Causality-Imposed (Kramers-Kronig) Relationships Between Attenuation and Dispersion

Abstract

Causality imposes restrictions on both the time-domain and frequency-domain responses of a system. The Kramers-Kronig (K-K) relations relate the real and imaginary parts of the frequency-domain response. In ultrasonics, K-K relations often are used to link attenuation and dispersion. We review both integral and differential forms of the frequency-domain K-K relations that are relevant to theoretical models and laboratory measurements. We consider two methods for implementing integral K-K relations for the case of finite-bandwidth data, namely, extrapolation of data and restriction of integration limits. For the latter approach, we discuss the accuracy of K-K predictions for specific classes of system behavior and how the truncation of the integrals affects this accuracy. We demonstrate the accurate prediction of attenuation and dispersion using several forms of the K-K relations relevant to experimental measurements of media with attenuation coefficients obeying a frequency power law and media consisting of resonant scatterers. We also review the time-causal relations that describe the time-domain consequences of causality in the wave equation. These relations can be thought of as time-domain analogs of the (frequency-domain) K-K relations. Causality-imposed relations, such as the K-K and time-causal relations, provide useful tools for the analysis of measurements and models of acoustic systems.

I. Introduction

Dispersion refers to the dependence of phase velocity on frequency, which can lead to the change in shape of a time-localized pulse as it propagates in a dispersive medium. The idea that phase velocity could exhibit a frequency dependence was appreciated early in the solutions of vibrations of spring-coupled point masses [1], [2]. The fact that attenuation and dispersion were related was recognized at least by the 1870s. In the 1920s, Kronig [3] and Kramers [4] showed that, as a consequence of causality and linearity, the real part of the electromagnetic index of refraction (i.e., $c_0/c_p(\omega )$, the ratio of the speed of light to phase velocity) could be related to the Hilbert transform of the imaginary part. That is, the dispersion and loss in a medium are not independent of one another. The term Kramers-Kronig relations now refers to the relations between physical quantities that are fundamentally expressed through Hilbert transforms. In the simplest case, the two quantities may be taken as the real and imaginary parts of a complex function in which each of the pair may be expressed as an integral over the other. Expressions (1) and (2) show a pair of Kramers-Kronig (K-K) relations that relate the real and imaginary parts of some complex function $G(\omega )$:

$$\mathrm{R}\mathrm{e}G(\omega )=\frac{2}{\pi }𝒫{\int }_0^{\mathrm{\infty }}\frac{{\omega }^{\prime }\mathrm{I}\mathrm{m}G\left({\omega }^{\prime }\right)}{{\omega }^{\prime 2}-{\omega }^2}d{\omega }^{\prime },\text{and}$$ (1)

$$\mathrm{I}\mathrm{m}G(\omega )=-\frac{2\omega }{\pi }𝒫{\int }_0^{\mathrm{\infty }}\frac{\mathrm{R}\mathrm{e}G\left({\omega }^{\prime }\right)}{{\omega }^{\prime 2}-{\omega }^2}d{\omega }^{\prime }.$$ (2)

Here the integrals are taken over all positive frequencies ${\omega }^{\prime }$, and $𝒫$ indicates that the Cauchy principal value sense of the integral is evaluated [5]. One key observation is that the determination of $\mathrm{R}\mathrm{e}G(\omega )$ (or $\mathrm{I}\mathrm{m}G(\omega )$) at any particular frequency requires knowledge of $\mathrm{I}\mathrm{m}G(\omega )$ (or $\mathrm{R}\mathrm{e}G(\omega )$) at all frequencies.

Since their introduction in the 1920s, the K-K relations have played a role in the study of virtually all forms of wave propagation, including electromagnetics and optics [6], particle physics [7], electronics [8], quantum mechanics [9], nuclear magnetic resonance [10], and acoustics [11], [12], which is of particular interest here. One common application of the acoustic K-K relations has been the investigation of the relationship between the frequency dependence of the real and imaginary parts of the complex wave number $(\kappa (\omega )=\omega /c(\omega )+i\alpha (\omega ))$ that is expressible in terms of the phase velocity $c(\omega )$ and attenuation coefficient $\alpha (\omega )$, respectively.

We continue in Section II with a brief review of the origin and development of the K-K relations. We then survey various forms of the K-K dispersion relations in Section III, and consider applications to ultrasonic propagation and scattering in Section IV. In Section V we extend our review of the consequences of causality by considering a time-domain representation of the K-K relations known as the time-causal relations. Then we consider how recent applications of the K-K relations may play a role in future ultrasonic research.

II. A Brief History of the Kramers-Kronig Relations

The K-K relations are based on the notions of linearity and causality, suggesting that they are general and independent of the physical details such as the scattering or propagation mechanism. The work of Kramers [4] showed that the existence of electromagnetic dispersion relations alone implies that no signal can propagate in a medium faster than the vacuum speed of light. Eventually, the general causal basis for the dispersion relations was appreciated [13].

The work of Kronig [3] and Kramers [4] concerned the absorption of light scattered by atoms. They were able to relate the absorption of light to the index of refraction as a consequence of causal propagation in a medium, essentially rediscovering the optical theorem [14]–[16]. However, the fact that the relations were a consequence of causality permitted their application to a host of physical problems. Consequently, the K-K relations have since become a check on the causal consistency of a dispersion model or measurement.

Various forms of the K-K dispersion relations can be found in many areas of physics and engineering. The first use of the K-K relations for the scattering of quantum-mechanical particles was developed independently by Feenberg [9] in the early 1930s. Bode [8] developed a relation between the frequency-dependent gain of an amplifier and the corresponding phase shift, which later proved useful for the development of an approximation to the ultrasonic K-K relations [17]. Significant contributions to the study of dispersion relations also have come from the particle-physics community [7], [13], [18]. Researchers in high-energy physics have used a differential form of the K-K dispersion relations [19], [20], in addition to the integral forms. There also has been application of the relations in elastic theory [21]. Furthermore, we find the K-K transforms are central to the widely used experimental technique of determining the complex refractive index from reflection measurements [22], [23].

