Nonlinear imaging with dual band pulse complexes

Abstract

The invention presents methods and instrumentation for measurement or imaging of a region of an object with waves of a general nature, for example electromagnetic (EM) and elastic (EL) waves, where the material parameters for wave propagation and scattering in the object depend on the wave field strength. The invention specially addresses suppression of 3.sup.rd order multiple scattering noise, referred to as pulse reverberation noise, and also suppression of linear scattering components to enhanced signal components from nonlinear scattering. The pulse reverberation noise is divided into three classes where the invention specially addresses Class I and Class II 3.sup.rd order multiple scattering that are generated from the same three scatterers, but in opposite sequence.

Claims

1. A method for measurement or imaging of a region of an object with waves where object material parameters for wave propagation are modified by the wave field strength, comprising the steps of: a) transmitting at least two pulse wave complexes in each of at least two transmit directions towards the region, where the pulse complexes are equal for each transmit direction, each of said pulse complexes comprising a low frequency (LF) pulse with frequencies in an LF band transmitted from an LF aperture, and a high frequency (HF) pulse with frequencies in an HF band transmitted from an HF aperture, wherein the pulse length of the transmitted HF pulse is less than half an oscillation period of the transmitted LF pulse; wherein the transmitted HF pulse propagates spatially so close to or within the transmitted LF pulse that the HF pulse observes a manipulation of the object material parameters by the co-propagating LF pulse for at least a part of a propagation depth of the HF pulse; and wherein at least for each transmit direction the LF pulse varies for each transmitted pulse complex to produce different manipulations of the object where the LF pulse is nonzero for at least one pulse complex and the LF pulse can also be zero, and for each transmit direction at least one LF pulse is non-zero and the transmitted LF pulse can also be zero; b) picking for each transmit direction up HF receive signals from one or both of scattered and transmitted HF components from each transmitted pulse complex and for each transmit direction, using a HF receive beam former which for selected image points forms HF receive beams with a receive beam aperture and focus centered at said image points; c) forming synthetic HF receive signals for each said selected image points as weighted sums of said HF receive signals from different transmit directions with HF receive beam focus at said selected image points, where said weighted sums for each said selected image points are selected so that said synthetic HF receive signals originates from synthetic transmit beams with close to the same aperture and focus as said receive beams centered at said image points; d) correcting said synthetic HF receive signals by at least one of i) delay corrections in a fast time with correction delays, and ii) speckle corrections, wherein one or both of said delay corrections and speckle corrections can be zero for at least one HF receive signal, to form corrected HF signals for said time interval; and e) combining said corrected HF signals from the at least two transmitted pulse complexes to form at least one noise suppressed HF signal with suppression of pulse reverberation noise.

2. A method according to claim 1, wherein the HF transmit and receive beams are selected to minimize a difference between the Class I and Class II 3.sup.rd order scattering noise for a zero LF pulse.

3. A method according to claim 2, further comprising imaging of more than one depth region wherein each depth region is imaged with different HF receive signals from different ones of said transmitted pulse complexes with HF transmit and receive beam focus selected within said each depth region.

4. A method according to claim 1, wherein in said step a) the pulse wave complexes are for each transmit direction transmitted in beams with the same HF transmit aperture and HF transmit focus depth; in said step b) the HF receive beam former uses the same HF receive aperture and HF receive focus as the HF transmit aperture and HF transmit focus; in said step c) said combining produces transversal filtering of the HF receive signals from the different transmit directions for at least one depth to obtain synthetic HF receive signals with synthetic HF image beam foci at said at least one depth; and the synthetic HF receive signals are processed according to said steps d) and e).

5. A method according to claim 1, wherein in said step a) the pulse wave complexes are transmitted in transmit beams that are wide in at least one direction for each of the image points; in said step b) the HF receive beam former forms HF receive signals with focus in each image point for a group of image points; in said step c) the HF receive signals for said image points are combined to form synthetic HF receive signals with synthetic HF transmit beams in said image points; and the synthetic HF receive signals are processed according to said steps d) and e).

6. A method according to claim 5, wherein the transmitted pulse complexes have wave fronts that are substantially planar in at least one direction.

7. A method according to claim 1, wherein at least one of said correction delays and said speckle corrections are estimated in a process that uses a mathematical model of LF pulse generated variations in at least one of i) HF receive signal delays, and ii) speckle of the multiple scattered noise signal components in the HF receive signals.

8. A method according to claim 7, further comprising the steps of: f) obtaining nonlinear propagation delay and pulse form distortion for a forward propagating HF pulse at multiple depths through simulation of a nonlinear wave equation for the forward propagating HF pulse with the actual transmitted LF/HF pulse complex and defined object material parameters; g) obtaining, from a measured HF receive signal, parameters relating to relative reflection coefficients as a function of depth along the beam; and h) estimating said correction delays and speckle corrections in processes that include combining one or both of the nonlinear propagation delays and pulse form distortions for the HF pulse obtained in said step f) with the parameters relating to relative reflection coefficients as a function of depth obtained in said step g) in a mathematical model for multiple scattered HF noise signals.

9. A method according to claim 8, where the method of said step d) is performed applying said steps f) to h), wherein said defined object material parameters are adjusted to increase the suppression of one or both of i) pulse reverberation noise and ii) 1.sup.st order linear scattering components in the processed HF receive signal, and wherein said adjustment is performed as one or both of a) a manual adjustment based on observation of a displayed image, and b) an automatic adjustment to optimize a functional on a processed HF receive signal in a region of an image.

10. A method according to claim 9, wherein said functional is total power of a processed HF receive signal in a region of an image.

11. A method according to claim 9, wherein the simulation of a nonlinear wave equation for the forward propagating HF pulse with an actual transmitted LF/HF pulse complex and defined object material parameters is performed for a set of multiple defined parameters and pre-stored in an electronic memory before said estimating of the correction delays and speckle corrections starts, and in the estimation process the simulation for defined object material parameters is retrieved from the memory based on a memory address related to said defined parameters.

12. A method according to claim 8, wherein in said step f) coefficients of series expansions of the nonlinear propagation delay and pulse form distortions dependency on the LF pulse and defined object material parameters are calculated and stored so that, for an LF pulse and defined object material parameters, estimates of the nonlinear propagation delays and pulse form distortions can be calculated by said series expansions utilizing said stored coefficients of the series expansions.

13. A method according to claim 12, wherein said series expansions are one or both of Taylor series and Fourier series expansions.

14. A method according to claim 1, wherein the LF and HF transmit apertures are defined so that the nonlinear propagation delay (t) for the HF pulse has a substantially linear variation with the time t, and wherein for suppression of pulse reverberation noise the correction delays are approximated by (t)/2.

15. A method according to claim 1, wherein for suppression of pulse reverberation noise in echo weak regions, the same HF receive signal obtained with a non-zero LF pulse is delay corrected with two different correction delays to provide two corrected HF signals, and said two corrected HF signals are combined to provide a noise suppressed HF signal.

16. A method according to claim 1, wherein transmitting LF array elements are selected so that an LF array aperture has a region around its center with no transmission of the LF pulse, so that manipulation of the object material parameters by the LF pulse observed by the co-propagating HF pulse is very low in a near field region of am HF array aperture.

17. A method according to claim 1, wherein said noise suppressed HF signals are used for estimating corrections for wave front aberrations in materials with spatial variations in wave propagation velocity.

18. A method according to claim 1, wherein said step c) further comprises constructing said synthetic HF transmit beams so that a difference between Class I and Class II 3.sup.rd order scattered noise-signals in said synthetic HF receive signals for said pulse complexes with zero LF pulse is substantially minimized, using the transmit directions in said set of image points.

19. A method according to claim 1, wherein the object material parameters for scattering are modified by the wave field strength, and wherein for a set of transmit directions steps a) and b) include transmitting at least 3 of said pulse complexes, each with different LF pulses towards the object producing at least 3 HF receive signals, steps c) through e) include combining, correcting and combining the HF receive signals to form at least two noise suppressed HF signals, and said at least two noise suppressed HF signals are further corrected by at least one of i) delay corrections using correction delays, and ii) speckle corrections using speckle corrections, to form at least two corrected noise suppressed HF signals; and the method further comprising the step of combining said at least two corrected noise suppressed HF signals to form at least one nonlinear scattering HF signal with linear scattering components and pulse reverberation noise substantially suppressed.

20. A method according to claim 1 or 19, wherein said speckle corrections are effected with a filter.

21. A method according to claim 20, wherein said filter is a bandpass filter that reduces the bandwidth of the signals.

22. An instrument for measurement or imaging of a region of an object with waves, wherein material parameters of the object for wave propagation are modified by the waves field strength, the instrument comprising: a) transmit means for transmitting at least two pulse wave complexes in each of at least two transmit directions towards the region, where the transmitted pulse complexes are equal for each transmit direction, each of said pulse complexes comprising a low frequency (LF) pulse with frequencies in an LF band transmitted from an LF aperture, and a high frequency (HF) pulse with frequencies in an HF band transmitted from an HF aperture, wherein a pulse length of the HF pulse being less than half an oscillation period of the LF pulse, and the HF pulse propagating spatially so close to or within the LF pulse that the HF pulse observes a manipulation of the object material parameters by the co-propagating LF pulse for at least a part of a propagation depth of the HF pulse, and for each transmit direction the LF pulse varies for each transmitted pulse complex to produce different manipulations of the object, where the LF pulse is nonzero for at least one pulse complex and the LF pulse can also be zero; b) HF receive and beam forming means that for each transmit direction picks up HF receive signals from one or both of scattered and transmitted HF components from the at least two transmitted pulse complexes for each transmit direction, where said HF receive beam forming means forms HF receive signals for selected image points with a HF receive beam with defined aperture and focus centered at said image points; c) means for forming synthetic HF receive signals for each said selected image points as weighted sums of said HF receive signals from different transmit directions with HF receive beams focused at said selected image points, where said weighted sums for each said selected image points are selected so that said synthetic HF receive signals originates from synthetic HF transmit beams with close to the same aperture and focus as said HF receive beams centered at said image points; d) correction means for correcting said synthetic HF receive signals for each said image point by at least one of i) delay corrections with correction delays, and ii) speckle corrections, wherein one or both of said delay corrections and speckle corrections can be omitted for at least one HF receive signal, to form corrected HF signals for each said image point; and e) processing means for further combining said corrected HF signals to form at least one noise suppressed HF signal with suppression of pulse reverberation noise for each said image point.

