although the theory of the interaction between electromagnetic fields and bodies in movement dates back to Einstein s times, from a practical point of view, a lot remains to be done. On the other hand, the electromagnetic scattering from moving bodies not only is a relevant theoretical problem, but also has important applications in many branches of physics and engineering. In particular, a great interest has been devoted to problems involving axially moving cylinders with stationary boundaries. In most cases, the interaction model is developed in the time domain []. In this particular context, however, the movement takes place in such a way that it is still possible to work in the frequency domain [], [2]. To cite just a few examples of applications, there are works related to the reconstruction of the dielectric and velocity profiles of ionized meteor trails [3], axially moving The authors are with the Department of Electrical, Electronic, Telecommunications Engineering and Naval Architecture, University of Genoa, Via Opera Pia a, I 645, Genoa, Italy, massimo.brignone@unige.it, gianluigi.gragnani@unige.it, matteo.pastorino@unige.it, mirco.raffetto@unige.it, andrea.randazzo@unige.it

2 2 Applied Electromagnetic Laboratory: report no plasma columns [4], [5], [6], jet exhausts [7], and mass flows in pneumatic pipes [8]. In recent years, several techniques [8] have been considered to determine the dielectric and velocity profiles of axially moving cylinders. In particular, the use of electromagnetic waves is suggested in many cases [4], [5], [8] and various electromagnetic techniques have been exploited so far. One can refer to techniques based on signals in the microwave or optical bands [8] or to approaches based on the Doppler effects. Many other possibilities have been considered and can be found in the open literature [8]. Microwave inverse scattering techniques [9] have been proved to possess excellent properties in the reconstruction of permittivity profiles in many fields of technology, and in a previous paper [], an effective two-step algorithm was developed, to reconstruct both the dielectric properties and the related velocities. In simple cases, for velocity v z <.c such an approach provided interesting results. In particular, this algorithm was tested on the reconstruction of multilayer elliptic scatterers, where all layers were moving at possibly different velocities []. Such an algorithm is based on a two step procedure: in the first step, by using a TM (with respect to the cylinder axis z) interrogating wave, the dielectric profile is reconstructed by using only the information provided by the z component E z of the scattered fields. This amounts to deal with the cylinder as it would be at rest in the laboratory reference frame (actually, it is well known that, for the considered range of v z, E z is basically independent of the velocity []). The reconstructed profile of ε r is then used in the second phase of the algorithm, which accounts for the movement, and makes only use of the measures of the H z component. This component is strictly related to the motion [] and would be zero if the scatterer was truly at rest in the laboratory frame. Should one consider a TE illumination, the roles of the axial components must be exchanged, but the algorithm remains the same. Although the simulations carried out by using the above algorithm have provided very interesting results and the above two step procedure proved to be very efficient for an impressive range of axial speed values, all previous works were lacking a study on the performances of the algorithm in more realistic situations, where one has to deal with the limitations posed by real world measurement instruments and sensors. For this reason, in the present work, the analysis of the capabilities of the algorithm is extended, with the aim of better evaluating its limitations. In particular, the present work is focused on the capabilities of the second step of the algorithm. Since the possible inability of the methodology is pointed out in a clearer way if the performances of the algorithm are investigated under particularly simple and good conditions, in the first part of our analysis we consider a simple flow, a multitude of illuminating fields and assume that the first step is able to recover the profile of ε r without errors. Our results show that the reconstruction of the average value of β = vz c is possible only when such a value belongs to a range [ x, 2], where x (> ) is inversely proportional to the cross-polarization isolation of the measurement system (the more the isolation, the larger the range). When β (,x) the reconstruction of its average value and, then, of the entire profile of the axial velocity, completely fails. In the second part of our analysis we show that, on the contrary, for β [x, 2 ], which is determined only by the cross-polarization isolation of the measurement system, the reconstructions of the average value of β is particularly reliable and this is shown to hold true for different types of flows inside the pipes, simple illumination systems, simple measurement systems for the second step of the algorithm, and in the presence of errors on the profile of ε r reconstructed by the first step and on the measurements of H z (provided the last two sources of errors are not too large).