Acoustical forms of the K-K relations do not appear to have been proposed until the 1950s [11] with verification following in the 1960s [24], [25] and 1970s [17], [26]. This timing may be related to the availability of techniques for accurate measurement of attenuation and dispersion. Prior to the 1940s, ultrasonic dispersion in liquids had not yet been established, with the exception of work by the Soviet physicist Shpakovskii [27], who had reported dispersion in an acetic acid in 1938. However, this would not have been known to the Western world, as has been noted elsewhere [28]. It was only after World War II that pulse-measurement techniques adapted from the radar community provided acousticians with the techniques necessary to perform accurate ultrasonic measurements of attenuation and dispersion in liquids. Consequently, the K-K dispersion relations were not applied to acoustical problems until 30 years after their introduction by Kronig and Kramers and their application in a number of other fields involving wave-based physics.

III. Forms of the Acoustic Kramers-Kronig Dispersion Relations

The Soviet physicist Ginzberg [11] appears to be the first to publish a derivation of the acoustic K-K relations between phase velocity and attenuation coefficient. In the following decades, a variety of researchers extended the analysis to areas such as geophysical acoustics [24], [25] and underwater acoustics [26]. In the late 1970s, a differential approximation to the K-K dispersion relations was introduced [17], [29], which has been of use for materials with attenuation that is linear with frequency. The acoustic K-K relations have since matured, and they are applicable in a wide variety of theoretical and experimental contexts. This includes measurements of soft tissue [29], liquids [30], [31], and suspensions [32]–[34], models and measurements for sediments [26], [35], measurements of polymer and composite media [31], [36]–[39], and other solid materials [40]–[42]. Furthermore, several forms of the K-K relations have been applied, both analytical [29]–[31] and numerical [32], [33], [43], including integral and differential forms. New mathematical forms and methods of implementation for acoustic K-K continue to be developed in an effort to extend their utility to a wider class of problems. We now consider various explicit forms of the K-K relations, and examine their accuracy and methods of implementation when applied to a diversity of data.

A. Integral Relations

One of the pair of standard K-K relations expresses the real part of a complex function in terms of an infinite integral involving the imaginary part of the complex function, as shown in (1). It is possible to use such an equation for examining theoretical models in which the form of the system response is given or known for all frequencies. In contrast, the application of such K-K relations to experimentally measured properties immediately presents the issue of how one handles finite-bandwidth data using equations that assume infinite bandwidth. This has led to the use of possible alternatives, including the extrapolation of measured data outside the measured bandwidth [30] and finite-integral forms of the K-K relations for which the impact of the truncation of the infinite integrals has been modeled [32], [33], [43].

1. Exact Forms:

Generally two issues arise when evaluating the K-K integrals, namely the handling of singularities in the K-K integrand, and the convergence of the integral. Expressions (1) and (2) indicate that the Cauchy principal value sense of the K-K integrals is evaluated, so as to emphasize how one evaluates the integrand at the singularity ${\omega }^{\prime }=\omega$. This singularity can be eliminated in the limit of ${\omega }^{\prime }\to \omega$ by exploiting the fact that the integral over the K-K kernel is zero:

$$𝒫{\int }_0^{\mathrm{\infty }}\frac{d{\omega }^{\prime }}{{\omega }^{\prime 2}-{\omega }^2}=0(\omega \ne 0)$$ (3)

This permits the use of subtraction constants in the numerator of the K-K integrand. One then often subtracts the value of the numerator at the frequency of evaluation $\omega$.

A second issue of concern is the convergence of the K-K integral. In order to ensure convergence, one may use a subtracted form of the complex function. Here we consider the complex wave number $\kappa (\omega )$. The general idea is to express the wave number as the first $n$ terms of its Taylor expansion plus a remainder term:

$$\begin{gathered} \kappa (\omega )={\left.\sum _{m=0}^{n-1}\frac{1}{m!}{\left(\omega -{\omega }_0\right)}^m\frac{d^m}{d{\omega }^m}\kappa (\omega )\right|}_{\omega ={\omega }_0} \\ +\frac{1}{n!}{\left(\omega -{\omega }_0\right)}^n{D}_n\left(\omega ,{\omega }_0\right) \end{gathered}$$ (4)

The quantity $D_n\left(\omega ,{\omega }_0\right)$ from the remainder term is referred to as the $n^{\text{th}}$-order subtracted function. Instead of developing K-K relations for the complex wave number, one forms K-K relations for the subtracted function $D_n\left(\omega ,{\omega }_0\right)$ that then can be used to relate the real and imaginary parts of the wave number. Terms of the Taylorseries expansion of the complex wave number are subtracted then normalized by the leading behavior of the remainder term (i.e., ${\left(\omega -{\omega }_0\right)}^n$). An advantage of using a subtracted form of the wave number is that we improve the convergence of the integral. However, this comes at the expense of having to specify values of the wave number and its derivatives at a frequency ${\omega }_0$, known as the subtraction frequency. The integral K-K relations for the second-order subtracted function $D_2(\omega ,0)$ are shown in Table I. We emphasize that these relationships between the attenuation coefficient $\alpha (\omega )$ and phase velocity $c(\omega )$ exist regardless of their corresponding physical mechanism.

TABLE I.

Integral and Differential Forms of the Kramers-Kronig Dispersion Relations with Two Subtractions between the Attenuation Coefficient and Phase Velocity.

Direction	Integral	Differential
$c\to \alpha$	$\alpha (\omega )=-\frac{2}{\pi }{\omega }^2{\int }_0^{\mathrm{\infty }}\left[\frac{1}{c\left({\omega }^{\prime }\right)}-\frac{1}{c(\omega )}\right]\frac{d{\omega }^{\prime }}{{\omega }^{\prime 2}-{\omega }^2}$	$\alpha (\omega )=-\omega \mathrm{t}\mathrm{a}\mathrm{n}\left(\frac{\pi }{2}\frac{d}{d\mathrm{l}\mathrm{n}\omega }\right)\frac{1}{c(\omega )}$
$\alpha \to c$	$\frac{1}{c(\omega )}-\frac{1}{c_0}=+\frac{2}{\pi }{\omega }^2{\int }_0^{\mathrm{\infty }}\left[\frac{\alpha \left({\omega }^{\prime }\right)}{{\omega }^{\prime 2}}-\frac{\alpha (\omega )}{{\omega }^2}\right]\frac{d{\omega }^{\prime }}{{\omega }^{\prime 2}-{\omega }^2}$	$\frac{1}{c(\omega )}-\frac{1}{c_0}=+\omega \mathrm{t}\mathrm{a}\mathrm{n}\left(\frac{\pi }{2}\frac{d}{d\mathrm{l}\mathrm{n}\omega }\right)\frac{\alpha (\omega )}{{\omega }^2}$