23. An instrument according to claim 22, wherein said transmit means transmits pulse wave complexes with the same HF transmit aperture and HF transmit focus depth for each transmit direction; wherein said HF receive and beam forming means uses the same HF receive aperture and HF receive focus as the HF transmit aperture and HF transmit focus; wherein said means for combining produces transversal filtering of the HF receive signals from the different transmit directions for at least one depth to obtain synthetic HF receive signals with synthetic HF image beam foci at the at least one depth; and wherein the synthetic HF receive signals are processed by the correction means of element d) and the processing means of element e).

24. An instrument according to claim 22, wherein said transmit means transmits pulse wave complexes in transmit beams that are wide in at least one direction for said image points; wherein said HF receive and beam forming means form HF receive signals with focus in each image point for a group of image points wherein said means for combining combines said HF receive signals for different transmit directions for said image points to form synthetic HF receive signals with synthetic HF transmit beams in said image points; and wherein the synthetic HF receive signals are processed by the correction means of element d) and the processing means of element e).

25. An instrument according to claim 24, wherein said transmit means is operable for transmission of LF and HF pulse waves with wave fronts that are substantially planar in at least one direction.

26. An instrument according to claim 22, wherein said transmit means allow the LF and HF transmit apertures to be defined so that the nonlinear propagation delay (t) for the transmitted HF pulse has a substantially linear variation with the time t, so that for suppression of pulse reverberation noise at time t said correction delays can be approximated by (t)/2.

27. An instrument according to claim 22, wherein said correction means delay corrects the same HF receive signal obtained with a non-zero LF pulse with two different correction delays to provide two corrected HF signals, and said two corrected HF signals are combined to provide a noise suppressed HF signal.

28. An instrument according to claim 22, further comprising means for selecting the transmitting LF array aperture to obtain an LF array aperture that has a region around its center with no transmission of the LF pulse, so that manipulation of the object material parameters by the LF pulse observed by the co-propagating HF pulse is very low in a near field region of the HF array aperture.

29. An instrument according to claim 22, further comprising: means for estimation of corrections for wave front aberrations in materials with spatial variations in the wave propagation velocity; and means for correction of the HF receive element signals for wave front aberrations.

30. An instrument according to claim 22, wherein said synthetic HF transmit beams in said element c) are constructed so that a difference between Class I and Class II 3.sup.rd order scattered noise-signals in the synthetic HF receive signals from said pulse complexes with zero LF pulse is substantially minimized using the transmit directions in said image points.

31. An instrument according to claim 22, wherein material parameters of the object for scattering are modified by the wave field strength; wherein said transmit means comprises means for transmitting at least 3 of said pulse complexes for a set of transmit directions with different LF pulses towards the object; wherein said means for combining, correction means and processing means include means for combining, correcting and combining the HF receive signals for said set of transmit directions to form at least two noise suppressed HF signals; and wherein the instrument further comprises: means for further processing of said at least two noise suppressed HF signals for said set of transmit directions, including at least one of i) delay corrections using correction delays (depth-time), and ii) speckle corrections, to form at least two corrected noise suppressed HF signals for said set of transmit directions; and means for combining said corrected noise suppressed HF signals to suppress linear scattering components to form nonlinear measurement or imaging HF signals for said set of transmit directions.

32. An instrument according to claim 31, wherein the HF transmit, HF receive and HF beam forming means comprise means for matching HF transmit and receive beams to substantially minimize a difference between Class I and Class II 3.sup.rd order scattering noise for a zero LF pulse.

33. An instrument according to claim 22 or 31, further comprising means for speckle corrections in the form of a filter.

34. An instrument according to claim 33, wherein said filter is a bandpass filter that reduces the bandwidth of the signals.

35. An instrument according to claim 22 or 31, further comprising estimation means for estimation of at least one of said correction delays and speckle corrections.

36. An instrument according to claim 35, wherein said estimation means uses a mathematical model of LF pulse generated variations in at least one of i) HF receive signal delays, and ii) speckle of the multiple scattered noise signal components in the HF receive signals.

37. An instrument according to claim 36, wherein said estimation means comprises: a) means for simulation of a nonlinear wave equation for a forward propagating HF pulse with an actual transmitted LF/HF pulse complex with defined object material parameters, and for obtaining the nonlinear propagation delay and pulse form distortion for the simulated HF pulsed wave; b) means for obtaining, from the measured HF receive signal, parameters relating to relative reflection coefficients as a function of depth along the beam; and c) means for estimation of said correction delays and speckle corrections in a process that includes combining one or both of said nonlinear propagation delays and pulse form distortions for the simulated HF pulsed wave obtained by said simulation means with said parameters relating to relative reflection coefficients as a function of depth obtained by said means for obtaining in a mathematical model for multiple scattered HF noise signals.

38. An instrument according to claim 37, wherein said estimation means further comprises: means for adjusting said defined object material parameters to increase the suppression of one or both of i) pulse reverberation noise and ii) 1.sup.st order linear scattering components in the processed HF receive signal, where said adjusting is carried out as one or both of i) manual adjustment based on observation of the displayed image, and ii) automatic adjustment to optimize a functional HF receive signal in a region of an image.

39. An instrument according to claim 37, wherein said means for simulation of a nonlinear wave equation for the forward propagating HF pulse with an actual transmitted LF/HF pulse complex with defined object material parameters comprises means for storage of a set of simulations with different defined object material parameters where said set of simulations are carried out before the measurements, and where in the estimation process the simulation for defined object material parameters is retrieved from memory based on a memory address related to said defined parameters.

40. An instrument according to claim 37, wherein said means for simulation comprises means for calculation and storage of coefficients of series expansions of the nonlinear propagation delay and pulse form distortions dependent on the LF pulse and defined object material parameters, so that for an LF pulse and defined object material parameters, estimates of the nonlinear propagation delays and pulse form distortions can be calculated by said series expansions utilizing said stored coefficients of the series expansions.

Description

4. SUMMARY OF THE DRAWINGS

(1) FIGS. 1a, 1b and 1c Show Prior Art dual band pulse complexes for suppression of multiple scattering noise and/or 1.sup.st order scattering to show nonlinear scattering with pulse wave measurements or imaging, and

(2) FIGS. 2a, 2b, 2c and 2d Show classifications of different types of multiple scattering noise that is found in a typical ultrasound image of an organ, and

(3) FIG. 3 Illustrates coordinates and volumes for calculating the received 1.sup.st and 3.sup.rd order scattered signals, and

(4) FIG. 4 Shows and illustration of image reconstruction from multi component transmit of broad beams, and

(5) FIG. 5 Shows example plane reflectors that generate strong 3.sup.rd order pulse reverberation noise, and

(6) FIG. 6 Shows coordinates for 1.sup.st to 3.sup.rd reflectors in the fast time domain, and

(7) FIGS. 7a and 7b Show interference between Class I and II pulse reverberation noise when Q>1, and

(8) FIG. 8 Shows illustration of the effect on noise correction delay for nonlinear variation with fast time of the nonlinear propagation delay, and

(9) FIG. 9 Shows a block diagram of an instrument according to the invention.

5. DETAILED DESCRIPTION OF THE INVENTION

(10) 5.1 Background of 1.sup.st and 3.sup.rd Order Scattering

(11) Example embodiments according to the invention, are presented in the following.

(12) The methods and structure of the instrumentation are applicable to both electromagnetic (EM) and elastic (EL) waves, and to a wide range of frequencies with a wide range of applications. For EL waves one can apply the methods and instrumentation to both shear waves and compression waves, both in the subsonic, sonic, and ultrasonic frequency ranges. We do in the embodiments describe by example ultrasonic measurements or imaging, both for technical and medical applications. This presentation is meant for illustration purposes only, and by no means represents limitations of the invention, which in its broadest aspect is defined by the claims appended hereto.

(13) The most damaging noise in pulse echo wave measurements and imaging has its origin in multiple scattering of the transmitted pulse. We refer to this noise as multiple scattering noise, or pulse reverberation noise. As the amplitude of the scattered pulse drops for each scattering, the 2.sup.nd and 3.sup.rd order scattering dominates the noise. With back scatter imaging we mainly have 3.sup.rd order scattering noise, where the 1.sup.st scatterer is inside the transmit beam, and the last (3.sup.rd) scatterer is inside the receive beam. With very wide transmit beams, like with plane wave imaging and similar, we can also observe 2.sup.nd order scattering noise. However, the major type of strong scatterers are layers of fat and muscles in the body wall, that are fairly parallel, which requires 3.sup.rd order scattering also with wide transmit beams. Bone structures and air in lungs and intestines can produce 2.sup.nd order scattering noise with synthetic multi beam imaging (e.g. plane wave imaging)

(14) The invention presents methods and instrumentation for suppression of such multiple scattering noise, and also for measurement and imaging of nonlinear scattering of waves. Such methods and instruments are also presented in U.S. Pat. No. 8,038,616 and US patent application Ser. Nos. 12/351,766, 12/500,518, and 13/213,965, where the current invention presents new and more advanced methods and instrumentation, utilizing transmission of similar dual band pulse complexes as in said applications.

(15) An example of such a dual band transmitted pulse complex is shown as 101 in FIG. 1a, composed of a low frequency (LF) pulse (102) and a high frequency (HF) pulse (103). The HF pulse is typically shorter than the period of the LF pulse, and the methods utilize the nonlinear manipulation of the object material parameters (e.g. object elasticity) by LF pulse as observed by the co-propagating HF pulse. One typically transmits at least two pulse complexes with differences in the LF pulse, and processes HF receive signals picked up from one or both of scattered and transmitted HF pulses of the at least two transmitted pulse complexes, to suppress the pulse reverberation noise and/or measure or image nonlinear scattering from the object.

