Spatial coherence of the nonlinearly generated second harmonic portion of backscatter for a clinical imaging system

Aberration correction in diagnostic ultrasound: A review of the prior field and current directions

Medical ultrasound images are reconstructed with simplifying assumptions on wave propagation, with one of the most prominent assumptions being that the imaging medium is composed of a constant sound speed. When the assumption of a constant sound speed are violated, which is true in most in vivoor clinical imaging scenarios, distortion of the transmitted and received ultrasound wavefronts appear and degrade the image quality. This distortion is known as aberration, and the techniques used to correct for the distortion are known as aberration correction techniques. Several models have been proposed to understand and correct for aberration. In this review paper, aberration and aberration correction are explored from the early models and correction techniques, including the near-field phase screen model and its associated correction techniques such as nearest-neighbor cross-correlation, to more recent models and correction techniques that incorporate spatially varying aberration and diffractive effects, such as models and techniques that rely on the estimation of the sound speed distribution in the imaging medium. In addition to historical models, future directions of ultrasound aberration correction are proposed.

Large-Array Deep Abdominal Imaging in Fundamental and Harmonic Mode

Deep abdominal images suffer from poor diffraction-limited lateral resolution. Extending the aperture size can improve resolution. However, phase distortion and clutter can limit the benefits of larger arrays. Previous studies have explored these effects using numerical simulations, multiple transducers, and mechanically swept arrays. In this work, we used an 8.8-cm linear array transducer to investigate the effects of aperture size when imaging through the abdominal wall. We acquired channel data in fundamental and harmonic modes using five aperture sizes. To avoid motion and increase the parameter sampling, we decoded the full-synthetic aperture data and retrospectively synthesized nine apertures (2.9-8.8 cm). We imaged a wire target and a phantom through ex vivo porcine abdominal samples and scanned the livers of 13 healthy subjects. We applied bulk sound speed correction to the wire target data. Although point resolution improved from 2.12 to 0.74 mm at 10.5 cm depth, contrast resolution often degraded with aperture size. In subjects, larger apertures resulted in an average maximum contrast degradation of 5.5 dB at 9-11 cm depth. However, larger apertures often led to visual detection of vascular targets unseen with conventional apertures. An average 3.7-dB contrast improvement over fundamental mode in subjects showed that the known benefits of tissue-harmonic imaging extend to larger arrays.

Adaptive Models for Multi-Covariate Imaging of Sub-Resolution Targets (MIST)

Multi-covariate imaging of sub-resolution targets (MIST) is a statistical, model-based image formation technique that smooths speckles and reduces clutter. MIST decomposes the measured covariance of the element signals into modeled contributions from mainlobe, sidelobes, and noise. MIST covariance models are derived from the well-known autocorrelation relationship between transmit apodization and backscatter covariance. During in vivo imaging, the effective transmit aperture often deviates from the applied apodization due to nonlinear propagation and wavefront aberration. Previously, the backscatter correlation length provided a first-order measure of these patient-specific effects. In this work, we generalize and extend this approach by developing data-adaptive covariance estimation, parameterization, and model-formation techniques. We performed MIST imaging using these adaptive models and evaluated the performance gains using 152 tissue-harmonic scans of fetal targets acquired from 15 healthy pregnant subjects. Compared to standard MIST imaging, the contrast-to-noise ratio (CNR) is improved by a median of 8.3%, and the speckle signal-to-noise ratio (SNR) is improved by a median of 9.7%. The median CNR and SNR gains over B-mode are improved from 29.4% to 40.4% and 24.7% to 38.3%, respectively. We present a versatile empirical function that can parameterize an arbitrary speckle covariance and estimate the effective coherent aperture size and higher order coherence loss. We studied the performance of the proposed methods as a function of input parameters. The implications of system-independent MIST implementation are discussed.

Point Detection in Ultrasound Using Prewhitening and Multilook Optimization

We investigate methods to improve the detection of point scatterers in ultrasound imaging using the standard delay-and-sum (DAS) image as our starting point. An optimized whitening transform can increase the spatial resolution of the image. By splitting an image's frequency spectrum into many subsets using the multilook technique, we can exploit the coherent properties of a point scatterer. We present three new multilook methods and evaluate their effect on point detection. The performances are compared to DAS using synthetic aperture Field II simulations of a point scatterer in uniform speckle background. The results show that optimized prewhitening of the images can significantly improve the point detection. The multilook methods have the potential to improve the detection performance further when a sufficient number of looks are used. If prior knowledge about the optimal spectrum limits is unavailable and a nonoptimal prewhitening is applied, applying that the new multilook methods can considerably improve the point detection.