2. Approximate Forms:

The principal difficulty in applying K-K relations directly to ultrasonic data is the finite bandwidth inherent in experimental measurements. Performing K-K analysis when a model or measurement of system response is known over only a finite bandwidth is often of interest. The notion that knowledge of the material properties over only a finite bandwidth may be sufficient for K-K analysis is based in part on the observation that the K-K kernel preferentially weights the portion of the integrand near ${\omega }^{\prime }\approx \omega$ [8], [29]. This turns out to be straightforward for systems with resonant behavior. Most of the data necessary to causally construct an attenuation peak from a dispersion peak/nadir pair or vice versa are from the bandwidth containing those features because these structures always appear in tandem. Regardless of the details, the ability to evaluate the causal consistency of a given measurement or model using only a finite bandwidth is attractive in that it avoids sometimes cumbersome infinite integrals, and it does not explicitly require information outside the experimentally available bandwidth. However, restricting the range of integration to the band limits of the measurement spectrum introduces artifacts that can seriously impact the accuracy of K-K transformations.

It is still an open question as to whether an accurate general procedure exists for applying K-K relations directly to any type of dispersion or attenuation data, or whether some information about the target system beyond the band-limited attenuation and dispersion data is required to suppress the artifacts. For data with resonant features in its dispersion and attenuation, it has been shown that these artifacts can be minimized sufficiently that accurate transformations can be performed with finite-interval relations.

Consider the case in which the phase velocity $c(\omega )$ is known over some finite bandwidth $\left[{\omega }_l,{\omega }_h\right]$, and we want to determine the attenuation coefficient $\alpha (\omega )$. We can write the finite-bandwidth form of the K-K relation with one subtraction for the attenuation coefficient as shown in (5) (see next page) [32], [33], where F indicates that the K-K estimate of attenuation is over a finite bandwidth and ${\omega }_0$ is the subtraction frequency. Here we have developed the K-K relations for $D_1\left(\omega ,{\omega }_0\right)$ of (4).

In general, artifacts will be introduced into our estimate of the attenuation by the truncation of the K-K integral. We can consider the finite-bandwidth K-K estimate of attenuation as being composed of the true attenuation modified by a multiplicative artifact term $\beta$ and an additive artifact term $\eta$, both of which in principle depend upon the bandwidth and a subtraction frequency as well as physical details of the system (e.g., resonance width and position):

$$\begin{array}{l}{\alpha }_{K-K}^F\left(\omega ,{\omega }_l,{\omega }_h\right)={\alpha }_{\mathit{true}}(\omega )\left(1-\beta \left(\omega ,{\omega }_0,{\omega }_l,{\omega }_h,\dots \right)\right)\\ -\eta \left(\omega ,{\omega }_0,{\omega }_l,{\omega }_h,\dots \right),\end{array}$$ (6)

where ${\alpha }_{\mathit{true}}$ refers to either the modeled or measured attenuation coefficient that we are trying to predict. It is possible to gain a measure of the artifacts if one has an analytical model of the system [43]. Furthermore, artifacts can be minimized by appropriate choice of a subtraction constant [32], [33]. An analogous approach is available for calculating dispersion from attenuation.

B. Differential Relations

Although it is not readily evident, it is possible to rewrite the K-K integral relations in terms of a differential expression [17], [29]. These have sometimes been referred to as derivative analyticity relations [19] or differential dispersion relations [31], [44]. As we discuss in more detail below, the differential dispersion relations are applicable to media with an attenuation coefficient obeying a frequency power law. For such a case, a closed-form solution is available. However, it remains a topic of ongoing research as to whether these differential dispersion relations are generally applicable to other forms of attenuation, such as a resonant loss.

$$\begin{array}{ll}{\alpha }_{K-K}^F\left(\omega ,{\omega }_l,{\omega }_h,{\omega }_0\right)& =\left(\frac{1}{\pi }{\int }_{{\omega }_l}^{{\omega }_h}\frac{{\omega }^{\prime }/c\left({\omega }^{\prime }\right)-\omega /c(\omega )}{{\omega }^{\prime }-\omega }d{\omega }^{\prime }+\frac{1}{\pi }{\int }_{{\omega }_l}^{{\omega }_h}\frac{{\omega }^{\prime }/c\left({\omega }^{\prime }\right)+\omega /c(\omega )}{{\omega }^{\prime }+\omega }d{\omega }^{\prime }\right)\\ & +\left(\alpha \left({\omega }_0\right)-\frac{1}{\pi }{\int }_{{\omega }_l}^{{\omega }_h}\frac{{\omega }^{\prime }/c\left({\omega }^{\prime }\right)-{\omega }_0/c\left({\omega }_0\right)}{{\omega }^{\prime }-{\omega }_0}d{\omega }^{\prime }-\frac{1}{\pi }{\int }_{{\omega }_l}^{{\omega }_h}\frac{{\omega }^{\prime }/c\left({\omega }^{\prime }\right)+{\omega }_0/c\left({\omega }_0\right)}{{\omega }^{\prime }+{\omega }_0}d{\omega }^{\prime }\right).\end{array}$$ (5)

1. Exact Forms:

Following a number of mathematical manipulations [31], [45], one can transform the K-K integral over all positive frequencies into a differential operator in which a derivative is the argument of a tangent function. A pair of differential dispersion relations that correspond to the second-order subtracted function $D_2(\omega ,0)$ is shown in Table I. These particular relations were derived by treating the complex wave number as a tempered distribution [31]. Tempered distributions are a type of generalized functions [46], [47] that permit one to relax mathematical restrictions on the growth behavior of the ultrasonic properties at high frequencies. In the context of the K-K relations, the use of tempered distributions leads to a generalized form of the K-K relations, which shares some features with subtracted forms of the K-K relations but without explicit need of subtraction constants. Further discussion of distributions is included in Section IV-C. These distributions also are used in the development of the time-causal relations, as discussed in Section V.