(16) In FIG. 1a is for example shown two pulse complexes (101, 104) with opposite polarities of the LF pulse, shown as 102 versus 105. The same is shown in FIG. 1b with the two polarities (108, 110) of the LF pulse. In FIG. 1a the HF pulse 103 is found at the crest of the LF pulse 102 with a temporal pulse length, T.sub.pH, so short in relation to the LF pulse oscillation period, T.sub.L, that it mainly observes a constant manipulation of the material parameters by the LF pulse observed by the co-propagating HF pulse, and hence both the HF pulse wave propagation velocity and the HF pulse scattering from spatial variations in the parameters. When switching polarity of the LF pulse to 105 so that the HF pulse 106 is found at the trough of the LF pulse, the HF pulse observes a different manipulation of the propagation velocity and scattering by the LF pulse. In FIG. 1b the BF pulses 107 and 109 are found at gradients of the LF pulses 108 and 110 so that the LF manipulation of the propagation velocity varies along the HF pulse, producing a pulse form distortion of the co-propagating HF pulses that accumulates with propagation distance. Due to diffraction of the LF pulse, the position of the HF pulse along the LF pulse oscillation can slide with propagation distance, indicated as the different positions 111, 112, 113 of the HF pulse along the LF pulse oscillation in FIG. 1c, providing mainly a pulse form distortion at positions 111 and 113, and a propagation advancement/delay at position 112.

(17) For more detailed analysis of the reverberation noise we refer to FIG. 2, which in FIG. 2a schematically illustrates the situation for 1.sup.st order scattering, and in FIG. 2b-c schematically illustrates 3 different classes of 3.sup.rd order multiple scattering noise, defined as Class I-III. The Figures all show an ultrasound transducer array 200 with a front transmit/receive and reflecting surface 201. The pulse propagation path and direction is indicated with the lines and arrows 202, where the 1.sup.st scatterer is indicated by the dot 203, and in FIG. 2a with 1.sup.st order scattering, the transmitted pulse complex propagates to the 1.sup.st scatterer 203 and back to the transducer where it is transformed to an electrical signal and further processed to form measurement or image signals.

(18) In FIG. 2b-c the backscattered signal from the 1.sup.st scatterer 203 is reflected/scattered by a 2.sup.nd scatterer 204 that can have multiple positions as indicated in the Figure. The pulse complex then propagates to a 3.sup.rd scatterer 205 where it is reflected back to the transducer 200 where the signal is picked up and further processed to provide measurement or image signals with strong suppression of pulse reverberation noise and/or suppression of linear scattering to enhance HF receive signal components from nonlinear scattering. Due to the multiple positions of the 2.sup.nd scatterers the two 1.sup.st and 3.sup.rd scatterers 203 and 205 then generates a tail of pulse reverberation noise 206 following the deepest of the 1.sup.st and 3.sup.rd scatterers. 207 shows for illustrative example a low echogenic region, for example in medicine the cross section of a vessel or a cyst, with a scattering surface 208 that we want to image with high suppression of pulse reverberation noise. The object has so low internal scattering that the pulse reverberation noise 206 generated in the different classes produces disturbances in the definition of the scattering surface 208.

(19) To obtain visible, disturbing reverberation noise in the image, the 1.sup.st-3.sup.rd scatterers must be of a certain strength, where the 1.sup.st scatterer must be inside the transmit beam, and the 3.sup.rd scatterer must be inside the receive beam. In medical ultrasound applications this is often found by fat layers in the body wall, while in technical applications one can encounter many different structures, depending on the application. In ultrasound applications, the strongest 2.sup.nd scatterer is often the ultrasound transducer array surface itself, as this can have a reflection coefficient that is 10 dB or more higher than the reflection coefficient from other soft tissue structures. The pulse reverberation noise that involves the transducer surface as the 2.sup.nd scatterer, is therefore particularly strong indicated by 209. The definition of Class I-III pulse reverberation noise is in more detail

(20) Class I (FIG. 2b) is the situation where the 1.sup.st scatterer (203) is closest to the transducer at fast time (depth time) of t.sub.1=2z.sub.1/c, where z.sub.1 is the depth of the 1.sup.st scatterer in the 3.sup.rd order scattering sequence of scatterers, and c is the elastic wave propagation velocity. For Class I pulse reverberations we have t.sub.1<t/2 and t=2z/c is the fast time (depth time) for the reception of the 1.sup.st order back scattered signal from a scatterer 703 at depth z in FIG. 2a. The 2.sup.nd scatterer (204) is at the depth z.sub.2 and fast time t.sub.2=2z.sub.2/c. The distance between the 3.sup.rd scatterer 205 and the noise point, which in the Figure coincides with the object surface 208, is the same as the distance d=t.sub.1-t.sub.2 between the 1.sup.st and the 2.sup.nd scatterers. With varying positions of the 2.sup.nd scatterer 204, we get a tail 206 of pulse reverberation noise. When the transducer surface 201 is the strongest 2.sup.nd scatterer, we get particularly strong noise at 209.

(21) Class II (FIG. 2c) Class II pulse reverberation noise has the same three scatterers as the Class I noise, with 1.sup.st and 3.sup.rd scatterers interchanged. Class I and II pulse reverberation noise hence always co-exist, and for Class II noise we have a fast time for the 1.sup.st scatterer 203 t.sub.1>t/2=z/c. The reasoning for the tail of pulse reverberation noise 206 is the same as that for Class I, where a strongly reflecting transducer surface gives the noise 209. When the 2.sup.nd scatterer is the ultrasound transducer array itself, we can define as follows:

(22) Class I reverberations: 1.sup.st scatterer z.sub.1<z/2 and 3.sup.rd scatterer z.sub.3>z/2, z.sub.3=zz.sub.1

(23) Class II reverberations: 1.sup.st scatterer z.sub.3<z/2 and 3.sup.rd scatterer z.sub.1>z/2, z.sub.1=zz.sub.3

(24) Harmonic imaging will mainly suppress Class I pulse reverberation noise where z.sub.1 is close to the transducer, where we in the following will show that methods according to this invention suppresses Class I and II pulse reverberation noise combined, regardless of the value of z.sub.1, as the nonlinear propagation delay and pulse form distortion of the combined Class I and II noise is quite different from that for the 1.sup.st order scattering.

(25) Class III (FIG. 2d) This is a situation where Class I and Class II merges, i.e. the 1.sup.st and the 3.sup.rd scatterers are at the same depth, and the 1.sup.st scatterer is close to z, i.e. all 3 scatterers are close to z. The nonlinear propagation delay and pulse form distortion for the Class III is therefore similar to that for 1.sup.st order scattering at the depth the noise is observed, and it is then difficult to suppress Class III noise relative to 1.sup.st order scattering. When Class III noise co-exists with Class I and II noise and interferes at the same depth, the model based estimation of correction delays and speckle corrections for the noise is able to suppress Class I, II and III noise combined.

(26) We shall first start by analyzing the situation with linear wave propagation.

(27) 5.2 Signal Models for 1.sup.st and 3.sup.rd Order Scattering in a Linear Material

(28) A. Models for 1.sup.st Order Scattering

(29) We first analyze the HF receive signal from the 1.sup.st order scattered wave components within the realm of pressure independent elasticity of the object, i.e. linear elasticity. With reference to FIG. 3 we get the linearly transmitted field as [1]

(30) $\begin{array}{cc}{P}_t\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)=P_t\left(\right)ik{H}_t\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)H_t\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)={}_{S_t}{d}^2{r}_{0}2G\left({\underset{\_}{r}}_0,{\underset{\_}{r}}_1;\right)e^{-i{}_t\left({\underset{\_}{r}}_0,{\underset{\_}{r}}_t\right)}{w}_t\left({\underset{\_}{r}}_0\right)H_{\mathrm{abc}}\left({\underset{\_}{r}}_0,;{\underset{\_}{r}}_t\right)c{}_t\left({\underset{\_}{r}}_0,{\underset{\_}{r}}_t\right)=r_t-.Math.{\underset{\_}{r}}_0-{\underset{\_}{r}}_t.Math.& \left(1\right)\end{array}$
where is the angular frequency, k=/c is the wave number with c as the wave propagation velocity, P.sub.t () is the Fourier transform of the transmitted pressure waveform at the transducer surface, .sub.T(r.sub.0, r.sub.T) is the transmit delays for focusing the transmit beam at r.sub.t in a homogeneous material, G(r.sub.0, r.sub.1;) is the Greens function for wave propagation from a source at r.sub.0 to a field point at r.sub.1 in the linear, heterogeneous material (i.e. spatial variation in wave propagation velocity), w.sub.t(r.sub.0) is an apodization function, and H.sub.abc(r.sub.0,; r.sub.t) is an aberration correction filter for spatial variations in the wave propagation velocity of the heterogeneous material for focusing at r.sub.t. In a homogeneous material where the wave propagation velocity parameters are constant in space, we have H.sub.abc (r.sub.0,; r.sub.t)=1. If H.sub.abc(r.sub.0;; r.sub.t) is set equal to unity in a heterogeneous material, the beam focusing will be sub-optimal, but the theory presented below will still be valid.

(31) The contribution to the 1.sup.st order scattered field from scatterer density (r.sub.1) of the infinitesimal volume element d.sup.3r.sub.1 at location r.sub.1 in R.sub.1 in the object (See FIG. 3) is [1]
dP.sub.s1(r,;r.sub.1)=k.sup.2P.sub.t()ikG(r,r.sub.1;)H.sub.t(r.sub.1,;r.sub.t)(r.sub.1)d.sup.3r.sub.1(2)

(32) For back scattering we have (r.sub.1)=.sub.1(r.sub.1)(r.sub.1), where .sub.1 and are the relative spatial variation of the material compressibility and mass density [1]. The total 1.sup.st ordered scattered field is then found by integration of dP.sub.s1(r,; r.sub.1) over r.sub.1 across R.sub.1. The 1.sup.st order received signal for transducer element or sub aperture located at r on the receiver transducer
dS.sub.1(r,;r.sub.1)=k.sup.2U()G(r,r.sub.1;)H.sub.t(r.sub.1,;r.sub.1)(r.sub.1)d.sup.3r.sub.1(3)
where U()=H.sub.rt()ikP.sub.t() is the received pulse for the element at r from a unit plane reflector [1]. Beam forming this signal over the array with receive beam focus at r.sub.r, we get the following received 1.sup.st order scattered signal from (r.sub.1) as [BA]