3 Noise limitations on the recovery of average values of velocity profiles Limitations of our approach As it was pointed out above, to find the limitations of our approach we consider a realistic bestcase scenario. Such a scenario is obtained by considering a circular pipe of inner radius r = m and thickness t =. m. In the following we will consider a Cartesian reference system whose z axis is the axis of the cylinder. The pipe is made up of a lossy medium having ε r = 2 j, µ r = and it is hosted in a homogeneous material with ε r = j., µ r =. The flow inside the pipe is simple. It is analogous to the stratified flow considered in [8] and is characterized by µ r = and ε r = C +C 2 y () β = β m ( +a(x x ) 2 +a(y y ) 2), (2) where C = 5, C 2 = 3, β m = m, m = 2,3,...,7, a = , x =, y = 3 4. In order to define a best-case scenario we assume that the first step of our algorithm is able to recover the profile of ε r without errors. The performances of the second step of our methodology are analyzed by assuming, moreover, that it can process the measurement data obtained by eight different illuminating TM-polarized fields at GHz and whose directions of propagation are given by the angles n π 4 radians, n =,...,8, in the (x,y) plane, with respect to the x axis. Finally, the measurement data collected by the receiving antennas provide the magnitudes and the phases of H z. We have considered p = 44 uniformly distributed sensors located on a circle centered on the z axis and of radius.8 m. Realistic transmitting antennas and sensors are not able to guarantee a perfect polarization purity and an ideal isolation of the polarizations, however. This aspect was not considered in our former analyses and is the target of the first part of the present work. In order to understand what happen under the indicated conditions, some simulations are carried out by assuming that the values of H z at all sensor positions are corrupted by additive random terms of magnitude proportional to a small fraction of that of the transverse magnetic field, evaluated at the same positions. In particular, in the simulations, the set of measures {H z,...,h zp } is corrupted by a white Gaussian noise whose power is from 3 to db smaller than the power of {H t,...,h tp }. The range of the scaling factor ([3, ] db) is wider than usually considered in applications requiring high polarization purity and very large cross-polarization isolation values [], [2]. This is useful to understand the intrinsic limitations of our algorithm. Our results will be provided in terms of the relative error on the weighted mean value of β, given by b = β(µ r ε r )dxdy. (3) x 2 +y 2 r 2 This is done because b takes into account that the effects of motion are scaled by µ r ε r and, in particular, that vacuum does not give any contribution, independently of its velocity [], [3]. In the reconstruction process we consider a flow with a constant axial speed β c. The optimal value β of β c is determined by minimizing a cost function which compares the measured H z values with the ones obtained when β = β c and the profile of ε r is the one recovered by the first step of our algorithm. Once β is determined, the corresponding b can be computed by replacing β with β in equation (3), so that the relative error b b b can be obtained. In Figure the average value of the relative errors on b are shown. For any configuration considered we have run five independent reconstruction processes. The standard deviation of the relative errors on b is provided in Figure, too. When the scaling factor is equal to 3 db the errors are completely unacceptable unless the peak value of the β profile, β p, is equal to

4 4 Applied Electromagnetic Laboratory: report no (that is the motion is taking place with the largest possible speed, among those which are considered by our two step algorithm). The results are similar also for a scaling factor of 4 db (not shown in the figure). The quality of the reconstruction becomes better and better for higher and higher isolation values. For example, for a scaling factor of 6 db, the graph shows that, although for β p = 4 the results are still unacceptable, when the same value is equal to 3, the average relative error is smaller than. with a negligible standard deviation. In general, the results get better as β p increases. Analogously, the results obtained for a cross-polarization Avg. of the relative error on b db: avg. 3 db: std. 6 db: avg. 6 db: std. 8 db: avg. 8 db: std. db: avg. db: std Std. dev. of the relative error on b e-6 e-5 e-4 e-3 e-2 β p Figure : Behaviour of the average value of the relative errors on b versus the peak value of the β profile, β p, for the best case scenario and for different performances of the antennas in terms of cross-polarization isolation. The standard deviation of the relative errors on b is shown as well. isolation of 8 db (respectively, db) show a very good accuracy with negligible deviations when β p 4 (respectively, 5 ). The other results computed are not shown in the figure (for cross-polarization isolation of 5, 7 and 9 db) but confirm the general behaviour described above. These results could be explained in a qualitative way by the following considerations. For any given amplitude of the transverse magnetic part of the incident field we get a transverse magnetic part of the total field having the same order of magnitude. We know that the isolation determines the power of the additive random noise contaminating the H z data. In particular, the noise to be added to H z has a magnitude roughly equal to that of H t multiplied by q, with q {,...,5} (q = when the isolation is 2 db, q = 5 when it is db). But in [] we showed that, from a qualitative point of view, for small values of β we can expect H z to have an integral norm proportional to the one of H t multiplied by β. Thus, for β = r the