2. Approximate Forms:

At first glance, the evaluation of a derivative appears simpler than that of an infinite integral. However, the presence of a derivative as an argument of a tangent function presents a difficulty. If the function of interest can be expressed as a power law or power series, a closed-form solution is available [31]. However, this generally is not the case. When a closed-form solution is not available, one may consider an approximation to the exact differential dispersion relations by expanding the tangent function in a power series.

The original nearly local approximation [17]:

$$\alpha (\omega )-\alpha (0)\approx -\frac{\pi }{2}{\omega }^2\frac{d}{d\omega }\left(\frac{1}{c(\omega )}\right),$$ (7)

is now known to be recoverable from the exact differential dispersion relation (e.g., see the $c\to \alpha$ form of Table I) as the lowest-order term when the tangent term is expanded as a power series. This approximation is often integrated to form an expression for the dispersion in terms of a finite integral of the attenuation over the experimental bandwidth. Consequently, the approximation is referred to as nearly local in contrast to the nonlocal aspect of the infinite integrals of the standard K-K relations. Most successful applications of the nearly local approximation given by (7) have been for cases in which the attenuation grows roughly proportional to frequency. Similar nearly local approximations can be developed for media with attenuation of more rapid growth (e.g., ${\omega }^y$, where $y>1$) if one considers a subtracted form of the complex wave number [31].

IV. Applications of the Acoustic Kramers-Kronig Dispersion Relations

We investigate two cases often considered in ultrasonic research: media with an attenuation coefficient obeying a frequency power law, and suspensions with resonant scattering properties. The first case of power-law attenuation is often considered for propagation in many soft tissues [29], [48], [49] and liquids [26], [50]. The resonant case encompasses a variety of physical environments, including ultrasonic contrast agents [32], [33], [51] and other gas-liquid [33], [52] or solid-liquid [53], [54] suspensions.

A. Power-Law Attenuation

The classical mechanism for sound absorption in liquids is shear viscosity resulting from some relaxation process, as has been described in several texts discussing ultrasonic propagation in liquids [55], [56]. Absorptions in excess of the classical mechanism also have been introduced based on empirical evidence in which excess absorptions often are couched in terms of a bulk or volume viscosity due to one of a variety of relaxation processes. A significant feature of the classical and excess sound-absorption mechanisms is that they produce an attenuation proportional to the square of the ultrasonic frequency, assuming that the ultrasonic frequency is well below the relaxation frequency of the liquid.

The ${\omega }^2$-dependence of the classical attenuation assumes a frequency-independent viscosity coefficient [57]. However, it often is observed that the attenuation in many liquids (and other media) does not exhibit an ${\omega }^2$-dependence. In such cases, the attenuation coefficient often is found to be well described by a power law:

$$\alpha (\omega )={\alpha }_0{\omega }^y,$$ (8)

where $\omega$ is angular frequency, and ${\alpha }_0$ and $y$ are materialdependent parameters with $y$ typically between 1 and 2, inclusive.

Two approaches have been investigated when performing K-K analysis on measurements of such media. The first approach extrapolates the measured ultrasonic properties beyond the experimental bandwidth, and it applies the exact integral or differential forms of the K-K dispersion relations shown in Table I. The corresponding dispersions are shown in Table II. The second approach limits the K-K analysis to only the available bandwidth, and it minimizes artifacts introduced by the truncation of the K-K integrals, as shown in (5).

TABLE II.

Dispersion Relations for Media with Attenuation Obeying a Frequency Power $\alpha (\omega )={\alpha }_0{\omega }^y$, where $0\le y\le 2$.

Power Law	Dispersion
$y=1$	$\frac{1}{c\left(\omega \right)}-\frac{1}{c\left({\omega }_0\right)}=-\frac{2}{\pi }{\alpha }_0\ln \left|\frac{\omega }{{\omega }_0}\right|$
$y\ne 1$	$\frac{1}{c\left(\omega \right)}-\frac{1}{c\left({\omega }_0\right)}={\alpha }_0\tan \left(\frac{\pi }{2}y\right)\left({\omega }^{y-1}-{\omega }_0^{y-1}\right)$

1. Extrapolation Approach:

In Figs. 1 and 2 we consider experimental measurements [30] of the attenuation coefficient and phase velocity for two values of the power-law exponent that have led to some discussion in the literature: a poly(methyl methacrylate) polymer with $y=1$, and a silicone oil with $y\cong 2$. We compare the measured phase velocity to the corresponding infinite integral K-K prediction, which uses a fit of the attenuation model of (8) to the experimentally measured attenuation coefficient. Here, we assume that the power-law form of the attenuation coefficient is valid outside the measured bandwidth. We find good agreement between the measured and predicted dispersions, indicating that the power-law model for the attenuation appears to be appropriate over an extended bandwidth. The K-K predictions also have been verified for intermediate values of $y$ [31].

Fig. 1.

Measurements of (a) attenuation coefficient and (b) phase velocity over a bandwidth of 1 to 16 MHz with standard deviation bars for a poly(methyl methacrylate) polymer. The phase velocity predicted using the K-K relation with one subtraction also is shown. (These results were originally published in Waters et al. [31].)

Fig. 2.

Measurements of (a) attenuation coefficient and (b) phase velocity over a bandwidth of 2 to 19 MHz with standard deviation bars for a silicone fluid. The phase velocity predicted using the K-K relation with two subtractions also is shown. (These measurements were originally published in Waters et al. [30].)

2. Finite-Bandwidth Approach:

We also consider a finite-bandwidth K-K relation with one subtraction [43] applied in a manner suited to predicting power-law exponents for the attenuation coefficient. Because this approach was developed for the analysis of data with localized resonant structures, it is not immediately clear whether this is applicable to the power-law problem. As shown in (6), finite-bandwidth predictions for $\alpha$ display two types of artifacts: a multiplicative factor $\beta$ and an additive factor $\eta$. In the case of power-law attenuation, we have found that the artifact is purely additive (i.e., $\beta =0$). However, the artifact can be quite large and, consequently, limits the usefulness of this approach for the power-law problem.