(33) $\begin{array}{cc}{\mathrm{dY}}_1\left(;{\underset{\_}{r}}_1\right)={}_{S_r}{d}^2{r}_4{e}^{-i{}_r\left({\underset{\_}{r}}_4,{\underset{\_}{r}}_r\right)}{w}_r\left({\underset{\_}{r}}_4\right)H_{\mathrm{abc}}\left({\underset{\_}{r}}_4,;{\underset{\_}{r}}_r\right){\mathrm{dS}}_1\left({\underset{\_}{r}}_4,;{\underset{\_}{r}}_1\right)c{}_r\left({\underset{\_}{r}}_4,{\underset{\_}{r}}_r\right)=r_r-.Math.{\underset{\_}{r}}_4-{\underset{\_}{r}}_r.Math.& \left(4\right)\end{array}$
where r.sub.4 is the integration coordinate over the receive array surface S.sub.r, and .sub.r(r.sub.4, r.sub.r) is the beam forming delays for the receiver beam focused at the receiver beam focus r.sub.r, calculated on the assumption of constant propagation velocity c in the tissue, w.sub.r(r.sub.4) is an apodization function, and H.sub.abc(r.sub.4,; r.sub.r) is the aberration correction filter that compensates for material heterogeneity to focus the receive beam at r.sub.r. As for the transmit beam, we have H.sub.abc(r.sub.4,; r.sub.r)=1 for the homogeneous material. Inserting Eq. (3) we can express the received 1.sup.st order scattered signal as

(34) $\begin{array}{cc}{\mathrm{dY}}_1\left(;{\underset{\_}{r}}_1\right)=-k^{2}U\left(\right)H_r\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_r\right)H_t\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)\left({\underset{\_}{r}}_1\right)d^3{r}_1{H}_r\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_r\right)={}_{S_r}{d}^2{r}_{4}2G\left({\underset{\_}{r}}_4,{\underset{\_}{r}}_1;\right)e^{-i{}_r\left({\underset{\_}{r}}_4-{\underset{\_}{r}}_r\right)}{w}_r\left({\underset{\_}{r}}_4\right)H_{\mathrm{abc}}\left({\underset{\_}{r}}_4,;{\underset{\_}{r}}_r\right)& \left(5\right)\end{array}$

(35) where H.sub.r(r.sub.1,; r.sub.r) is the aberration corrected receive beam at the field point r.sub.1 when the beam is focused at r.sub.r.

(36) B. Models for 3.sup.rd Order Scattering

(37) Multiple scattering is produced because the 1.sup.st order scattered field in Eq. (2) is in turn scattered from a scatterer at r.sub.2 in R.sub.2 to produce a 2.sup.nd order scattered field

(38) $\begin{array}{cc}{\mathrm{dP}}_{s2}\left(\underset{\_}{r},;{\underset{\_}{r}}_1\right)={}_{R_2}{d}^3{r}_{2}G\left(\underset{\_}{r},{\underset{\_}{r}}_2;\right)\left({\underset{\_}{r}}_2,\right){\mathrm{dP}}_{s1}\left({\underset{\_}{r}}_2,;{\underset{\_}{r}}_1\right)& \left(6\right)\end{array}$
where (r.sub.2, ) typically is either a strong volume scatterer distribution (r.sub.2), for example structured fatty layers, where we have
(r.sub.2,)=k.sup.2(r.sub.2)(7)
or reflection from a layer surface or the transducer surface as
(r.sub.2,)=ik2R(r.sub.2,)(S.sub.R(r.sub.2))(8)
where R(r.sub.2,) is the reflection coefficient of the reflecting surface S.sub.R defined by S.sub.R(r.sub.2)=0. In technical applications one often have strongly reflecting surfaces at a distance from the transducer, for example a surface of a metal layer, which can be modeled as Eq. (8) where S.sub.R(r.sub.2)=0 now defines the layer surface. Strong volume scatterers are modeled as in Eq. (7).

(39) The 2.sup.nd order scattered field is then scattered a 3.sup.rd time from the volume scatterer distribution (r.sub.3) and produces a received signal for transducer element or sub aperture located at r.sub.4 on the receiver transducer
dS.sub.3(r.sub.4,;r.sub.1,r.sub.3)=k.sup.2H.sub.rt(r.sub.4,)2G(r.sub.4,r.sub.3;)dP.sub.s2(r.sub.3,;r.sub.1)(r.sub.3(9)
where H.sub.rt() is defined in relation to Eq. (3). Introducing the expression for P.sub.s1(r.sub.2,) and P.sub.s2(r.sub.3,) and from Eqs. (2,6) we can express the received 3.sup.rd order reverberation element signal as

(40) $\begin{array}{cc}{\mathrm{dS}}_3\left({\underset{\_}{r}}_4,;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)=k^{4}U\left(\right)2G\left({\underset{\_}{r}}_4,{\underset{\_}{r}}_3;\right)\left({\underset{\_}{r}}_3\right)d^3{r}_3{H}_{\mathrm{rev}}\left({\underset{\_}{r}}_3,{\underset{\_}{r}}_1;\right)H_t\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)\left({\underset{\_}{r}}_1\right)d^3{r}_1{H}_{\mathrm{rev}}\left({\underset{\_}{r}}_3,{\underset{\_}{r}}_1;\right)={}_{R_2}{d}^3{r}_{2}G\left({\underset{\_}{r}}_3,{\underset{\_}{r}}_2;\right)\left({\underset{\_}{r}}_2,\right)G\left({\underset{\_}{r}}_2,{\underset{\_}{r}}_1;\right)& \left(10\right)\end{array}$
where R.sub.2 is the region of the second scatterers U()=H.sub.rt()ikP.sub.t() is the received pulse for the element at r for a reflected wave front conformal to the array surface, defined in relation to Eq. (3). Beam forming this signal over the array with receive beam focus at r.sub.r, we get the following received reverberation beam signal for the reverberations as

(41) $\begin{array}{cc}{\mathrm{dY}}_3\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)={\left({}_{S_r}d\right)}^2{r}_4{e}^{-{}_r\left({\underset{\_}{r}}_4,{\underset{\_}{r}}_r\right)}{w}_r\left({\underset{\_}{r}}_4\right)H_{\mathrm{abc}}\left({\underset{\_}{r}}_4,;{\underset{\_}{r}}_r\right){\mathrm{dS}}_3\left({\underset{\_}{r}}_4,;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)c{}_r\left({\underset{\_}{r}}_4,{\underset{\_}{r}}_r\right)=r_r-.Math.{\underset{\_}{r}}_4,{\underset{\_}{r}}_r.Math.& \left(11\right)\end{array}$
where r.sub.4 is the integration coordinate over the receive array surface S.sub.r, and the other functions are as defined in relation to Eqs. (4,5). Inserting Eq. (10) we can express the 3.sup.rd order reverberation signal as
dY.sub.3(;r.sub.1,r.sub.3)=k.sup.4U()H.sub.r(r.sub.3,;r.sub.r)(r.sub.3)d.sup.3r.sub.3H.sub.rev(r.sub.3,rr.sub.1;)H.sub.t(r.sub.1,;r.sub.t)(r.sub.1)d.sup.3r.sub.1(12)
where H.sub.r(r.sub.3,; r.sub.r) is the aberration corrected receive beam at the field point r.sub.3 when the beam is focused at r.sub.r as defined in Eq. (5)
5.3 Nonlinear Propagation and Scattering
A. Wave Equation for 2.sup.nd Order Nonlinear Elasticity

(42) Nonlinear material parameters influences both the wave propagation velocity and the scattering from spatial variations in the nonlinear parameters. For acoustic waves in fluids and solids, the bulk elasticity can generally be approximated to the 2.sup.nd order in the pressure, i.e. the volume compression V of a small volume V, is related to the pressure p

(43) $\begin{array}{cc}\frac{V}{V}=-{}_{\underset{\_}{}}=\left(1-{}_p\left(\underset{\_}{r}\right)p\right)\left(\underset{\_}{r}\right)p+h_{\mathrm{ab}}\underset{t}{.Math.}\left(\underset{\_}{r}\right)p& \left(13\right)\end{array}$
where (r, t) is the acoustic particle displacement vector, p(r,t) is the acoustic pressure, (r) is the linear bulk compressibility of the material, and .sub.p(r)=.sub.n(r)(r) where .sub.n(r)=1+B(r)/2A(r) is a nonlinearity parameter of the bulk compressibility, and h.sub.ab(r,t) is a convolution kernel that represents absorption of wave energy to heat. The 2.sup.nd order approximation holds for soft tissue in medical imaging, fluids and polymers, and also rocks that show special high nonlinear bulk elasticity due to their granular micro-structure. Gases generally show stronger nonlinear elasticity, where higher order terms in the pressure often might be included. Micro gas-bubbles with diameter much lower than the acoustic wavelength in fluids, also show a resonant compression response to an oscillating pressure, which is discussed above. When the material parameters can be approximated to 2.sup.nd order in the wave fields, we can for example for ultrasonic pressure waves formulate a wave equation that includes nonlinear forward propagation and scattering phenomena as [2,3]

(44) $\begin{array}{cc}\underset{\underset{\left(1\right)\mathrm{Linear}\mathrm{propagation}}{}}{{}^{2}p\left(\underset{\_}{r},t\right)-\frac{1}{c_0^2\left(\underset{\_}{r}\right)}\frac{{}^{2}p\left(\underset{\_}{r},t\right)}{t^2}}+\underset{\underset{\left(2\right)\mathrm{Nonlinear}\mathrm{propagation}}{}}{\frac{{}_p\left(\underset{\_}{r}\right)}{c_0^2\left(\underset{\_}{r}\right)}\frac{{}^2{p\left(\underset{\_}{r},t\right)}^2}{t^2}}-\underset{\underset{3)\mathrm{Absorption}}{}}{h_{\mathrm{ab}}\left(\underset{\_}{r},t\right)\underset{t}{.Math.}\frac{1}{c_0^2}\frac{{}^{2}p\left(\underset{\_}{r},t\right)}{t^2}}=\underset{\underset{\left(4\right)\mathrm{Linear}\mathrm{scattering}\mathrm{source}\mathrm{terms}}{}}{\frac{{}_l\left(\underset{\_}{r}\right)}{c_0^2\left(\underset{\_}{r}\right)}\frac{{}^{2}p\left(\underset{\_}{r},t\right)}{t^2}+\left(\left(\underset{\_}{r}\right)p\left(\underset{\_}{r},t\right)\right)}-\underset{\underset{\underset{\mathrm{source}\mathrm{term}}{\left(5\right)\mathrm{Nonlinear}\mathrm{scattering}}}{}}{\frac{{}_n\left(\underset{\_}{r}\right)}{c_0^2\left(\underset{\_}{r}\right)}\frac{{}^2{p\left(\underset{\_}{r},t\right)}^2}{t^2}}& \left(14\right)\end{array}$
where c.sub.0(r) is the linear wave propagation velocity for low field amplitudes, .sub.1(r) and (r) are linear scattering parameters given by the relative spatial variation of the material compressibility and mass density, and .sub.n(r) is a nonlinear scattering parameter. The left side propagation parameters vary with r on a scale >approximately the wavelength, while the right side scattering parameters vary with r on a scale <approximately the wave length. A similar equation for shear waves and electromagnetic waves can be formulated that represents similar nonlinear propagation and local scattering phenomena for the shear and EM waves.
B. Nonlinear Interaction in Dual Band Pulse Complexes

(45) The different terms of Eq. (14) have different effects on the wave propagation and scattering: The linear propagation terms (1) guide the forward spatial propagation of the incident wave without addition of new frequency components. The linear scattering source terms (4) produce local scattering of the incident wave without addition of new frequency components, i.e. the linearly scattered wave has the same frequency components as the incident wave, with an amplitude modification .sup.2 produced by 2.sup.nd order differentiation in the scattering terms.