5 Noise limitations on the recovery of average values of velocity profiles... 5 fundamental data for the second step of our algorithm are r smaller than H t. It is now clear that we should not expect good results when r is larger than q. The previous example indicates the need for very accurate measurement systems. In particular, the imaging approach based on two steps can provide accurate reconstructions of the axial velocity profiles only if the maximum value of β is larger than the reciprocal of the crosspolarization isolation (expressed without using decibels). On the other hand, it was already shown [] that the two step algorithm has an intrinsic upper bound corresponding to β p = 2 (this is essentially due to the fact that the motion is neglected in the reconstruction of ε r, in the first step of our procedure). Hence, we are left with a range of average β values which can be empty unless sensors with a very good cross-polarization isolation are used. 3 Reliability of our approach in the presence of good sensors While, to assess the limitations of the algorithm, a best case scenario can be sufficient, in order to test for the capabilities of the proposed approach, some worst cases have been considered. They take account of: a simple illuminating system, different types of flows inside the pipes, errors in the reconstruction of the ε r profile, provided by the first step of our algorithm, different measurement systems for the second step of the algorithm, noise on the measurements of H z, that accounts for realistic disturbances. In particular, in any case we have considered a single illuminating TM-polarized field at GHz propagating in the ŷ direction. Moreover, three profiles of β and ε r have been considered: in the first two we have a classical parabolic profile of β [8] and a constant or a linear profile of ε r ; in the third a roping fluid profile [8] is analyzed. In all tests we refer to in this section, the profile of ε r, provided as an input quantity to the second step of our procedure, is affected by errors, as it will be the case in practical applications in which this field is reconstructed by the first step of the algorithm. In all cases considered in this section, the errors on ε r have been obtained by adding a Gaussian white noise to the true profile. The added noise determines a SNR on ε r of 2 db. For each test, two different measurement set-ups were considered: the first was the same of the best case, with the sensors placed all around the pipe, whereas the second was reproducing a limited view angle of 9 degrees, that was on the back scattering zone, symmetric with respect to the illumination. For both the measurement set-ups, we considered the reconstructions obtained by using complete data (amplitude and phase) or simply the measured amplitudes. Finally, the measurements of H z have been considered as affected by noise. In order to be able to compare our new results with those of the previous section, we define its magnitude in terms of cross-polarization isolation. Taking account of our former analysis and of the target of this section, we have considered noise levels determined by cross-polarization isolation of 6, 8 and db. In the following subsections the results are presented and discussed. In all cases the same pipe and the same outer medium as in our former best case scenario have been considered. Moreover,

6 6 Applied Electromagnetic Laboratory: report no for each set of parameters the related simulation was repeated again five times and the average value of the relative errors on b and the corresponding standard deviation have been evaluated. Finally, all media are characterized by µ r = in all tests. 3. Test In the first test, a parabolic profile for β and a constant value for ε r have been assumed. They are defined by equations (2) and (), respectively. The considered parameters are: C = 5, C 2 =, β m = m, m = 2,3,...,7, a =.55, x =, y =. The behaviour of the average relative error on the weighted mean value of β versus the peak value of the β profile, β p, for different levels of noise on the measurements of H z are shown in Figure 2. The measurements are supposed to be performed with a simple system able to collect amplitude-only data, on an arc of a circle of radius.8 m and subtending an angle of 9 degrees in the back scattering zone. The results of Figure 2 are, from a qualitative point of view, similar to those of Figure. As for a quantitative analysis, the reader can notice that even a very significant reduction of the information processed (single view instead of eight views and data related to H z on one fourth of a circle instead of H z on the whole circle) and a non-negligible error on the profile of ε r provided as input datum, our algorithm is very robust in providing good estimates of the weighted mean value of β. In particular, with respect to the behaviours shown in Figure, the reliability of the estimate is achieved starting from a peak value of β which is ten times as large as the one we have got for the best case scenario and this is rather independent of the noise level. As an example, the results obtained for a noise level corresponding to a cross-polarization isolation of 8 db are very good with negligible deviations when β p 4 for the best case scenario and when β p 3 for the present case. In the presence of more complicated measurement systems the results are better than those shown in Figure 2 and in general get closer and closer to those shown in Figure as the processed data become more and more complete. Some of these intermediate results are shown just for test number three for space reasons. 3.2 Test 2 Also for this second test, a parabolic profile for β have been assumed. In this case, however, ε r has a linear profile. They are defined by equations (2) and (), respectively. The considered parameters (C, C 2, β m, a, x and y ) are the same as in Section 2. The results for this case are shown in Figure 3. They refer to the same conditions considered to get the results shown in Figure 2. The reader can observe a limited deterioration of the quality of the results achieved. This is expected to be related to the more challenging flow considered in test number two with respect to the one of test number one. In this case, too, we have verified that an increase of the information recovered by the measurement system does correspond to a reduction of the reconstruction errors on b. 3.3 Test 3 The last case study aims at the reconstruction of the features of a roping flow, in which all moving media are confined to a subregion of the pipe [8]. In particular, we have considered a profile of β given by equations (2) with β m = m, m = 2,3,...,7, a = 7 2, x =.3, y =.3, when (x x ) 2 +(y y ) 2.6 and β = otherwise. As for the relative permittivity we have ε r = 8( (x x ) (y y ) 2 ) when (x x ) 2 +(y y ) 2.6 and ε r = otherwise.