To illustrate this limitation, we focus our attention on one aspect of the problem, namely, the prediction of the power-law exponent $y$. Using a finite-integral version of the dispersion relation from Table I $(c\to \alpha )$ [43], we examine the prediction of the power-law exponent at different frequency scales over bandwidths that cover several orders of magnitude (e.g., ${\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left({\omega }_h/{\omega }_l\right)>3$). The frequency scales of interest are described by the frequency $\omega$, in which we assume $\omega \gg {\omega }_0$ (subtraction frequency) and $\omega \gg {\omega }_l$ (lower limit of integration). Consequently, the finite-bandwidth estimate ${\alpha }_{K-K}^F$ of the attenuation coefficient is independent of ${\omega }_0$ and ${\omega }_l$. The objective is to identify regions of the spectrum in which there is agreement between the modeled and predicted power-law exponents expressed in terms of the frequency $\omega$ and upper limit of the bandwidth ${\omega }_h$.

Fig. 3 compares ${\alpha }_{K-K}^F\left(\omega ,{\omega }_h\right)$ and ${\alpha }_{\mathit{true}}(\omega )$ on a log-log plot covering three decades in frequency up to the upper limit of integration ${\omega }_h$ for the case $y=1$. We find that the prediction agrees well with the model for $\omega /{\omega }_h<0.05$.Furthermore, to match the proper power law to within 1% in this case using the finite-bandwidth K-K relations requires knowledge of the dispersion for 1.5 orders of magnitude above the frequency scale of interest. As the exponent grows from 1 toward 2, the bandwidth requirements become even more demanding, as discussed in [43].

Fig. 3.

Comparison of attenuation coefficient model $(\alpha (\omega )=\omega )$ and K-K prediction. Results are shown on a log-log scale. The prediction used a finite-bandwidth K-K relation with one subtraction over three decades in frequency. (This figure was originally published in Mobley et al. [43].)

For media with attenuation obeying a frequency power law, the variations in the acoustic properties are globally monotonic (i.e., no local extrema). We observe that predicting the power-law exponent with reasonable accuracy using the finite-bandwidth approach requires very wide bandwidths. When predicting attenuation from dispersion, the extrapolation approach has proven to be the best choice for performing accurate K-K transforms for such media. However, recent work suggests that the prediction of dispersion from power-law attenuation may be performed accurately for bandwidths well within spectral bounds commonly obtained in the laboratory. Quantifying the relationship between band-limited attenuation data and the accuracy of the K-K dispersion calculations is currently under investigation.

3. Comments:

Media with a power-law exponent $y=1$, as in Figs. 1 and 3, have led to some theoretical concern. These media are sometimes referred to as having a hysteretic loss, with this type of phenomenological loss mechanism having been cited for polymeric materials [58] as well as biological media [59]. The general concern is that hysteretic systems are acausal based on general mathematical restrictions to causal systems [60]. However, these systems can be analyzed by the K-K relations as seen above, if one considers subtracted forms of the relations. The equivalence of causality and the K-K dispersion relations supports the observation that hysteretic-loss systems are indeed causal.

For the case of silicone fluid with $y\cong 2$ in Fig. 2, we observe an essentially dispersionless system. The small discrepancy (~1 m/s) between the measured and predicted phase velocities of the silicone fluid near the high end of the experimental bandwidth may be due to proximity of the relaxation frequency of the silicone fluid, as has been observed in a similar fluid with a relaxation frequency of 40 MHz [61]. Classical fluids are modeled as exhibiting an attenuation with ${\omega }^2$-dependence and a frequency-independent phase velocity, with the assumption that the ultrasonic frequencies of interest are well below the relaxation frequency of the fluid. From the viewpoint of the K-K analysis, however, a frequency-dependent attenuation generally corresponds to a frequency-dependent phase velocity. The case of ${\omega }^2$-attenuation is a special case for which a frequency-dependent attenuation does not correspond to frequency-dependent phase velocity, as appears to have been originally predicted as a consequence of causality by Horton [26]. Nevertheless, the point may be moot in that a power-law attenuation is only an approximation to a medium with a relaxation loss, albeit in many cases a very good approximation.

B. Resonant Scattering

We next consider two suspensions that exhibit resonant scattering behavior: encapsulated microbubbles and solid polymer microspheres. The first suspension consists of a 0.1% solution of air microbubbles that are stabilized by encapsulating shells of the denatured protein albumin [62]. The majority of the microbubble volume comes from shells with radii from 2 to 5 μm. These suspensions behave like a band-reject filter for ultrasound and have an attenuation peak near 2 MHz, in which the encapsulated microbubbles act as damped resonators. The resonant mode corresponds to a uniform radial (monopolar) oscillation of the encapsulated bubble. The second suspension consists of solid polymer microspheres narrowly distributed in size with a mean diameter near 80 μm [51]. They are suspended in saline with some added surfactant. In this case, the resonances arise from interface modes propagating around the microsphere/liquid boundary.

In Figs. 4 and 5, we compare experimental measurements and K-K predictions of the attenuation coefficient and phase velocity using a finite-bandwidth form of the K-K relations with two subtractions (i.e., using $D_2\left(\omega ,{\omega }_0\right)$). Only experimentally determined values of the attenuation and dispersion are used in the calculations. For the encapsulated microbubble suspension in Fig. 4, the bandwidth is restricted from 1 to 15 MHz, and the subtraction frequency ${\omega }_0$ is 8.6 MHz for calculation of the attenuation coefficient, and 3.6 MHz for calculation of the phase velocity. For the solid microsphere suspension in Fig. 5, the bandwidth is 5 to 22 MHz. For attenuation coefficient calculations, the subtraction frequency ${\omega }_0$ is 5.55 MHz, and for the phase velocity predictions ${\omega }_0$ is 11.11 MHz. The proper choice of the subtraction frequency ${\omega }_0$ is critical in order to achieve reasonable agreement between the experimental and predicted values. As shown in previous work [32], [33], one can treat ${\omega }_0$ as a fitting parameter to minimize the artifact errors due to the restriction of the K-K integral to the available bandwidth. Good agreement is found between the experimentally measured and K-K predicted attenuation. A modest discrepancy between the measured and predicted dispersions at the high end of the bandwidth occurs in both types of suspensions. However, the total dispersion (difference between the maximum and minimum phase velocity) across the bandwidth is consistent between the measured and predicted values.

