^{1,a)}, Mark Andrews

^{1}, Ninos Donabed

^{1}, Ameya Galinde

^{1}, Carey Rappaport

^{1}and Douglas Fenneman

^{2}

### Abstract

An analytic model is developed for the time-dependent ultrasound field reflected off a randomly rough vibrating surface for a continuously scanning ultrasound vibrometer system in bistatic configuration. Kirchhoff’s approximation to Green’s theorem is applied to model the three-dimensional scattering interaction of the ultrasound wave field with the vibrating rough surface. The model incorporates the beam patterns of both the transmitting and receiving ultrasound transducers and the statistical properties of the rough surface. Two methods are applied to the ultrasound system for estimating displacement and velocity amplitudes of an oscillating surface: incoherent Doppler shift spectra and coherent interferometry. Motion of the vibrometer over the randomly rough surface leads to time-dependent scattering noise that causes a randomization of the received signal spectrum. Simulations with the model indicate that surface displacement and velocity estimation are highly dependent upon the scan velocity and projected wavelength of the ultrasound vibrometer relative to the roughness height standard deviation and correlation length scales of the rough surface. The model is applied to determine limiting scan speeds for ultrasound vibrometer measuring ground displacements arising from acoustic or seismic excitation to be used in acoustic landmine confirmation sensing.

I. INTRODUCTION

II. FULL-FIELD MODEL FOR ULTRASOUND SIGNAL REFLECTED OFF MOVING SURFACE WITH RANDOM ROUGHNESS

III. ULTRASOUND SIGNAL ANALYSIS FOR SURFACE DISPLACEMENT AND VELOCITY ESTIMATION

A. Incoherent Doppler shift spectra

B. Coherent interferometry

IV. GENERATING TWO-DIMENSIONAL SPATIAL REALIZATIONS OF THE ROUGH SURFACE

V. NUMERICAL SIMULATION

A. Performance of scanning ultrasound vibrometer for varying surface types

B. Random spectral broadening in the continuously scanning vibrometer

C. Effect of ultrasound frequency, measurement time, insonification area, angle of incidence, and scanning speed

D. Limiting scan speed for acoustic landmine imaging

E. Anisotropic rough surfaces

VI. CONCLUSION

### Key Topics

- Ultrasonography
- 94.0
- Rough surfaces
- 80.0
- Surface scattering
- 33.0
- Mining
- 25.0
- Medical image noise
- 23.0

## Figures

Setup of a bistatic ultrasound vibrometer system. The origin of the coordinate system is located at the mean surface, at the intersection of the source and receiver beam-pattern axes.

Setup of a bistatic ultrasound vibrometer system. The origin of the coordinate system is located at the mean surface, at the intersection of the source and receiver beam-pattern axes.

Magnitude of ultrasound beam pattern in decibels at intercepted by a surface for (a) , and (b) incidence of beam axis with surface. The transducer is a circular disk of diameter and is positioned from the surface. The equivalent beamwidth of the transducer is approximately .

Magnitude of ultrasound beam pattern in decibels at intercepted by a surface for (a) , and (b) incidence of beam axis with surface. The transducer is a circular disk of diameter and is positioned from the surface. The equivalent beamwidth of the transducer is approximately .

Realizations of randomly rough (a) medium sand and (b) gravel surfaces. Spectrum of received ultrasound field, , reflected and scattered off harmonically oscillating (c) medium sand and (d) gravel surfaces for an ultrasound vibrometer continuously scanning the surface at a speed of . The corresponding results for the stationary vibrometer are plotted for comparison. (e) Medium sand and (f) gravel are the corresponding coherently processed spectrum of low-pass filtered mixed signal, . The spectra are plotted for an ultrasound source level of re at ( , , , , , ).

Realizations of randomly rough (a) medium sand and (b) gravel surfaces. Spectrum of received ultrasound field, , reflected and scattered off harmonically oscillating (c) medium sand and (d) gravel surfaces for an ultrasound vibrometer continuously scanning the surface at a speed of . The corresponding results for the stationary vibrometer are plotted for comparison. (e) Medium sand and (f) gravel are the corresponding coherently processed spectrum of low-pass filtered mixed signal, . The spectra are plotted for an ultrasound source level of re at ( , , , , , ).

Spectrum of ultrasound field at reflected and scattered at normal incidence off a (a) smooth and nonoscillating surface for the stationary vibrometer (same result for scanning system), (b) smooth and harmonically oscillating surface for stationary vibrometer (same result for scanning system), (c) nonoscillating rough gravel surface with stationary vibrometer, (d) oscillating rough gravel surface with stationary vibrometer, (e) nonoscillating rough gravel surface with scanning vibrometer, and (f) harmonically oscillating rough gravel surface with scanning vibrometer. In (a), (b), (e), and (f) the scanning vibrometer is moving at . Unless otherwise specified, the sonar, measurement, and surface parameters are the same as in Fig. 3(d) .

