Page 65 - Summer 2021

as the sensor beam is scanned across the aggregation of scatterers, there will be some echoes in which the inter- ference is constructive and the echoes are relatively large. Other echoes will be small due to destructive interference. For a dense aggregation of scatterers, the echo histogram will tend to be distributed according to the Rayleigh PDF, which corresponds to the echo variability in the limit of a large number of randomly distributed scatterers. However, the histogram will deviate significantly from the Rayleigh distribution under a number of conditions, including for the case when the density of the scatterers is lower (Figure 4). Also, in some applications, all echo histograms are plot- ted on a log-log scale (whereas the histograms in Figure 4 are on a lin-lin scale). On a log-log scale, the “tail” of an echo histogram (that is, the portion of the histogram asso- ciated with the highest echo values), can readily be seen as significantly higher than that of the Rayleigh curve when the scatterer density is low.
There are broadly similar effects as illustrated in Figure 4 when there are changes in the beamwidth and pulse dura- tion of the sensor system. When either of those quantities becomes smaller, there are fewer scatterers causing an echo for a given transmission, resulting in deviations from the Rayleigh distribution. And once there are vari- abilities in the acoustic properties of the medium, such as changes in density and/or the presence of boundar- ies, more variability in the echo is introduced. Details of these effects for a wide variety of conditions are given in Stanton et al. (2018).
What Type of Information Can be Obtained from Echo Statistics?
Statistical analyses of echoes are especially useful in dis- criminating between one type of scatterer and another. In medical ultrasound, one can potentially discriminate between cancerous tissue and normal tissue. In defense applications, one can potentially distinguish between an enemy submarine and a school of fish. As part of these discrimination techniques, echo statistics can be used to identify types of scatterers, at least in terms of their gross physical features. For example, objects that are spherical will cause echoes that vary differently from objects that are elongated (Figure 3).
The numerical density of the objects can also be esti- mated from the echo variability. This is due to the fact that echoes from dense aggregations of scatterers, where
all echoes overlap, will have different statistics from those involving sparse aggregations (Figure 4). This is useful for a wide range of applications, including the assessment of fish stocks for the management of food resources and estimating the density of scatterers within tissue. Fur- thermore, if two different types of scatterers are known to occur with different numerical densities, then those differences can be used to discriminate between them.
Physics-Based Echo Statistics Versus Generic Approaches
As discussed, a model can be used to interpret the sta- tistics of echoes for classification (Figure 2). There is a wide range of models that can be divided into two broad categories: (1) physics-based models and (2) generic sta- tistical functions. With the physics-based models, the scattering process that gives rise to the echo is formu- lated from fundamental physical principles (e.g., Eqs. 1 and 2) (Stanton et al., 2018). Through randomization of various parameters associated with the scatterer and the environment, parameters of the echo variability can be expressed explicitly in terms of those physical quantities and of the sensor system itself.
When generic statistical functions are used, a function that is generally not derived from physical principles is fit to experimental data and the echoes are classified in terms of the best fit parameters of that distribution. Classifica- tion is commonly verified by comparisons with controlled experimental (that is, empirically based) data rather than through physics-based modeling. There are many different types of functions used in this approach, including varia- tions of the K-distribution (Jakeman and Ridley, 2006; Destrempes and Cloutier, 2010). The K-distribution is based on a Rayleigh PDF but with a randomized mean square that is used to account for the echoes from a non- uniform spatial distribution of scatterers.
There are both advantages and disadvantages to each of the above two approaches. Because the physics- based approach is explicitly connected to the physical parameters of the object, quantitative information such as numerical density can potentially be extracted from the data. However, the analysis requires detailed knowl- edge of the environment and other physical information concerning the object, such as its shape. Conversely, if the shape is of interest, then that can be inferred if other information, such as the numerical density is
Summer 2021 • Acoustics Today 65