7 Noise limitations on the recovery of average values of velocity profiles Avg. of the relative error on b db: avg..2 6 db: std. 8 db: avg.. 8 db: std. db: avg. db: std. e-6 e-5 e-4 e-3 e-2 β p Std. dev. of the relative error on b Figure 2: Behaviour of the average value of the relative errors on b versus the peak value of the β profile, β p, for the worst case no. and for different levels of noise on the measurements of H z. These measurements are supposed to collect amplitude-only data. The sensors are placed on an arc of a circle of radius.8 m and subtending an angle of 9 degrees in the back scattering zone. The standard deviation of the relative errors on b is shown as well. The problem analyzed in this case is the most challenging one among those considered in the three tests, due to the discontinuities we have in the profiles of β and ε r. However, these additional difficulties are not enough to prevent our algorithm from getting good results, as shown in Figure 4. The reader can observe a limited deterioration of the results with respect to those of tests one and two. The deterioration is more evident if the comparison is done with Figure. However, even this challenging comparison shows that the deterioration is limited to a restriction of the range of β p allowing good reconstructions of a factor which remains of the order of ten, as in the former tests one and two. For example, the best case scenario allowed an average relative error on the weighted mean value of β smaller than.2 for β p 5, when the noise corresponds to a cross-polarization isolation of db. Figure 4 shows that the same is true in test number three provided that β p is larger or equal to 4. As indicated above, in Figure 5 we report the results obtained when the measurement system is able to recover the magnitude and the phase of H z, even though this is done on the same arc of a circle as before. As it was stated above, the reader can check that these results are better that those shown in Figure 4 and worse than those of Figure. In all tests the quality of the approximation was the worst when amplitude-only data are

8 8 Applied Electromagnetic Laboratory: report no Avg. of the relative error on b db: avg. 6 db: std. 8 db: avg. 8 db: std. db: avg. db: std Std. dev. of the relative error on b e-6 e-5 e-4 e-3 e-2 β p Figure 3: Behaviour of the average value of the relative errors on b versus the peak value of the β profile, β p, for the worst case no. 2 and for different levels of noise on the measurements of H z. These measurements are supposed to collect amplitude-only data. The sensors are placed on an arc of a circle of radius.8 m and subtending an angle of 9 degrees in the back scattering zone. The standard deviation of the relative errors on b is shown as well. collected on a limited arc. A slight increase of the quality is achieved when amplitude-only measurements are carried out on the whole circle. A further reduction of the errors is obtained when the magnitude and the phase are measured on a limited arc. Finally, the performances are very close to those of the best case scenario when the magnitude and the phase are measured on the whole circle. These considerations are confirmed by the data reported in Table, where the average relative errors on the weighted mean value of β for β p = 4 and for a noise corresponding to 8 db of cross-polarization isolation are reported. 4 Conclusions In this letter, a recently proposed two-step algorithm for reconstructing the dielectric and velocity profiles of axially moving cylinders has been further analyzed, with the aim of better defining its limitations and capabilities. It has been shown that, in practice, acceptable results can only be obtained when the indicated velocity is larger than a lower bound which is only related to the cross-polarization isolation of the measurement system. This bound is almost completely independent of the type of flow considered, of the configuration of the measurement system, and of the errors coming from the reconstruction of the dielectric profile. Finally, the same