Spatial Coherence in Medical Ultrasound: A Review

Traditional pulse-echo ultrasound imaging heavily relies on the discernment of signals based on their relative magnitudes but is limited in its ability to mitigate sources of image degradation, the most prevalent of which is acoustic clutter. Advances in computing power and data storage have made it possible for echo data to be alternatively analyzed through the lens of spatial coherence, a measure of the similarity of these signals received across an array. Spatial coherence is not currently explicitly calculated on diagnostic ultrasound scanners but a large number of studies indicate that it can be employed to describe image quality, to adaptively select system parameters and to improve imaging and target detection. With the additional insights provided by spatial coherence, it is poised to play a significant role in the future of medical ultrasound. This review details the theory of spatial coherence in pulse-echo ultrasound and key advances made over the last few decades since its introduction in the 1980s.

Synthetic Aperture Focusing for Multi-Covariate Imaging of Sub-Resolution Targets

Coherence-based imaging methods suffer from reduced image quality outside the depth of field for focused ultrasound transmissions. Synthetic aperture methods can extend the depth of field by coherently compounding time-delayed echo data from multiple transmit events. Recently, our group has presented the Multi-covariate Imaging of Sub-resolution Targets (MIST), an estimation-based method to image the statistical properties of diffuse targets. MIST has demonstrated improved image quality over conventional delay-and-sum, but like many coherence-based imaging methods, suffers from limited depth of field artifacts. This article applies synthetic aperture focusing to MIST, which is evaluated using focused, plane-wave, and diverging-wave transmit geometries. Synthetic aperture MIST is evaluated in simulation, phantom, and in vivo applications, demonstrating consistent improvements in contrast-to-noise ratio (CNR) over conventional dynamic receive MIST outside the transmit depth of field, with approximately equivalent results between synthetic transmit geometries. In vivo synthetic aperture MIST images demonstrated 16.8 dB and 16.6% improvements in contrast and CNR, respectively, over dynamic receive MIST images, as well as 17.4 dB and 32.3% improvements over synthetic aperture B-Mode. MIST performance is characterized in the space of plane-wave imaging, where the total plane-wave count is reduced through coarse angular sampling or total angular span. Simulation and experimental results indicate wide applicability of MIST to synthetic aperture imaging methods.

Extending Retrospective Encoding for Robust Recovery of the Multistatic Data Set

Robust recovery of multistatic synthetic aperture data from conventional ultrasound sequences can enable complete transmit-and-receive focusing at all points in the field of view without the drawbacks of virtual-source synthetic aperture and further enables more advanced imaging applications, such as backscatter coherence, sound speed estimation, and phase aberration correction. Recovery of the multistatic data set has previously been demonstrated on a steered transmit sequence for phased arrays using an adjoint-based method. We introduce two methods to improve the accuracy of the multistatic data set. We first modify the original technique used for steered transmit sequences by applying a ramp filter to compensate for the nonuniform frequency scaling introduced by the adjoint-based method. Then, we present a regularized inversion technique that allows additional aperture specification and is intended to work for both steered transmit and walking aperture sequences. The ramp-filtered adjoint and regularized inversion techniques, respectively, improve the correlation of the recovered signal with the ground truth from 0.9404 to 0.9774 and 0.9894 in steered transmit sequences and 0.4610 to 0.4733 and 0.9936 in walking aperture sequences.

Speckle coherence of piecewise-stationary stochastic targets

The van Cittert-Zernike (VCZ) theorem describes the propagation of spatial covariance from an incoherent source distribution, such as backscatter from stochastic targets in pulse-echo imaging. These stochastic targets are typically assumed statistically stationary and spatially incoherent with uniform scattering strength. In this work, the VCZ theorem is applied to a piecewise-stationary scattering model. Under this framework, the spatial covariance of the received echo data is demonstrated as the linear superposition of covariances from distinct spatial regions. This theory is analytically derived from fundamental physical principles, and validated through simulation studies demonstrating superposition and scaling. Simulations show that linearity is preserved over various depths and transmit apodizations, and in the presence of noise. These results provide a general framework to decompose spatial covariance into contributions from distinct regions of interest, which may be applied to advanced imaging methods. While the simulation tools used for validation are specific to ultrasound, this analysis is generally applicable to other coherent imaging applications involving stochastic targets. This covariance decomposition provides the physical basis for a recently described imaging method, Multi-covariate Imaging of Sub-resolution Targets.