(46) For materials with adequately high nonlinearity in the material parameters relative to the wave field amplitude, the nonlinearity affects both the propagation and local scattering of the wave. A slow variation (close to constant on scale >wave length) of the nonlinear parameters give a value to .sub.p(r) that provides a nonlinear forward propagation distortion of the incident waves that accumulates in amplitude with propagation distance through term (2). A rapid variation of the nonlinear material parameters (on scale <wavelength) gives a value to .sub.n(r) that produces a local nonlinear scattering of the incident wave through term (5).

(47) We study the situation where the total incident wave is the sum of the LF and HF pulses, i.e. p(r,t)=p.sub.L(r,t)+p.sub.H(r,t). The nonlinear propagation and scattering are in this 2.sup.nd order approximation both given by

(48) $\begin{array}{cc}\text{~}{p\left(\underset{\_}{r},t\right)}^2={\left(p_L\left(\underset{\_}{r},t\right)+p_H\left(\underset{\_}{r},t\right)\right)}^2=\underset{\underset{\underset{\mathrm{distortion}}{\mathrm{nonlin}\mathrm{self}}}{}}{{p_L\left(\underset{\_}{r},t\right)}^2}+\underset{\underset{\mathrm{nonlin}\mathrm{interaction}}{}}{2p_L\left(\underset{\_}{r},t\right)p_H\left(\underset{\_}{r},t\right)}+\underset{\underset{\underset{\mathrm{distortion}}{\mathrm{nonlin}\mathrm{self}}}{}}{{p_H\left(\underset{\_}{r},t\right)}^2}& \left(15\right)\end{array}$

(49) A multiplication of two functions in the temporal domain produces a convolution of the functions temporal Fourier transforms (i.e. temporal frequency spectra) introducing frequency components in the product of the functions that are sums and differences of the frequency components of the factors of the multiplication. The squares p.sub.L.sup.2, and p.sub.H.sup.2 then produces a self-distortion of the incident pulses that pumps energy from the incident fundamental bands into harmonic components of the fundamental bands, and is hence found as a nonlinear attenuation of the fundamental bands of both the LF and HF waves. However, the wave amplitude determines the self-distortion per wave length, and the LF pulse typically propagates only up to 30 LF wavelengths into the object which allows us to neglect both nonlinear self distortion and nonlinear attenuation for the LF wave. The HF wave, there-against, propagates up to 250 HF wavelengths into the object, so that both the nonlinear self-distortion an the nonlinear attenuation becomes observable. We return to these phenomena for simulations of the wave propagation in Section 5.8B.

(50) C. Nonlinear Propagation and Scattering for Dual Band Pulse Complexes

(51) When the temporal HF pulse length T.sub.pH is much shorter than half the period of the LF pulse, T.sub.L/2, i.e. the bandwidth of the HF pulse B.sub.H>.sub.L/2, where .sub.L=2/T.sub.L is the angular frequency of the LF wave, the sum and difference frequency spectra overlaps with each other and the fundamental HF spectrum. This is the situation illustrated in FIG. 1. For the further analysis we assume that |2.sub.p(r)p.sub.L(r,t)|=|x|<<1 which allows the approximation 1x1/(1+x) and the propagation terms of the left side of Eq. (14) can for the manipulation of the HF pulse by the co-propagating LF pulse be approximated as

(52) 0$\begin{array}{cc}\underset{\underset{\left(1\right)\mathrm{Linear}\mathrm{propagation}}{}}{{}^2{p}_H\left(\underset{\_}{r},t\right)-\frac{1}{c_0^2\left(\underset{\_}{r}\right)}\frac{{}^2{p}_H\left(\underset{\_}{r},t\right)}{t^2}}+\underset{\underset{\left(2\right)\mathrm{Nonlinear}\mathrm{propagation}}{}}{\frac{2{}_p\left(\underset{\_}{r}\right)p_L\left(\underset{\_}{r},t\right)}{c_0^2\left(\underset{\_}{r}\right)}\frac{{}^2{p}_H\left(\underset{\_}{r},t\right)}{t^2}}{}^2{p}_H\left(\underset{\_}{r},t\right)-\frac{1}{c_0^2\left(\underset{\_}{r}\right)\left(1+2{}_p\left(\underset{\_}{r}\right)p_L\left(\underset{\_}{r},t\right)\right)}\frac{{}^2{p}_H\left(\underset{\_}{r},t\right)}{t^2}& \left(16\right)\end{array}$

(53) The numerator in front of the temporal derivative in this propagation operator is the square propagation velocity, and we hence see that the LF pulse pressure p.sub.L modifies the propagation velocity for the co-propagating HF pulse p.sub.H as
c(r,p.sub.L)={square root over (c.sub.0.sup.2(r)(1+2.sub.p(r)p.sub.L))}c.sub.0(r)(1+.sub.p(r)p.sub.L)(17)
where p.sub.L is the actual LF field variable along the co-propagating HF pulse. The orthogonal trajectories of the HF pulse wave fronts are paths of energy flow in the HF pulse propagation. The propagation lag of the HF pulse along the orthogonal trajectories of the HF pulse wave-fronts, (r), can then be approximated as

(54) $\begin{array}{cc}t\left(\underset{\_}{r}\right)={}_{\left(\underset{\_}{r}\right)}\frac{\mathrm{ds}}{c\left(\underset{\_}{s},p.Math.p_L\left(\underset{\_}{s}\right)\right)}{}_{\left(\underset{\_}{r}\right)}\frac{\mathrm{ds}}{c_0}-p{}_{\left(\underset{\_}{r}\right)}\frac{\mathrm{ds}}{c_0}{}_p\left(\underset{\_}{s}\right)p_L\left(\underset{\_}{s}\right)=t_0\left(\underset{\_}{r}\right)+p\left(\underset{\_}{r}\right)t_0\left(\underset{\_}{r}\right)={}_{\left(\underset{\_}{r}\right)}\frac{\mathrm{ds}}{c_0\left(\underset{\_}{s}\right)}p\left(\underset{\_}{r}\right)=-p{}_{\left(\underset{\_}{r}\right)}\frac{\mathrm{ds}}{c_0\left(\underset{\_}{s}\right)}{}_p\left(\underset{\_}{s}\right)p_L\left(\underset{\_}{s}\right)& \left(18\right)\end{array}$
where p.Math.p.sub.L(s) is the LF pressure at the center of gravity at s of the co-propagating HF pulse, and p is a scaling/polarity variable for the LF pulse that is extracted to more clearly show results of scaling the amplitude and changing the polarity of the LF pulse. The propagation lag without manipulation of the propagation velocity by the LF pulse is t.sub.0(r). (r) is the added nonlinear propagation delay produced by the nonlinear manipulation of the propagation velocity for the HF pulse by the LF pressure p.sub.L(s). We note that when the 1.sup.st scattering/reflection occurs, the LF pressure p.sub.L drops so much that the LF modification of the propagation velocity is negligible for the scattered HF wave. This means that we only get contribution in the integral for (r) up to the 1.sup.st scattering, an effect that we will use to suppress multiple scattered waves in the received signal and to enhance the nonlinear scattering components.

(55) Variations of the LF pressure along the co-propagating HF pulse produces a variation of the propagation velocity along the HF pulse, that in turn produces a nonlinear pulse from distortion of the HF pulse that accumulates with propagation distance. When B.sub.HF>.sub.LF/2 the nonlinear propagation distortion can be described by a filter, as shown below in Eqs. (19,23).

(56) In summary, the nonlinear propagation term, i.e. term (2) in Eq. (14) produces:

(57) i) A nonlinear propagation delay (r) produced by the LF pressure at the center of gravity of the co-propagating HF pulse according to Eq. (18), and is accumulated up to the 1.sup.st scattering, where the amplitude of the LF pulse drops so much that the effect of the LF pulse can be neglected after this point.
ii) A nonlinear propagation distortion of the HF pulse produced by the variation of the LF pulse field along the co-propagating HF pulse that accumulates with propagation distance up to the 1.sup.st scattering, where the amplitude of the LF pulse drops so much that the effect of the LF pulse can be neglected after this point.
iii) A nonlinear self-distortion of the HF pulse which up to the 1.sup.st scattering transfers energy from the fundamental HF band to harmonic components of the fundamental HF band, and is hence found as a nonlinear attenuation of the fundamental band of the HF pulse.

(58) When B.sub.HF>.sub.LF/2 the transmitted HF pulse field P.sub.t(r.sub.1,; r.sub.t) hence observes a nonlinear propagation distortion as described by point i)-iii) above and takes the form P.sub.tp(r.sub.1,; r.sub.t)

(59) $\begin{array}{cc}{P}_{\mathrm{tp}}\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)=V_p\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)P_t\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)V_p\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)=\frac{P_{\mathrm{tp}}\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)}{P_t\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)}={\tilde{V}}_p\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right){}^{-p\left({\underset{\_}{r}}_1\right)}& \left(19\right)\end{array}$
where the subscript p designates the amplitude/polarity/phase of the LF pulse. P.sub.t(r.sub.1,; r.sub.t) is given in Eq. (1), and V.sub.p includes all nonlinear forward propagation distortion, where the linear phase component of V.sub.p is separated out as the nonlinear propagation delay p(r.sub.1) up to the point r.sub.1. {tilde over (V)}.sub.p hence represents the nonlinear pulse form distortion of the HF pulse by the co-propagating LF pulse, and also the nonlinear attenuation produced by the nonlinear self distortion of the HF pulse.