9 Noise limitations on the recovery of average values of velocity profiles Avg. of the relative error on b db: avg..2 6 db: std. 8 db: avg.. 8 db: std. db: avg. db: std. e-6 e-5 e-4 e-3 e-2 β p Std. dev. of the relative error on b Figure 4: Behaviour of the average value of the relative errors on b versus the peak value of the β profile, β p, for the worst case no. 3 and for different levels of noise on the measurements of H z. These measurements are supposed to collect amplitude-only data. The sensors are placed on an arc of a circle of radius.8 m and subtending an angle of 9 degrees in the back scattering zone. The standard deviation of the relative errors on b is shown as well. simulations have confirmed that, whenever the axial velocity is larger than the above indicated lower bound, the reliability of our algorithm is very good and its accuracy gets better and better as the axial velocity is larger and larger. Our previous works establish, however, that in any case such a velocity should be bounded to about one hundredth of the speed of light in vacuum. References [] J. G. Van Bladel. Electromagnetic Fields. IEEE Press, Piscataway, NJ, USA, 2nd edition, 27. [2] D. K. Cheng and J.-A. Kong. Covariant descriptions of bianisotropic media. Proceedings of the IEEE, 56(3):248 25, March 968. [3] J. V. Parker, J. C. Nickel, and R. W. Gould. Resonance oscillations in a hot nonuniform plasma. Physics of Fluids, 7(9):489 5, September 964. [4] A. M. Messiaen and P. E. Vandenplas. High-frequency effect due to the axial drift velocity of a plasma column. Physical Review, 49:3 4, September 966.

10 Applied Electromagnetic Laboratory: report no Avg. of the relative error on b db: avg. 6 db: std. 8 db: avg. 8 db: std. db: avg. db: std Std. dev. of the relative error on b e-6 e-5 e-4 e-3 e-2 β p Figure 5: Behaviour of the average value of the relative errors on b versus the peak value of the β profile, β p, for different levels of noise on the measurements of H z. These results refer to the presence of a roping flow but in this case the measurements are supposed to be able to get the magnitude and the phase of H z. The sensors are placed on an arc of a circle of radius.8 m and subtending an angle of 9 degrees in the back scattering zone. The standard deviation of the relative errors on b is shown as well. [5] C. Yeh. Scattering obliquely incident microwaves by a moving plasma column. Journal of Applied Physics, 4(3): , December 969. [6] T. Shiozawa and S. Seikai. Scattering of electromagnetic waves from an inhomogeneous magnetoplasma column moving in the axial direction. IEEE Transactions on Antennas and Propagation, 2(4): , July 972. [7] D. Censor. Scattering of electromagnetic waves by a cylinder moving along its axis. IEEE Transactions on Microwave Theory and Techniques, 7(3):54 58, March 969. [8] Y. Yan. Mass flow measurement of bulk solids in pneumatic pipelines. Measurement Science and Technology, 7(2):687, 996. [9] M. Pastorino. Microwave Imaging. Wiley, New York, 2. [] M. Pastorino, M. Raffetto, and A. Randazzo. Electromagnetic inverse scattering of axially moving cylindrical targets. IEEE Transactions on Geoscience and Remote Sensing, 53(3): , March 25.

11 Noise limitations on the recovery of average values of velocity profiles... H z circle H z arc H z circle H z arc Test Test Test Table : Average relative errors on the weighted mean value of β for β p = 4 and for a noise corresponding to 8 db of cross-polarization isolation. The data refer to all tests performed and all measurements systems adopted. [] R. Touzi, P. W. Vachon, and J. Wolfe. Requirement on antenna cross-polarization isolation for the operational use of C-band SAR constellations in maritime surveillance. IEEE Geoscience and Remote Sensing Letters, 7(4):86 865, October 2. [2] T. J. Miller, O. Kilic, and M. S. Mirotznik. Antenna cross-polarization isolation and calibration of hybrid-polarization radars. IEEE Antennas and Wireless Propagation Letters, 2:2 23, 23. [3] M. Brignone and M. Raffetto. Well posedness and finite element approximability of twodimensional time-harmonic electromagnetic problems involving non-conducting moving objects with stationary boundaries. ESAIM: Mathematical Modelling and Numerical Analysis, 49(4):57 92, July-August 25.