Multi-covariate Imaging of Sub-resolution Targets

Conventional B-mode ultrasound imaging assumes that targets consist of collections of point scatterers. Diffraction, however, presents a fundamental limit on a scanner's ability to resolve individual scatterers in most clinical imaging environments. Well-known optics and ultrasound literature has characterized these diffuse scattering targets as spatially incoherent and statistically stationary. In this paper, we apply a piecewise-stationary statistical model to diffuse scattering targets, in which the covariance of backscattered echoes can be described as the linear superposition of constituent components corresponding to echoes from distinct spatial regions in the field. Using this framework, we present Multi-covariate Imaging of Sub-resolution Targets (MIST), a novel estimation-based method to image the statistical properties of diffuse scattering targets, based on a decomposition of aperture domain spatial covariance. The mathematical foundations of the estimator are analytically derived, and MIST is evaluated in phantom, simulation, and in vivo studies, demonstrating consistent improvements in contrast-to-noise ratio and speckle statistics across imaging targets, without an apparent loss in resolution.

Lag-One Coherence as a Metric for Ultrasonic Image Quality

Reliable assessment of image quality is an important but challenging task in complex imaging environments such as those encountered in vivo. To address this challenge, we propose a novel imaging metric, known as the lag-one coherence (LOC), which leverages the spatial coherence between nearest-neighbor array elements to provide a local measure of thermal and acoustic noise. In this paper, we derive the theory that relates LOC and the conventional image quality metrics of contrast and contrast-to-noise ratio (CNR) to channel noise. Simulation and phantom studies are performed to validate this theory and compare the variability of LOC to that of conventional metrics. We further evaluate the performance of LOC using matched measurements of contrast, CNR, and temporal correlation from in vivo liver images formed with varying mechanical index (MI) to assess the feasibility of adaptive acoustic output selection using LOC feedback. Simulation and phantom results reveal a lower variability in LOC relative to contrast and CNR over a wide range of clinically relevant noise levels. This improved stability is supported by in vivo measurements of LOC which show an increased monotonicity with changes in MI compared to matched measurements of contrast and CNR (88.6% and 85.7% of acquisitions, respectively). The sensitivity of LOC to stationary acoustic noise is evidenced by positive correlations between LOC and contrast ( ) and LOC and CNR ( ) at high acoustic output levels in the absence of thermal noise. Results indicate that LOC provides repeatable characterization of patient-specific trends in image quality, demonstrating feasibility in the selection of acoustic output using LOC and its application for in vivo image quality assessment.

Improved Sensitivity in Ultrasound Molecular Imaging With Coherence-Based Beamforming

Ultrasound molecular imaging (USMI) is accomplished by detecting microbubble (MB) contrast agents that have bound to specific biomarkers, and can be used for a variety of imaging applications, such as the early detection of cancer. USMI has been widely utilized in preclinical imaging in mice; however, USMI in humans can be challenging because of the low concentration of bound MBs and the signal degradation caused by the presence of heterogenous soft tissue between the transducer and the lesion. Short-lag spatial coherence (SLSC) beamforming has been proposed as a robust technique that is less affected by poor signal quality than standard delay-and-sum (DAS) beamforming. In this paper, USMI performance was assessed using contrast-enhanced ultrasound imaging combined with DAS (conventional CEUS) and with SLSC (SLSC-CEUS). Each method was characterized by flow channel phantom experiments. In a USMI-mimicking phantom, SLSC-CEUS was found to be more robust to high levels of additive thermal noise than DAS, with a 6dB SNR improvement when the thermal noise level was +6dB or higher. However, SLSC-CEUS was also found to be insensitive to increases in MB concentration, making it a poor choice for perfusion imaging. USMI performance was also measured in vivo using VEGFR2-targeted MBs in mice with subcutaneous human hepatocellular carcinoma tumors, with clinical imaging conditions mimicked using a porcine tissue layer between the tumor and the transducer. SLSC-CEUS improved the SNR in each of ten tumors by an average of 41%, corresponding to 3.0dB SNR. These results indicate that the SLSC beamformer is well-suited for USMI applications because of its high sensitivity and robust properties under challenging imaging conditions.