Spectrum of ultrasound field at reflected and scattered at normal incidence off a (a) smooth and nonoscillating surface for the stationary vibrometer (same result for scanning system), (b) smooth and harmonically oscillating surface for stationary vibrometer (same result for scanning system), (c) nonoscillating rough gravel surface with stationary vibrometer, (d) oscillating rough gravel surface with stationary vibrometer, (e) nonoscillating rough gravel surface with scanning vibrometer, and (f) harmonically oscillating rough gravel surface with scanning vibrometer. In (a), (b), (e), and (f) the scanning vibrometer is moving at . Unless otherwise specified, the sonar, measurement, and surface parameters are the same as in Fig. 3(d) .

Spectrum of received ultrasound field reflected and scattered off an oscillating (a) medium sand and (b) gravel surfaces. Results were for ten independent realizations of the sand and gravel surfaces, respectively. The average spectrum is obtained by incoherently averaging the intensity of the individual spectra. The ultrasound scan speed is . Unless otherwise specified, the sonar, measurement, and surface parameters are the same as in Figs. 3(c) and 3(d) , respectively.

Spectrum of received ultrasound field reflected and scattered off an oscillating (a) medium sand and (b) gravel surfaces. Results were for ten independent realizations of the sand and gravel surfaces, respectively. The average spectrum is obtained by incoherently averaging the intensity of the individual spectra. The ultrasound scan speed is . Unless otherwise specified, the sonar, measurement, and surface parameters are the same as in Figs. 3(c) and 3(d) , respectively.

Time domain wave form of ultrasound received field reflected and scattered off harmonically oscillating (a) sand and (b) gravel surface for the continuously scanning vibrometer. These were used to compute the spectra shown in Figs. 3(c) and 3(d) for the continuously scanning system.

Time domain wave form of ultrasound received field reflected and scattered off harmonically oscillating (a) sand and (b) gravel surface for the continuously scanning vibrometer. These were used to compute the spectra shown in Figs. 3(c) and 3(d) for the continuously scanning system.

Effect of adjusting the ultrasound source frequency on the spectrum of the received signal. The measurement setup is the same as in Fig. 3(d) . The axis should be scaled according to the ultrasound frequency for each of the cases.

Effect of adjusting the ultrasound source frequency on the spectrum of the received signal. The measurement setup is the same as in Fig. 3(d) . The axis should be scaled according to the ultrasound frequency for each of the cases.

Effect of adjusting the (a) measurement time , (b) vibrometer stand-off distance , and (c) angle of incidence , on the spectrum of the received signal. Unless otherwise specified, the measurement setup is the same as Fig. 3(d) .

Effect of adjusting the (a) measurement time , (b) vibrometer stand-off distance , and (c) angle of incidence , on the spectrum of the received signal. Unless otherwise specified, the measurement setup is the same as Fig. 3(d) .

Effect of adjusting the vibrometer scan velocity on the incoherently averaged spectrum of received signal from (a) sand and (b) gravel surfaces. All other parameters are the same as in Figs. 5(a) and 5(b) .

Effect of adjusting the vibrometer scan velocity on the incoherently averaged spectrum of received signal from (a) sand and (b) gravel surfaces. All other parameters are the same as in Figs. 5(a) and 5(b) .

Averaged spectrum of scattered noise level vs scan speed in (a) sand and (b) gravel for sidebands corresponding to ground resonance frequencies of 140 and for a ultrasound vibrometer. The scattered noise level is converted to equivalent displacement amplitude using Eq. (27) .

Averaged spectrum of scattered noise level vs scan speed in (a) sand and (b) gravel for sidebands corresponding to ground resonance frequencies of 140 and for a ultrasound vibrometer. The scattered noise level is converted to equivalent displacement amplitude using Eq. (27) .

(a) Anisotropic rough surface with five times longer correlation length in the direction than in the direction. (b) Similar to Fig. 3(d) but for the vibrometer continuously scanning in the direction or direction at .

(a) Anisotropic rough surface with five times longer correlation length in the direction than in the direction. (b) Similar to Fig. 3(d) but for the vibrometer continuously scanning in the direction or direction at .

## Tables

Estimates of surface displacement amplitude from incoherent processing for a ultrasound vibrometer continuously scanning the surface at a speed of . The estimates are obtained by either using data from a single scan, averaging estimates over ten independent scans at a given location, or estimating from the average spectrum for ten independent realizations. The quantities in parentheses are the accuracies for the estimation.

Estimates of surface displacement amplitude from incoherent processing for a ultrasound vibrometer continuously scanning the surface at a speed of . The estimates are obtained by either using data from a single scan, averaging estimates over ten independent scans at a given location, or estimating from the average spectrum for ten independent realizations. The quantities in parentheses are the accuracies for the estimation.

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