(60) The nonlinear scattering term (5) in Eq. (14) can for the nonlinear interaction term be approximated within the same regime as

(61) $\begin{array}{cc}\underset{\underset{\underset{\mathrm{source}\mathrm{term}}{\left(5\right)\mathrm{Nonlinear}\mathrm{scattering}}}{}}{\frac{{}_n\left(\underset{\_}{r}\right)}{c_0^2\left(\underset{\_}{r}\right)}\frac{{}^2{p\left(\underset{\_}{r},t\right)}^2}{t^2}}\frac{2{}_n\left(\underset{\_}{r}\right)p.Math.p_L\left(\underset{\_}{r},t\right)}{c_0^2\left(\underset{\_}{r}\right)}\frac{{}^2{p}_H\left(\underset{\_}{r},t\right)}{t^2}& \left(20\right)\end{array}$

(62) We hence see that when the temporal pulse length of the HF pulse T.sub.pH<T.sub.L/2, the local LF pulse pressure at the co-propagating HF pulse exerts an amplitude modulation of the scattered wave, proportional to p.Math.2.sub.n(r)p.sub.L(r,t). For the nonlinear self-distortion, term (5) produces harmonic components in the scattered wave, and is not noticed in the fundamental HF band.

(63) The effect of nonlinearity of material parameters on the scattered signal, can hence be split into two groups:

(64) Group A originates from the linear scattering, i.e. term (4) of Eq. (14), of the forward, accumulative nonlinear propagation distortion of the incident wave, i.e. combination of term (2) and term (4) in Eq. (14), which is split into a nonlinear propagation delay according to Eq. (18) (i above), a nonlinear pulse form distortion (ii above), and a nonlinear attenuation of the HF pulse (iii above), and
Group B originates directly in the local nonlinear scattering, i.e. term (5), of the original frequency components in the incident wave, i.e. interaction between term (1), the nonlinear propagation delay of term (2) and term (5), where the local LF pulse pressure at the co-propagating HF pulse exerts an amplitude modulation of the scattered wave, proportional to p.Math.2.sub.n(r)p.sub.L(r,t).

(65) There is also a Group C found as local nonlinear scattering from term (5) of the forward accumulative nonlinear propagation distortion components in the incident wave, i.e. interaction between the pulse distortion component of term (2) and term (5) in Eq. (14), but typical nonlinear material parameters are so low that this group is negligible.

(66) 5.4 Signal Models for Nonlinear Propagation and Scattering

(67) A. Models for 1.sup.st order scattering The 1.sup.st order linear and nonlinear scattered wave that hits the receiver array at r is then from Eq. (2, 19, 20)
dP.sub.l1(r,;r.sub.1)=k.sup.2G(r,r.sub.1;)P.sub.tp(r.sub.1,;r.sub.t)(r.sub.1)d.sup.3r.sub.1
dP.sub.n1(r,;r.sub.1)=k.sup.2G(r,r.sub.1;)P.sub.tp(r.sub.1,;r.sub.t)2p.sub.L(r.sub.1).sub.n(r.sub.1)d.sup.3r.sub.1(21)

(68) The received signal from the 1.sup.st order linear and nonlinear scattering of the forward nonlinear distorted pressure pulse for the focused and aberration corrected receive beam is hence
dY.sub.l1(;r.sub.1)=k.sup.2H.sub.r(r.sub.1,;r.sub.r)U.sub.p(r.sub.1,;r.sub.t)(r.sub.1)d.sup.3r.sub.1
dY.sub.n1(;r.sub.1)=k.sup.2H.sub.r(r.sub.1,;r.sub.r)U.sub.p(r.sub.1,;r.sub.t)2p.sub.L(r.sub.1).sub.n(r.sub.1)d.sup.3r.sub.1
U.sub.p(r.sub.1,;r.sub.t)=H.sub.rt()P.sub.tp(r.sub.1,;r.sub.t)=V.sub.p(r.sub.1,;r.sub.t)U()H.sub.t(r.sub.1,;r.sub.t)(22)
where we have assumed that the pressure to signal transfer function H.sub.rt () of the transducer elements, is the same for all receiver array elements. Introducing the scaling variable p for the LF field, p.sub.L(r.sub.1).fwdarw.p.Math.p.sub.L(r.sub.1), the total 1.sup.st order scattered wave takes the form

(69) $\begin{array}{cc}{\mathrm{dY}}_{1p}\left(;{\underset{\_}{r}}_1\right)=V_p\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)\left\{{\mathrm{dX}}_l\left(;{\underset{\_}{r}}_1\right)+{\mathrm{pdX}}_n\left(;{\underset{\_}{r}}_1\right)\right\}{\mathrm{dX}}_l\left(;{\underset{\_}{r}}_1\right)=-k^{2}U\left(\right)H_r\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_r\right)H_t\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)\left({\underset{\_}{r}}_1\right)d^3{r}_1{\mathrm{dX}}_n\left(;{\underset{\_}{r}}_1\right)=-k^{2}U\left(\right)H_r\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_r\right)H_t\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)2{}_n\left({\underset{\_}{r}}_1\right)p_L\left({\underset{\_}{r}}_1\right)d^3{r}_1{V}_p\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)=\frac{U_p\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)}{U\left(\right)H_t\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)}={\tilde{V}}_p\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)e^{-p\left({\underset{\_}{r}}_1\right)}& \left(23\right)\end{array}$

(70) The magnitude of the nonlinear scattering component is generally much lower than the linear scattering component. To estimate this nonlinear scattering component one could typically transmit two pulse complexes with opposite polarities of the LF pulse, giving the HF receive signals
dY.sub.+(;r.sub.1)=V.sub.+(r.sub.1,;r.sub.t)dX.sub.l(;r.sub.1)+V.sub.+(r.sub.1,;r.sub.t)dX.sub.n(;r.sub.1)
dY.sub.(;r.sub.1)=V.sub.(r.sub.1,;r.sub.t)dX.sub.l(;r.sub.1)V.sub.(r.sub.1,;r.sub.t)dX.sub.n(;r.sub.1)(24)
The nonlinear scattering component is then obtained as
2dX.sub.n(;r.sub.1)=V.sub.+(r.sub.1,;r.sub.t).sup.1dY.sub.+(;r.sub.1)V.sub.(r.sub.1,;r.sub.t).sup.1dY.sub.(,r.sub.1)(25)
where V.sub.(r.sub.1,; r.sub.t).sup.1={tilde over (V)}.sub.(r.sub.1,;r.sub.t).sup.1e.sup.ip(r.sup.1.sup.) are the inverse filters of V.sub.(r.sub.1,; r.sub.t) in Eq. (24). The inverse filter is hence composed of a delay correction p(r.sub.1), and a pulse distortion correction {tilde over (V)}.sub.(r.sub.1,;r.sub.t).sup.1.

(71) The signal is received as a function of the fast time (or depth-time) t, where we assign a depth z in the image as z=ct/2. When one have a distribution of scatterers, one must sum over the scatterer coordinate r.sub.1, and the LF pressure can in addition to the non-linear propagation delay produce a LF pressure dependent speckle in the received signal that we discuss in Section 5.7. One then needs to generalize the correction for nonlinear propagation delay to correction with a correction delay that is an average nonlinear propagation delay, and generalize the pulse form correction to a speckle correction, as further discussed in Sections 5.7 and 5.8.

(72) C. Models for 3.sup.rd Order Scattering

(73) For 3.sup.rd order multiple scattering, the nonlinear propagation delay and pulse form distortion will be a function of the first scatterer coordinate r.sub.1,3, as both the LF and HF pulse amplitudes drop so much in this first scattering that we get linear propagation after the 1.sup.st scattering. The nonlinear effect with dual band pulse complexes as above, hence modifies the 3.sup.rd order scattered signals in Eq. (12) to
dY.sub.Ip(;r.sub.1,r.sub.3)=k.sup.4H.sub.r(r.sub.3,;r.sub.r)(r.sub.3)d.sup.3r.sub.3H.sub.rev(r.sub.3,r.sub.1;)U.sub.p(r.sub.1,;r.sub.1)(r.sub.1)d.sup.3r.sub.1
dY.sub.IIp(;r.sub.1,r.sub.3)=k.sup.4H.sub.r(r.sub.1,;r.sub.r)(r.sub.1)d.sup.3r.sub.1H.sub.rev(r.sub.1,r.sub.3;)U.sub.p(r.sub.3,;r.sub.t)(r.sub.3)d.sup.3r.sub.3(26)
where dY.sub.Ip(;r.sub.1, r.sub.3) and dY.sub.IIp(;r.sub.1,r.sub.3) are Class I and II multiple scattering noise with non-zero LF pulse, U.sub.p is the HF receive pulse from a plane reflector. The sum of the Class I and Class II pulse reverberation noise can then be written as

(74) $\begin{array}{cc}\begin{array}{c}{\mathrm{dY}}_p\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)={\mathrm{dY}}_{\mathrm{Ip}}\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)+{\mathrm{dY}}_{\mathrm{IIp}}\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)\\ =\{\frac{H_r\left({\underset{\_}{r}}_3,;{\underset{\_}{r}}_t\right)U_p\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)}{H_r\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)H_t\left({\underset{\_}{r}}_3,;{\underset{\_}{r}}_t\right)U\left(\right)}+\\ \frac{U_p\left({\underset{\_}{r}}_3,;{\underset{\_}{r}}_t\right)}{H_t\left({\underset{\_}{r}}_3,;{\underset{\_}{r}}_t\right)U\left(\right)}\}{\mathrm{dY}}_{3\mathrm{II}}\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)\end{array}& \left(27\right)\end{array}$
where dY.sub.3II(;r.sub.1,r.sub.3) is Class II multiple scattering with zero LF pulse, i.e. linear wave propagation as given from Eq. (12) with r.sub.3>r.sub.1
dY.sub.3II(;r.sub.1,r.sub.3)=k.sup.4U()H.sub.r(r.sub.1,;r.sub.r)(r.sub.1)d.sup.3r.sub.1H.sub.rev(r.sub.1,r.sub.3;)H.sub.t(r.sub.3,;r.sub.t)(r.sub.3)d.sup.3r.sub.3(28)