Efficient Strategies for Estimating the Spatial Coherence of Backscatter

The spatial coherence of ultrasound backscatter has been proposed to reduce clutter in medical imaging, to measure the anisotropy of the scattering source, and to improve the detection of blood flow. These techniques rely on correlation estimates that are obtained using computationally expensive strategies. In this paper, we assess the existing spatial coherence estimation methods and propose three computationally efficient modifications: a reduced kernel, a downsampled receive aperture, and the use of an ensemble correlation coefficient. The proposed methods are implemented in simulation and in vivo studies. Reducing the kernel to a single sample improved computational throughput and improved axial resolution. Downsampling the receive aperture was found to have negligible effect on estimator variance, and improved computational throughput by an order of magnitude for a downsample factor of 4. The ensemble correlation estimator demonstrated lower variance than the currently used average correlation. Combining the three methods, the throughput was improved 105-fold in simulation with a downsample factor of 4- and 20-fold in vivo with a downsample factor of 2.

Resolution and brightness characteristics of short-lag spatial coherence (SLSC) images

We previously described a novel beamforming method that images the spatial correlation of an echo wave field with demonstrated applications to clutter reduction in high-noise environments. In this paper, several characteristics of the resolution and brightness of short-lag spatial coherence (SLSC) images formed by this method are compared with B-mode images formed by conventional delay-and-sum beamforming methods. Point target widths were measured to estimate resolution, the autocorrelation of image texture was measured to estimate texture size, and the contrast (i.e., brightness ratio) of clinically relevant targets was assessed. SLSC images demonstrate improved resolution and contrast with increasing values of channel noise and clutter, whereas B-mode resolution was degraded in the presence of high noise (i.e., > -12 dB channel noise-to-signal ratios) and high clutter magnitudes (i.e., > -21 dB relative to point target magnitude). Lateral resolution in SLSC images was improved with increasing lag value, whereas axial resolution was degraded with increasing correlation kernel length. The texture size of SLSC images was smaller than that of matched B-mode images. Results demonstrate that the resolution and contrast of coherence-based images depend on a range of parameters, but are generally superior to those of matched B-mode images under challenging imaging conditions.

In vivo visualization of prostate brachytherapy seeds with photoacoustic imaging

We conducted a canine study to investigate the in vivo feasibility of photoacoustic imaging for intraoperative updates to brachytherapy treatment plans. A fiber coupled to a 1064-nm Nd:YAG laser was inserted into high-dose-rate brachytherapy needles, which diffused light spherically. These needles were inserted through the perineum into the prostate for interstitial light delivery and the resulting acoustic waves were detected with a transrectal ultrasound probe. Postoperative computed tomography images and ex vivo photoacoustic images confirmed seed locations. Limitations with insufficient light delivery were mitigated with short-lag spatial coherence (SLSC) beamforming, providing a 10-20 dB contrast improvement over delay-and-sum (DAS) beamforming for pulse energies ranging from 6.8 to 10.5 mJ with a fiber-seed distance as large as 9.5 mm. For the same distance and the same range of energy densities, signal-to-noise ratios (SNRs) were similar while the contrast-to-noise ratio (CNR) was higher in SLSC compared to DAS images. Challenges included visualization of signals associated with the interstitial fiber tip and acoustic reverberations between seeds separated by ≤ 2 mm. Results provide insights into the potential for clinical translation to humans.

Short-lag spatial coherence imaging of cardiac ultrasound data: initial clinical results

Short-lag spatial coherence (SLSC) imaging is a novel beamforming technique that reduces acoustic clutter in ultrasound images. A clinical study was conducted to investigate clutter reduction and endocardial border detection in cardiac SLSC images. Individual channel echo data were acquired from the left ventricle of 14 volunteers, after informed consent and institutional review board approval. Paired B-mode and SLSC images were created from these data. Contrast, contrast-to-noise, and signal-to-noise ratios were measured in paired images, and these metrics were improved with SLSC imaging in most cases. Three cardiology fellows rated the visibility of endocardial segments in randomly ordered B-mode and SLSC cine loops. SLSC imaging offered 22%-33% improvement (p < 0.05) in endocardial border visibility when B-mode image quality was poor (i.e., 80% or more of the endocardial segments could not be visualized by the three reviewers). The percentage of volunteers with poor-quality images was decreased from 21% to 7% with the SLSC beamformer. Results suggest that SLSC imaging has the potential to improve clinical cardiac assessments that are challenged by clutter.