Fig. 4.

Comparison of experimentally measured and K-K predicted (a) attenuation coefficient and (b) dispersion of an encapsulated bubble suspension (0.1% concentration). The K-K predictions use a twice-subtracted relation over a bandwidth of 1 to 15 MHz and subtraction frequencies of 8.6 MHz and 3.6 MHz for the attenuation coefficient and dispersion, respectively. (This figure was originally published in Mobley et al. [32].)

Fig. 5.

Comparison of experimentally measured and K-K predicted (a) attenuation coefficient and (b) dispersion of a suspension of solid polymer microspheres (80-μm diameter). The K-K predictions use a twice-subtracted relation over a bandwidth of 5 to 22 MHz and subtraction frequencies of 5.55 MHz and 11.11 MHz for the attenuation coefficient and dispersion, respectively.

C. Further Comments on the Validity of the K-K Relations

As we mentioned previously, the K-K dispersion relations are based upon the notions of linearity and causality. The mathematician Titchmarsh [63] showed that these assumptions were sufficient for the real and imaginary parts of a causal transform to be related as a Hilbert transform pair, or equivalently, by the K-K relations. Furthermore, for systems in which the causality condition is satisfied (i.e., effect does not precede cause), the transfer function of a system has a regular analytic continuation in the upper half of the complex frequency plane [7] (or the right-half of the s-plane using the Laplace transform convention of network analysis). Another restriction that follows from Titchmarsh’s theorem is known as the Paley-Wiener condition [60], which states that the transfer function $H(\omega )~\mathrm{e}\mathrm{x}\mathrm{p}(i\kappa (\omega ))$ of a linear system is causal if:

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\left|\mathrm{l}\mathrm{n}\right|H\left({\omega }^{\prime }\right)\left|\right|}{1+{\omega }^{\prime 2}}d{\omega }^{\prime }<\mathrm{\infty }.$$ (9)

This has been discussed for general linear systems [64] as well as acoustic systems [30]. Based on the Paley-Wiener condition, some have argued that systems with attenuation that grow proportional to ${\omega }^y$ for $y\ge 1$ are acausal [65]. However, as we have seen, K-K dispersion relations are available that agree well with experimental data for media with attenuation proportional to ${\omega }^y$ with $1\le y\le 2$. Consequently, empirical observation indicates that such systems are indeed causal even though the transfer function may not be a causal transform. We elaborate further on this point.

For cases in which the transfer function is square integrable, classical K-K relations are sufficient. However, not all physically realizable quantities need be square integrable. For example, a transfer function of a linear system need only be bounded [7]. In such cases, the transfer function still can be analytically continued into the upper half of the complex frequency plane, but it no longer is a causal transform. However, for a transfer function that is bounded, the transfer function with one subtraction can be shown to be a causal transform. Knowing the subtraction constant, the real and imaginary parts of the transfer function then determine each other. We remark that there is an added complexity regarding causality and analyticity when the K-K relations are applied to the complex wave number rather than the transfer function. This case has been addressed in the work by Burge et al. [66] as well as Weaver and Pao [67].

We can extend this to cases in which the attenuation grows as a power law ${\omega }^y$ for $y\ge 1$. In such cases higher-order subtractions are necessary. We reiterate that a transfer function with attenuation that grows as $\omega$ is not a causal transform. Nevertheless, it still can be analytically continued into the upper half of the complex frequency plane. One then can show that the transfer function with two subtractions is a causal transform. Knowing the appropriate subtraction constants and its behavior (i.e., derivative) around the subtraction frequency (or frequencies) permits determination of the real and imaginary parts of transfer function in terms of one another. Furthermore, these transfer functions with subtractions satisfy the Paley-Wiener condition and, hence, are causal.

The method of subtractions is a straight-forward approach for finding a causal transform, and has led to the physical interpretation of subtraction constants, as in the case of coupling constants that characterize interactions in high-energy particle experiments [68]. The technique can be used with experimental data as well as model fits to the data. Although the use of higher-order subtraction functions can be useful for improving the convergence of the K-K integrals, they also can become unwieldy. This in part led to consideration of physical quantities as distributions rather than point functions. The theory of distributions (or generalized functions) was formalized by Schwartz [46], [47]. Lighthill [69] made significant contributions to the analysis of tempered distributions that are useful for the analysis of media with ultrasonic attenuation obeying a frequency power law.

Beltrami and Wohlers [70] developed generalized K-K dispersion relations in the context of distributions rather than point functions. One form of the generalized K-K relations valid for tempered distributions is written in terms of a derivative of arbitrary order of the distribution. Because derivatives of tempered distributions also are tempered distributions [71], K-K relations can be developed for derivatives of the quantities of interest. For example, one can relate the derivative of the attenuation coefficient to the group velocity. Furthermore, for the case of media with attenuation obeying a power law, one can derive the same dispersion equations using either the classical K-K relations with the method of subtractions for point functions or the generalized K-K relations for tempered distributions [31]. An interesting observation is that there is no explicit use of subtraction constants for the generalized K-K relations. The correspondence of these two approaches is a topic of continuing investigation.

Regardless of the approach used, the application of any particular form of the K-K relations to experimental data evaluates the consistency of the measurement with the assumptions of the K-K analysis, namely, linearity, causality, and the accuracy of a physical or empirical model if a model is being fit to data. Discrepancy between experiment and the K-K analysis indicates that one or more of the assumptions are not satisfied. In the case of fitting a model to experimental measurement, extrapolation beyond the known bandwidth always must be done with care. Discrepancies can arise, as was found for the case of the original nearly-local approximation applied to ultrasonic measurements of cancellous bone [72].

V. Causality-Imposed Relations in the Time Domain

The K-K dispersion relations are developed in the context of the frequency-domain response of a linear and causal medium. Based on the logical connection between the dispersion relations and causality, it is notable that there has been little discussion of the consequences of these relations in the time domain. Whereas time-domain analogs of the optical K-K relations [73] and the optical theorem [74], [75] have been previously proposed, it was not until the mid-1990s that time-domain relations imposed by causality were considered for ultrasonic propagation [48], [65].