(75) Comparing with Eqs. (19, 22, 23) we can rewrite Eq. (27) as

(76) $\begin{array}{cc}{\mathrm{dY}}_p\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)=K_p\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right){\mathrm{dY}}_{3\mathrm{II}}\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)K_p\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)=Q\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)V_p\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)+V_p\left({\underset{\_}{r}}_3,;{\underset{\_}{r}}_t\right)Q\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)=\frac{H_t\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)H_r\left({\underset{\_}{r}}_3,;{\underset{\_}{r}}_r\right)}{H_r\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_r\right)H_t\left({\underset{\_}{r}}_3,;{\underset{\_}{r}}_t\right)}=\frac{\stackrel{.Math.}{Q}\left(;{\underset{\_}{r}}_3\right)}{\stackrel{.Math.}{Q}\left(;{\underset{\_}{r}}_1\right)}\stackrel{.Math.}{Q}\left(;\underset{\_}{r}\right)=\frac{H_r\left(\underset{\_}{r},;{\underset{\_}{r}}_r\right)}{H_t\left(\underset{\_}{r},;{\underset{\_}{r}}_t\right)}& \left(29\right)\end{array}$

(77) The nonlinear modification by the LF pulse of the HF pulse U.sub.p from the linear form UH.sub.t, is hence given by V.sub.p that is defined in Eqs. (19, 22, 23), while Q represents the difference between Class I and Class II pulse reverberations in the linear propagation regime. Note that when r.sub.1.fwdarw.r.sub.3, we have Q.fwdarw.1 and V.sub.p3.fwdarw.V.sub.p1. The linear phase component of V.sub.p represents the nonlinear propagation delay p(r.sub.i) up to the point r.sub.i, i=1, 3, of the 1.sup.st scatterer where the amplitude of both the LF and HF pulses drops so much that one have basically linear wave propagation after this point. The nonlinear pulse form distortion due to variations of the LF pulse along the co-propagating HF pulse and also the nonlinear attenuation of the fundamental band of the HF pulse due to self distortion of the HF pulse, is included in {tilde over (V)}.sub.p of Eqs. (19,23). This nonlinear attenuation reduces the magnitude of {tilde over (V)}.sub.p (r,; r.sub.t) with depth up to the 1.sup.st scatterer, hence reducing |{tilde over (V)}.sub.p(r.sub.3,;r.sub.t)| compared to |{tilde over (V)}.sub.p(r.sub.1,;r.sub.t)| in Eq. (29)

(78) In Eq. (29) the total pulse reverberation noise with nonzero LF pulse is hence a filtered version of the Class II reverberation with zero LF pulse, dY.sub.3II (;r.sub.1,r.sub.3), where the filter depends on the LF pulse in relation to the co-propagating HF pulse and has the form

(79) $\begin{array}{cc}\begin{array}{c}{K}_p\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)=Q\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right){\tilde{V}}_p\left({\underset{\_}{r}}_1,;{\underset{\_}{r}}_t\right)e^{-p\left({\underset{\_}{r}}_1\right)}+\\ {\tilde{V}}_p\left({\underset{\_}{r}}_3,;{\underset{\_}{r}}_t\right)e^{-p\left({\underset{\_}{r}}_3\right)}\\ =e^{-p\left[\left({\underset{\_}{r}}_1\right)+\left({\underset{\_}{r}}_3\right)\right]/2}{\tilde{K}}_p\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)\end{array}{\tilde{K}}_p\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)=Q\left(;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right){\tilde{V}}_p\left(;{\underset{\_}{r}}_1\right)e^{p\left[\left({\underset{\_}{r}}_3\right)-\left({\underset{\_}{r}}_1\right)\right]/2}+{\tilde{V}}_p\left(;{\underset{\_}{r}}_3\right)e^{-p\left[\left({\underset{\_}{r}}_3\right)-\left({\underset{\_}{r}}_1\right)\right]/2}& \left(30\right)\end{array}$

(80) The effect of this filter for a distribution of scatterers is discussed in Section 5.7B below.

(81) 5.5 Synthetic Transmit and Receive Beams

(82) A. 1.sup.st Order Scattering with Multi-Component Transmit

(83) With synthetic transmit and receive beams it is possible to obtain Q=1 in a large group of image points without loosing frame rate. With such methods, one transmits a set of wide transmit beams {H.sub.tj(r,); j=1, . . . , J} with different directions. With a linear array shown as 400 in FIG. 4, H.sub.tj(r,) can for example be beams with close to plane wave fronts with different directions in the azimuth direction, while the elevation variation of the beams are the same as with standard fixed focusing. With two-dimensional arrays, H.sub.tj(r,) can have close to plane wave fronts in both the azimuth and elevation directions. An example of such a plane wave front in the azimuth direction is shown in 401 of FIG. 4. The plane wave is transmitted at an angle .sub.j to the array, and we define t=0 so that the center element transmits at t=0. For each transmit beam direction, the HF receive signal is picked up in parallel at all array elements.

(84) The signal is received on all elements for each transmit beam that gives the following set {Y.sub.jk(); j=1, . . . , J; k=1, . . . , K} HF receive element signals from element # k with transmit beam # j. The 1.sup.st order scattered HF receive signal is in its linear approximation given from Eq. (5) as
dY.sub.jk(;r.sub.1)=k.sup.2U()H.sub.rk(r.sub.1,)H.sub.tj(r.sub.1,)(r.sub.1)d.sup.3r(31)
where H.sub.rk(r.sub.1,) is the receive beam of element # k at the location r.sub.1 (405) of a scatterer. For the image reconstruction we use the time signal of the HF receive element signal obtained as the inverse Fourier transform of Eq. (31)

(85) $\begin{array}{cc}{\mathrm{dy}}_{\mathrm{jk}}\left(t;{\underset{\_}{r}}_1\right)=c^{-2}\stackrel{\mathrm{.Math.}}{u}\left(t\right)\underset{t}{.Math.}{h}_{\mathrm{rk}}\left({\underset{\_}{r}}_1,t\right)\underset{t}{.Math.}{h}_{\mathrm{ij}}\left({\underset{\_}{r}}_1,t\right)\left({\underset{\_}{r}}_1\right)d^3{r}_1& \left(32\right)\end{array}$
where h.sub.rk(r.sub.1,t) is the spatial impulse response of array element # k, and h.sub.tj(r.sub.1,t) is the spatial impulse response for transmit beam # j, both obtained by inverse Fourier transform of the spatial frequency responses H.sub.rk(r.sub.1,) and H.sub.tj(r.sub.1,) in Eq. (31).

(86) We reconstruct an image signal in a spatial location r (402) of the image with a synthetic receive beam and a synthetic transmit beam that are both focused at r as

(87) $\begin{array}{cc}\mathrm{dz}\left(\underset{\_}{r};{\underset{\_}{r}}_1\right)=\mathrm{dy}\left(2r/c;{\underset{\_}{r}}_1\right)=\underset{j,k}{.Math.}{a}_k\left(\underset{\_}{r}\right)b_j\left(\underset{\_}{r}\right){\mathrm{dy}}_{\mathrm{jk}}\left(2r/c-{}_{\mathrm{rk}}\left(\underset{\_}{r}\right)-{}_{\mathrm{ij}}\left(\underset{\_}{r}\right);{\underset{\_}{r}}_1\right)c{}_{\mathrm{rk}}\left(x,z\right)=z-\sqrt{{\left(x_k-x\right)}^2+z^2}c{}_{\mathrm{tj}}\left(x,z\right)=x\sin {}_j+z\left(\cos {}_j-1\right)& \left(33\right)\end{array}$
where r=|r| and .sub.rk(r) is the focusing delay for HF receive element # k (404) for focusing the receive beam centered around x onto r=xe.sub.x+ze.sub.z, a.sub.k(r) is a receive beam apodization function, .sub.tj(r) is a delay for component transmit beam # j to provide a synthetic transmit beam focused at r, and b.sub.j(r) is a transmit beam apodization function. Introducing the HF receive element signals from Eq. (32), the reconstructed image signal is

(88) 0$\begin{array}{cc}\begin{array}{c}{\mathrm{dz}}_1\left(\underset{\_}{r};{\underset{\_}{r}}_1\right)=c^{-2}\stackrel{\mathrm{.Math.}}{u}\left(2r/c\right)\underset{t}{.Math.}{.Math.}_k{a}_k{h}_{\mathrm{rk}}\left({\underset{\_}{r}}_1,r/c-{}_{\mathrm{rl}}\left(\underset{\_}{r}\right)\right)\underset{t}{.Math.}\\ {.Math.}_j{b}_j{h}_j\left({\underset{\_}{r}}_1,r/c-{}_{\mathrm{rj}}\left(\underset{\_}{r}\right)\right)\left({\underset{\_}{r}}_1\right)d^3{r}_1\\ =c^{-2}\stackrel{\mathrm{.Math.}}{u}\left(2r/c\right)\underset{t}{.Math.}{h}_r\left({\underset{\_}{r}}_1,t;\underset{\_}{r}\right)\underset{t}{.Math.}{h}_t\left({\underset{\_}{r}}_1,t;\underset{\_}{r}\right)\left({\underset{\_}{r}}_1\right)d^3{r}_1\end{array}{h}_r\left({\underset{\_}{r}}_1,t;\underset{\_}{r}\right)={.Math.}_k{a}_k{h}_k\left({\underset{\_}{r}}_1,t-{}_{\mathrm{rk}}\left(\underset{\_}{r}\right)\right)h_t\left({\underset{\_}{r}}_1,t;\underset{\_}{r}\right)={.Math.}_j{b}_j{h}_{\mathrm{ij}}\left({\underset{\_}{r}}_1,t-{}_{\mathrm{tj}}\left(\underset{\_}{r}\right)\right)& \left(34\right)\end{array}$
h.sub.t(r.sub.1,t; r) and h.sub.r(r.sub.1,t; r) are the spatial impulse responses of the synthetic transmit and receive beams for a linear material. Out of the total array width, D.sub.arr, one can hence operate with a synthetic receive aperture D.sub.r determined by how many elements one sums over, and a synthetic transmit aperture determined by how many transmit beam components one sums over. With a non-zero LF pulse we get the synthetic linearly and nonlinearly scattered signals as in Eq. (23), with the synthetic transmit and receive beams of Eq. (34). We should note that the LF pressure that produces the nonlinear propagation delay and the pulse form distortion is the actual LF pressure for each transmitted component H.sub.tj(r,).