Effects of dual apodization with cross-correlation on tissue harmonic and pulse inversion harmonic imaging in the presence of phase aberration

Dual apodization with cross-correlation (DAX) is a relatively new beamforming technique which can suppress side lobes and clutter to enhance ultrasound image contrast. However, previous studies have shown that with increasing aberrator strength, contrast enhancements with DAX diminish and DAX becomes more prone to image artifacts. In this paper, we propose integrating DAX with tissue harmonic imaging (THI) or pulse inversion harmonic imaging (PIHI) to overcome their shortcomings and achieve higher image contrast. Compared with conventional imaging, our experimental results showed that DAX with THI allows for synergistic enhancements of image contrast with improvements of more than 231% for a 5-mm pork aberrator and 703% for a 12-mm pork aberrator. With PIHI, improvements of 238% and 890% were observed for the two pork tissue samples. Our results suggest that the complementary contrast enhancement mechanism employed by the proposed method may be useful in improving imaging of technically difficult patients in clinics.

Short-lag spatial coherence beamforming of photoacoustic images for enhanced visualization of prostate brachytherapy seeds

Prostate brachytherapy, administered by implanting tiny radioactive seeds to treat prostate cancer, currently relies on transrectal ultrasound imaging for intraoperative visualization of the metallic seeds. Photoacoustic (PA) imaging has been suggested as a feasible alternative to ultrasound imaging due to its superior sensitivity to metal surrounded by tissue. However, PA images suffer from poor contrast when seeds are distant from the light source. We propose a transperineal light delivery method and investigate the application of a short-lag spatial coherence (SLSC) beamformer to enhance low-contrast photoacoustic signals that are distant from this type of light source. Performance is compared to a conventional delay-and-sum beamformer. A pure gelatin phantom was implanted with black ink-coated brachytherapy seeds and the mean contrast was improved by 3-25 dB with the SLSC beamformer for fiber-seed distances ranging 0.6-6.3 cm, when approximately 10% of the receive aperture elements were included in the short-lag sum. For fiber-seed distances greater than 3-4 cm, the mean contrast-to-noise ratio (CNR) was approximately doubled with the SLSC beamformer, while mean signal-to-noise ratios (SNR) were mostly similar with both beamformers. Lateral resolution was decreased by 2 mm, but improved with larger short-lag values at the expense of poorer CNR and SNR. Similar contrast and CNR improvements were achieved with an uncoated brachytherapy seed implanted in ex vivo tissue. Results indicate that the SLSC beamformer has potential to enhance the visualization of prostate brachytherapy seeds that are distant from the light source.

Short-lag spatial coherence of backscattered echoes: imaging characteristics

Conventional ultrasound images are formed by delay-and-sum beamforming of the backscattered echoes received by individual elements of the transducer aperture. Although the delay-and-sum beamformer is well suited for ultrasound image formation, it is corrupted by speckle noise and challenged by acoustic clutter and phase aberration. We propose an alternative method of imaging utilizing the short-lag spatial coherence (SLSC) of the backscattered echoes. Compared with matched B-mode images, SLSC images demonstrate superior SNR and contrast-to-noise ratio in simulated and experimental speckle-generating phantom targets, but are shown to be challenged by limited point target conspicuity. Matched B-mode and SLSC images of a human thyroid are presented. The challenges and opportunities of real-time implementation of SLSC imaging are discussed.

Transmit beams adapted to reverberation noise suppression using dual-frequency SURF imaging

A method that uses dual-frequency pulse complexes of widely separated frequency bands to suppress noise caused by multiple scattering or multiple reflections in medical ultrasound imaging is presented. The excitation pulse complexes are transmitted to generate a second order ultrasound field (SURF) imaging synthetic transmit beam. This beam has reduced amplitude near the transducer, which illustrates the multiple scattering suppression ability of the imaging method. Field simulations solving a nonlinear wave equation are used to calculate SURF imaging beams, which are compared with beams for pulse inversion (PI) and fundamental imaging. In addition, a combined SURF and PI beam generation is described and compared with the beams mentioned above. A quality ratio, relating the energy within the near-field to that within the imaging region, is defined and used to score the multiple scattering and multiple reflection suppression abilities when imaging with the different beams. The realized combined SURF-PI beam scores highest, followed by SURF, PI (that score equally well), and the fundamental. The amplitude in the imaging region and therefore also the SNR is highest for the fundamental followed by SURF, PI, and SURF-PI. The work hence indicates that when substituting PI for SURF, one may trade increased SNR into use of increased imaging frequencies without loss of multiple scattering and multiple reflection noise suppression.