The need to know the frequency response of a system over all frequencies is generally why the K-K analysis can be difficult. In an effort to avoid these infinite integrals in the frequency domain, Szabo [65] proposed a pair of time-causal relations that transformed the investigation of loss and dispersion in the frequency domain to the time domain. The attractiveness of a time-domain solution for the study of causal, ultrasonic propagation is that all the physically realizable pressure fields are temporally bounded. In other words, they have finite extent in the time domain.

The time-causal theory of Szabo [65] is based upon a one-dimensional lossy wave equation:

$${\nabla }^{2}p(z,t)-\frac{1}{c_0}\frac{\partial }{\partial t}\left({ℒ}_{\gamma }\left(t\right)\mathrm{*}p(z,t)\right)-\frac{1}{c_0^2}\frac{{\partial }^{2}p(z,t)}{\partial t^2}=s(z,t)$$ (10)

where $p(z,t)$ is the pressure field, and the typical constant damping term has been replaced by a convolution loss operator, ${ℒ}_{\gamma }\left(t\right)$, that includes the effects of both attenuation and dispersion. The original development [48] considered those media with attenuation of the form given by (8), but more recently has been extended to a broader range of attenuation forms [76]. The time-causal theory assumes that the ultrasonic parameters are represented by tempered distributions as opposed to conventional point functions. We remark that the functional notation of the tempered distributions is implicit [76].

The convolution loss operator ${ℒ}_{\gamma }\left(t\right)$ is defined as being proportional to the inverse Fourier transform of the attenuation coefficient. By using the action of the inverse Fourier transform on the Hilbert transform [70], [77], the time-causal relations are then:

$${ℒ}_k\left(t\right)=-i\mathrm{s}\mathrm{g}\mathrm{n}\left(t\right){ℒ}_{\alpha }\left(t\right)\text{and}$$ (11)

$${ℒ}_{\alpha }\left(t\right)=+i\mathrm{s}\mathrm{g}\mathrm{n}\left(t\right){ℒ}_k\left(t\right),$$ (12)

where ${ℒ}_{\gamma }\equiv {ℒ}_{\alpha }+i{ℒ}_k$. The time-causal relations are, in essence, the time-domain analog of the Hilbert-transform pairs in the frequency domain [76]. However, the recurring issue of needing knowledge of the system frequency response over all frequencies arises once again, here when calculating the Fourier transform of the attenuation coefficient. Although the time-causal formalism resides in the time domain, it still is necessary to have knowledge of the frequency-domain behavior of the system of interest over the entire spectrum. In cases in which only a finite bandwidth is available, one must once again make assumptions about the form of the attenuation coefficient outside the available band-width, or use a bandwidth much broader than the frequency scale of interest. We discuss this in more detail in Section VI.

For the specific case of the attenuation coefficient obeying a frequency power law (8), the convolution loss operator is [48]:

$${ℒ}_{\gamma }(t)~\left\{\begin{array}{ll}{\delta }^{\left(y\right)}(t)& \left(y\text{an}\text{even}\text{integer}\right)\\ 1/|t{|}^{y+1}& \left(y\text{not}\text{an}\text{even}\text{integer}\right)\end{array},\right.$$ (13)

where ${\delta }^{\left(y\right)}(t)$ is the $y^{\text{th}}$ derivative of the Dirac delta distribution [47]. Here, the power-law form is assumed to exist over the entire spectrum. For the case in which $y$ is an even integer (e.g., $\alpha ~{\omega }^2$), the pressure-loss term of (10) depends only upon the pressure values at that instant in time. More specifically, ${\delta }^{\left(y\right)}(t)\mathrm{*}p(z,t)$ is simply ${\partial }^{y}p(z,t)/\partial t^y$ and, consequently, does not depend on the history of the pressure at that point in space. In contrast, if $y$ is not an even integer, then the pressure will depend upon the history of the pressure, as given by the convolution term $|t{|}^{-(y+1)}\mathrm{*}p(z,t)$.

The time-causal formalism can be used for modeling of ultrasonic propagation in the time domain. In addition, predictions for dispersion given a form of attenuation also are possible, based on the relations between the ultrasonic properties and the convolution loss operator as shown in Fig. 6. We observe that the real and imaginary parts of the convolution loss operator are related by the time-causal relations of (11) and (12), and the ultrasonic properties are related by the Hilbert transform. Fourier analysis moves between the time and frequency domains. A notable observation for the case of quadratic loss (i.e., $\alpha ~{\omega }^2$) is that the time-domain solution does not depend on the history of the pressure, and the phase velocity is independent of frequency (i.e., no dispersion). However, for other values of power-law attenuation (e.g., $\alpha ~\omega$), the existence of dispersion appears to correspond to a fading memory of the system, which was discussed previously for cases of granular materials [78] and viscoelastic media [79].

Fig. 6.

Relationships between the frequency-domain ultrasonic properties and time-domain loss distributions of the time-causal theory. The methods to transform between the representations are provided. Here, $\mathscr{H}$ represents the Hilbert transform, and $ℱ$ and ${ℱ}^{-1}$ represent the forward and inverse Fourier transforms, respectively. (This figure was originally published in Waters et al. [76].)

VI. A Prospective Look

The K-K dispersion relations are used in a variety of applications. We have reviewed a number of cases here in which various forms of the relations have been used to determine the consistency of a measurement. We also have looked at how artifacts can be minimized for the K-K analysis of models of acoustic systems when only a finite bandwidth is available. In addition, the K-K relations can be used for the investigation of apparent inconsistencies in measurements of the attenuation coefficient and phase velocity, as in reports on measurements of cancellous bone [72]. Such investigations may lead to improved models for the ultrasonic propagation in these anisotropic and heterogeneous systems. Extensions of the K-K relations also may prove useful in understanding and overcoming limitations to ultrasonic imaging arising from the effects of phase and amplitude aberration.