(89) For dual frequency pulse complexes in a nonlinear material, the HF receive signal from component transmit pulse # j at HF receive array element # k is

(90) $\begin{array}{cc}{\mathrm{dy}}_{\mathrm{jk}}\left(t;{\underset{\_}{r}}_1\right)=c^{-2}{h}_{\mathrm{rk}}\left({\underset{\_}{r}}_1,t\right)\underset{t}{.Math.}{h}_{\mathrm{tj}}\left({\underset{\_}{r}}_1,t\right)\underset{t}{.Math.}{\stackrel{\mathrm{.Math.}}{u}}_{\mathrm{pj}}\left({\underset{\_}{r}}_1,t\right)\left({\underset{\_}{r}}_1\right)d^3{r}_1{u}_{\mathrm{pj}}\left({\underset{\_}{r}}_1,t\right)=v_{\mathrm{pj}}\left({\underset{\_}{r}}_1,t\right)\underset{t}{.Math.}u\left(t\right)\underset{t}{.Math.}{h}_{\mathrm{ij}}\left({\underset{\_}{r}}_1,t\right)v_{\mathrm{pj}}\left({\underset{\_}{r}}_1,t\right)={\tilde{v}}_p\left({\underset{\_}{r}}_1,t-p\left({\underset{\_}{r}}_1\right)\right)& \left(35\right)\end{array}$
u.sub.pj(r.sub.1,t) is the forward propagating HF pressure pulse # j convolved with the receive element impulse response, similar to Eq. (23). u.sub.pj(r.sub.1,t) has undergone a nonlinear propagation delay and pulse form distortion by the co-propagating LF pulse, and also the nonlinear self distortion.

(91) Applying the synthetic image reconstruction as in Eq. (33,34) we get

(92) $\begin{array}{cc}d{y}_l\left(2r/c;{\underset{\_}{r}}_1\right)=c^{-2}{h}_r\left({\underset{\_}{r}}_1,2r/c;\underset{\_}{r}\right)\underset{t}{.Math.}{\stackrel{\mathrm{.Math.}}{u}}_{pt}\left({\underset{\_}{r}}_1,2r/c;\underset{\_}{r}\right)\left({\underset{\_}{r}}_1\right)d^3{r}_{1}d{y}_n\left(2r/c;{\underset{\_}{r}}_1\right)=c^{-2}{h}_r\left({\underset{\_}{r}}_1,2r/c;\underset{\_}{r}\right)\underset{t}{.Math.}{\stackrel{\mathrm{.Math.}}{u}}_{pt}\left({\underset{\_}{r}}_1,2r/c;\underset{\_}{r}\right)2{}_n\left({\underset{\_}{r}}_1\right)p_L\left({\underset{\_}{r}}_1\right)d^3{r}_1{u}_{pt}\left({\underset{\_}{r}}_1,2r/c;\underset{\_}{r}\right)={.Math.}_j{u}_{p,j}\left({\underset{\_}{r}}_1,t-{}_{\mathrm{tj}}\left(\underset{\_}{r}\right)\right)& \left(36\right)\end{array}$

(93) This expression has the same form as for the focused transmit and receive beams in Eq. (22), and We hence also get the same form for the total 1.sup.st order HF receive signal with transmitted dual frequency pulse complexes in a nonlinear material as in Eq. (23). Signal processing for the multi component transmit beam imaging hence takes the same form as for focused transmit and receive beams.

(94) B. 3.sup.rd Order Scattering with Multi-Component

(95) The image reconstruction is hence based on the assumption of 1.sup.st order scattering. However, each HF receive element signal will contain multiple scattering noise, where the 3.sup.rd order scattering noise given by the formula in Eq. (26) is the most dominating component. The HF receive element signal for element # k and component transmit beam # j for the 3.sup.rd order scattered pulse reverberations of Class I and II with nonlinear propagation of the transmit pulse up to the 1.sup.st scatterer, is hence

(96) $\begin{array}{cc}{\mathrm{dy}}_{\mathrm{Ip},\mathrm{jk}}\left(t;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)=c^{-4}{h}_{\mathrm{rk}}\left({\underset{\_}{r}}_3,t\right)\left({\underset{\_}{r}}_3\right)d^3{r}_3\underset{t}{.Math.}{h}_{\mathrm{rev}}\left({\underset{\_}{r}}_3,{\underset{\_}{r}}_1;t\right)\underset{t}{.Math.}{u}_{p,j}^{\left(4\right)}\left({\underset{\_}{r}}_1,t\right)\left({\underset{\_}{r}}_1\right)d^3{r}_1{\mathrm{dy}}_{\mathrm{IIp},\mathrm{jk}}\left(t;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)=c^{-4}{h}_{\mathrm{rk}}\left({\underset{\_}{r}}_1,t\right)\left({\underset{\_}{r}}_1\right)d^3{r}_1\underset{t}{.Math.}{h}_{\mathrm{rev}}\left({\underset{\_}{r}}_3,{\underset{\_}{r}}_1;t\right)\underset{t}{.Math.}{u}_{p,j}^{\left(4\right)}\left({\underset{\_}{r}}_1,t\right)\left({\underset{\_}{r}}_3\right)d^3{r}_3{\mathrm{dy}}_{3p,\mathrm{jk}}\left(t;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)={\mathrm{dy}}_{\mathrm{Ip},\mathrm{jk}}\left(t;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)+{\mathrm{dy}}_{\mathrm{IIp},\mathrm{jk}}\left(t;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right),& \left(37\right)\end{array}$
where u.sub.pj.sup.(4) is the transmitted nonlinear HF pulse component # j at r.sub.1, scattered twice from a point scatterer and received through the transducer element. Reconstructing an image signal at r according to the lines of Eqs. (33,34), the reconstructed signal will contain a 3.sup.rd order pulse reverberation component according to

(97) $\begin{array}{cc}\begin{array}{c}{\mathrm{dy}}_{1p}\left(2r/c;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)=\underset{j,k}{.Math.}{\mathrm{dy}}_{p,\mathrm{jk}}\left(2r/c-{}_{\mathrm{rk}}\left(\underset{\_}{r}\right)-{}_{\mathrm{tj}}\left(\underset{\_}{r}\right);{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)\\ ={\mathrm{dy}}_{\mathrm{Ip}}\left(2r/c;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)+{\mathrm{dy}}_{\mathrm{IIp}}\left(2r/c;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)\end{array}& \left(38\right)\end{array}$
where the Class I and Class II components are

(98) $\begin{array}{cc}d{y}_{\mathrm{Ip}}\left(t;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)=c^{-4}{h}_r\left({\underset{\_}{r}}_3,t;{\underset{\_}{r}}_t\right)\left({\underset{\_}{r}}_3\right)d^3{r}_3\underset{t}{.Math.}{h}_{\mathrm{rev}}\left({\underset{\_}{r}}_3,{\underset{\_}{r}}_1;t\right)\underset{t}{.Math.}{u}_p^{\left(4\right)}\left({\underset{\_}{r}}_1,t;{\underset{\_}{r}}_t\right)\left({\underset{\_}{r}}_1\right)d^3{r}_{1}d{y}_{\mathrm{IIp}}\left(t;{\underset{\_}{r}}_1,{\underset{\_}{r}}_3\right)=c^{-4}{h}_r\left({\underset{\_}{r}}_1,t;{\underset{\_}{r}}_r\right)\left({\underset{\_}{r}}_1\right)d^3{r}_1\underset{t}{.Math.}{h}_{\mathrm{rev}}\left({\underset{\_}{r}}_3,{\underset{\_}{r}}_1;t\right)\underset{t}{.Math.}{u}_p^{\left(4\right)}\left({\underset{\_}{r}}_3,t;{\underset{\_}{r}}_t\right)\left({\underset{\_}{r}}_3\right)d^3{r}_3{u}_p\left({\underset{\_}{r}}_1,t;\underset{\_}{r}\right)={.Math.}_j{b}_k{u}_{\mathrm{pj}}\left({\underset{\_}{r}}_1,t-{}_{\mathrm{ij}}\left(\underset{\_}{r}\right)\right)& \left(39\right)\end{array}$

(99) The pulse reverberation noise of the reconstructed HF receive signals hence has the same form as for the focused transmit and receive beams as in Eq. (26). By matching the number of transmit beam components and apodization weights that participates to the synthetic transmit beam to the aperture and apodization weights of the synthetic receive beam we can obtain
H.sub.r(r.sub.i,;r)=H.sub.t(r.sub.i,;r)i=1,3{circumflex over (Q)}(;r)=1Q(;r.sub.1,r.sub.3)=1(40)
in Eq. (29), and the difference between the Class I and II pulse reverberation noise is found in the nonlinear pulse form distortion produced by the LF pulse and the nonlinear self distortion attenuation of the fundamental HF band, produced by the HF pulse itself, both given by {tilde over (V)}.sub.p(r.sub.1,) and {tilde over (V)}.sub.p (r.sub.3,). The estimation schemes of Sections 5.7 to suppress pulse reverberation noise and estimate nonlinear scattering, are hence fully applicable to the reconstructed synthetic signals above.
C. Synthetic Dynamic Focusing with Transversal Filter

(100) We choose the z-axis along the beam axis with z=0 at the center of the array aperture. This gives a coordinate representation r=ze.sub.z+r.sub.=z+r.sub. where e.sub.z is the unit vector along the beam axis and r.sub. is the coordinate in the plane orthogonal to the z-axis, i.e. z.Math.r.sub.=0. The image is obtained by lateral scanning of the beam, where the 1.sup.st order scattered image signal can from Eq. (23) be modeled as a convolution in the transversal plane

(101) $\begin{array}{cc}{\mathrm{dY}}_{l1}\left(z_i,{\underset{\_}{r}}_{},\right)=-{\mathrm{dzk}}^2{V}_p\left(z_i,\right)U\left(\right){}_{Z_l}{d}^2{r}_1{H}_b\left(z,{\underset{\_}{r}}_1,\right)\left(z,{\underset{\_}{r}}_1\right)& \left(41\right)\end{array}$
where Z.sub.i is an interval around z.sub.i of scatterers that all interfere at z.sub.i, and
H.sub.b(z,r.sub.,)=H.sub.r(z,r.sub.,)H.sub.t(z,r.sub.,)(42)
is the composite beam spatial frequency response as defined in detail in Appendix B. We seek a whitening filter in the transversal coordinates with an impulse response w that minimizes the lateral point spread function for each depth z as