Two-dimensional coherence for measurement of asymmetry in postural steadiness

Two-dimensional magnitude squared coherence (2D-MSC) is developed to compare the two-valued time series which represent the center of pressure (COP) under each foot. A sinusoidal multiple taper spectral estimator is used to reduce bias and improve spectral resolution. The measure is applied to evaluate symmetry in the dual-plate postural steadiness time series obtained from healthy young and elderly volunteers, and patients with Huntington's Disease (HD), a group in which asymmetries in postural steadiness are anticipated. The results demonstrate that the 2D-MSC is a robust measure of inter-limb coordination that may be of value in studies of aging and neurologic disease.

Aberration in nonlinear acoustic wave propagation

Theory and simulations are presented indicating that imaging at the second-harmonic frequency does not solve the problem of ultrasonic wave aberration. The nonlinearity of acoustic wave propagation in biological tissue is routinely exploited in medical imaging because the improved contrast resolution leads to better image quality in many applications. The major sources of acoustic noise in ultrasound images are aberration and multiple reflections between the transducer and tissue structures (reverberations), both of which are the result of spatial variations in the acoustic properties of the tissue. These variations mainly occur close to the body surface, i.e., the body wall. As a result, the nonlinearly generated, second harmonic is believed to alleviate both reverberation and aberration because it is assumed that the second harmonic is mainly generated after the body wall. However, in the case of aberration, the second harmonic is generated by an aberrated source. Thus the second harmonic experiences considerable aberration at all depths, originating from this source. The results in this paper show that the second harmonic experiences similar aberration as its generating source, the first harmonic.

Impact of propagation through an aberrating medium on the linear effective apodization of a nonlinearly generated second harmonic field

Techniques based on the nonlinearly generated second harmonic signal (tissue harmonic imaging) have rapidly supplanted linear (fundamental) imaging methods as the standard in two-dimensional echocardiography. Enhancements to the compactness of the nonlinearly generated second harmonic (2f) field component with respect to the fundamental (1f) field component are widely considered to be among the factors contributing to the observed image quality improvements. The objective of this study was to measure the impact of phase and amplitude aberrations resulting from propagation through an inhomogeneous tissue, on the beamwidths associated with: the fundamental (1f); the nonlinearly generated second harmonic (2f); and the linearly propagated, effective apodization signal at the same (21) frequency. Modifications to the transmit characteristics of a phased-array imaging system were validated with hydrophone measurements. Results demonstrate that the characteristics of the diffraction pattern associated with the linear-propagation effective apodization transmit case were found to be in good agreement with the detailed spatial characteristics of the nonlinearly generated second harmonic field. The effects of the abdominal wall tissue aberrators are apparent for all three of the beam profiles studied. Consistent with the improved image quality associated with harmonic imaging, the aberrated nonlinearly generated second harmonic beam was shown to remain more compact than the corresponding aberrated fundamental beam patterns in the presence of the interposed aberrator.

On the stability of the effective apodization of the nonlinearly generated second harmonic with respect to range

The concept of an effective apodization was introduced to describe the field pattern for the nonlinearly generated second harmonic (2f) within the focal zone using a linear propagation model. Our objective in this study was to investigate the validity of the concept of an effective apodization at 2f as an approach to approximating the field of the second harmonic over a wide range of depths. Two experimental setups were employed: a vascular imaging array with a water path and an adult cardiac imaging array with an attenuating liver path. In both cases the spatial dependencies of the ultrasonic fields were mapped by scanning a point-like hydrophone within a series of planes orthogonal to the propagation direction. The sampling distances were located before, within, and beyond the focal zone. The signals were Fourier transformed and the complex values at 2f were linearly backpropagated to the transmit plane in order to obtain an effective apodization. The measured results demonstrated a relatively constant effective apodization at 2f as a function of propagation distance. Finite amplitude computer simulations were found to be in agreement with these measurements. Thus the measure of the effective apodization at 2f provides an approximation to the second harmonic field outside the focal zone.