With the introduction of the time-causal relations, it is feasible to model the time-domain wave propagation in a causally consistent manner that includes the effects of both attenuation and dispersion. Chen and Holm [80] have applied a fractional derivative method to the time-causal wave equation in order to facilitate numerical solutions. Norton and Novarini [81] have performed finite-difference, time-domain simulations based on the time-causal formalism of acoustic propagation through an ocean sediment and a bubble cloud. These efforts appear promising for modeling wave propagation in a causally consistent manner for a variety of media. However, as mentioned above, assumptions must be made on how to treat finite-bandwidth data. These assumptions are similar in nature to those made for the K-K analyses.

For the case of propagation through ocean sediment, the attenuation coefficient is assumed to be linear with frequency. Norton and Novarini [81] applied a tapering window that left the frequency range of interest invariant but minimized the contribution at other frequencies. This is similar in spirit to the use of the nearly local approximation to the K-K relations in which knowledge over only the bandwidth of interest is necessary and performs well for attenuations that increase roughly proportional to frequency. For the case of propagation through a bubble cloud, the attenuation coefficient no longer exhibits a monotonic behavior, but rather has resonant features. Norton and Novarini [81] used broad bandwidth data (0 kHz to 800 kHz) for the analysis, but restricted their attention to a lower portion of the spectrum (0 kHz to 15 kHz). Here, the ratio $\omega /{\omega }_h$ of the high end of the bandwidth of interest (15 kHz) to the high end of the available bandwidth (800 kHz) is within the range that was found to work well for the finite-bandwidth forms of the integral K-K relations discussed in Section IV-A (i.e., $\omega /{\omega }_h<0.05$).

One could speculate that the corresponding treatments of finite-bandwidth data can be mutually beneficial to the two approaches (i.e., K-K and time-causal) for the study of causality-imposed restrictions on ultrasonic propagation. For example, time-domain modeling at present can be computationally intensive due to the convolution term of the lossy wave equation. As we observed in Section V for media with attenuation obeying a frequency power law, the propagation of the ultrasonic pressure field will, in general, depend upon the history of the pressure (i.e., the medium has a fading memory). However, it appears that the ultrasonic propagation in media with ${\omega }^2$-loss is independent of the pressure history. Consequently, the memory in some media may fade more quickly than in other media. This potentially could lead to the development of useful approximations, perhaps a recent memory form that would be similar in spirit to the nearly local approximation to the K-K relations.

VII. Summary

Causality places restrictions on the time-domain and frequency-domain responses of a system, and the K-K relations are the manifestation of these restrictions. The K-K relations provide the link between the real and imaginary parts of the frequency-domain response of a system, and they also can relate quantities derived from the response function. Their application to acoustical problems has resulted in several forms of the relations that relate the attenuation and dispersion, including conventional integral forms as well as differential forms. Furthermore, forms of the K-K relations can be developed for cases of infinite and finite bandwidth. For the latter case, the restriction to a finite bandwidth leads to artifacts in the accuracy of the K-K analysis, but in some cases these artifacts can be suppressed by the judicious choice of a subtraction frequency. We have applied both infinite- and finite-bandwidth K-K relations to experimental measurements of systems of resonant scatterers and media with attenuation obeying a frequency power law. Good agreement was observed between the experimental and K-K predicted ultrasonic properties for all cases. Additional consequences imposed by causality to acoustic propagation were considered using the time-causal relations, which can be thought of as time-domain analogs of the K-K relations. The combination of the K-K and time-causal relations provide useful tools for the analysis of models and measurements of acoustic systems.

Biographies

Kendall R. Waters (S’92–M’01) was born in 1970 in Chicopee, MA. He graduated from the University of Texas at Austin in 1993 with a B.S. in physics and a B.A. in mathematics. He received his M.S. and Ph.D. degrees in physics (ultrasonics) from Washington University in St. Louis, MO, in 1995 and 2000, respectively.

He worked as a postdoctoral researcher in the Laboratoire d’Imagerie Paramétrique and the Laboratoire Ondes et Acoustique in Paris, France. He is currently an NRC Research Associate at the National Institute of Standards and Technology in Boulder, CO. His research interests include biomedical acoustics, physical acoustics, and nondestructive evaluation.

Dr. Waters is an Associate Editor of the IEEE UFFC Society and a member of the Acoustical Society of America.

Joel Mobley (S’93–S’96–M’97) received his B.S. in physics and mathematics from the University of Kentucky, Lexington, KY, in 1989. He received his M.A. and Ph.D. degrees in physics from Washington University, St. Louis, MO, in 1991 and 1998, respectively. He was a postdoctoral fellow at Oak Ridge National Laboratory in Oak Ridge, TN, from 1997–2003. He is presently a Research Physicist at the U.S. Army Research Laboratory in Adelphi, MD.

Dr. Mobley is a recipient of a Research and Development R&D-100 award in 2003. He also serves as an Associate Editor for Acoustics Research Letters Online. His interests include biomedical ultrasound, photoacoustics, and atmospheric acoustics. He is a member of the IEEE-UFFC Society and the Acoustical Society of America.

James G. Miller (M’75–SM’79–F’98) received an A.B. degree in physics from St. Louis University, St. Louis, MO, in 1963, and M.A. and Ph.D. degrees in physics from Washington University, St. Louis, MO, in 1966 and 1969, respectively.

He is currently the Albert Gordon Hill Professor of Physics at Washington University, St. Louis, MO. He holds joint appointments as Professor of Medicine and Professor of Biomedical Engineering.

His research focuses on the physics of inherently inhomogeneous media, ultrasonic tissue and materials characterization, and ultrasonic transducers. Dr. Miller holds several patents and has coauthored approximately 148 manuscripts on ultrasonic topics. He was the recipient of Industrial Research IR-100 awards in 1974 and 1978. He served as a Sigma Xi National Lecturer from 1981 to 1982. He received a National Institute of Health (NIH) Method to Extend Research in Time (MERIT) Award in 1988. He is a Fellow of the American Institute of Ultrasound in Medicine, the Acoustical Society of America, the American Institute for Medical and Biological Engineering, and the IEEE. He has served three terms as a member of the Administrative Committee of the IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society. He has been a member of the Technical Program Committee for the IEEE Ultrasonics Symposia since 1975 and served as Chairman of the Technical Program Committee for the 1986 Symposium. He also serves as a member of the Technical Program committee for the annual International Symposium on Ultrasonic Imaging and Tissue Characterization.