Causality Analysis of Frequency Dependent Wave Attenuation

Kowar Richard; Scherzer Otmar; Bonnefond Xavier

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Causality Analysis of Frequency Dependent Wave Attenuation

Kowar Richard, Scherzer Otmar, Bonnefond Xavier

arXiv:0906.4678v2Jun 25, 2009math-phmath.APmath.MP

TL;DR

Standard attenuation models lack causality in the parameter range relevant to photoacoustic imaging. Using linear system theory, the paper derives causal attenuation equations and shows that Kramers–Kronig relations alone do not ensure finite wavefront speed. The analysis identifies limits of standard power-law and thermo-viscous models, while finite-front-speed models support stable numerical implementation.

Problem
Standard attenuation models lack causality in the parameter range relevant to photoacoustic imaging.
Source: §1 · p. 2
Method
The paper uses linear system theory to derive causal attenuation models, including equations formulated with homogeneous initial conditions.
Source: §1 · p. 2, §10 · p. 18
Results
Frequency power laws have finite wavefront speed only for γ∈(0, 1), while the thermo-viscous equation lacks a causal domain of influence.
Source: §6 · p. 15, §10 · p. 19
Takeaways & Limitations
Finite-front-speed models permit stable numerical implementation and are suitable for photoacoustic imaging inversion techniques.
Source: §10 · p. 18
The stated physical-process assumption of homogeneity and isotropy implies shift invariance and rotational symmetry of the operator response.
Source: §2 · p. 3

Abstract

from arXiv · show

The work is inspired by thermo-and photoacoustic imaging, where recent efforts are devoted to take into account attenuation and varying wave speed parameters. In this paper we derive and analyze causal equations describing propagation of attenuated pressure waves. We also review standard models, like frequency power laws, and the thermo-viscous equation and show that they lack causality in the parameter range relevant for biological photoacoustic imaging. To discuss causality in mathematical rigor we use the results and concepts of linear system theory. We present some numerical experiments, which show the physically unmeaningful behavior of standard attenuation models, and the realistic behavior of the novel models.

1. Introduction

Photoacoustic imaging motivates modeling attenuation in pressure-wave propagation, but standard models lack causality in the parameter range relevant to the application. The paper uses linear system theory to derive novel causal attenuation models.

1. Introduction

Photoacoustic imaging seeks to reconstruct absorption density from pressure measured outside the object, while recent work has considered attenuation and varying wave speed.The standard attenuation model captures faster attenuation of high-frequency wave components.Source: §1 · p. 2
Standard power-law and thermo-viscous attenuation models lack causality in the parameter range relevant to photoacoustic imaging.The paper reviews these models and motivates investigating novel equations.Source: §1 · p. 2
Linear system theory grounds the derivation of novel causal models, which are written as wave-equation-like, generally integrodifferential equations.They use inhomogeneous equations with homogeneous initial conditions, a formulation the authors identify as mathematically sensible for attenuation.Source: §1 · p. 2
The proposed models include bounded-frequency-range power laws relevant to biological specimens with γ∈(1, 2] and to castor oil with γ=1.66.The paper also notes that finite-front-speed models can be implemented stably, whereas noncausal wave equations lack the CFL condition.Source: §1 · p. 2

2. Linear System Theory

This section sets out linear system theory and definitions used to analyze causal wave propagation. It connects operator properties to wavefront travel time and to the Green-function description of the standard wave equation.

2. Linear System Theory

A causal function vanishes for t<0, and an operator is causal when it maps causal functions to causal functions.The section also defines translation invariance through compatibility with space-time shifts.Source: §2 · p. 3
A causal domain of influence bounds wavefront travel: when T exists and satisfies the stated condition, the wavefront speed is at most c0.T corresponds to the travel time of a wavefront initiated at the origin.Source: §2 · p. 3
Every linear, translation-invariant operator can be represented as space-time convolution with a causal kernel when the operator is causal.The kernel is called the operator's Green function.Source: §2 · p. 4
For the standard wave equation, the solution operator maps causal inputs to causal solutions, and its point-source solution is the Green function.That Green function is described as a spherical wave originating at the source at time zero.Source: §2 · p. 4

3. Attenuation

The section formulates attenuation through frequency-domain operators and examines the conditions for causality. It shows that Kramers–Kronig compliance alone does not establish a causal domain of influence, including for standard power laws.

3. Attenuation

The standard attenuation formulation multiplies the unattenuated Green function by exp(−β*(|x|, ω)), with α=Re(β*) called the attenuation law.The paper relates nonnegative real part of β* to attenuation as frequency-dependent energy dissipation.Source: §3 · p. 5
For the standard form of β*, travel time is T(|x|)=|x|/c for a constant 0<c≤c0.Source: §3 · p. 5
An attenuation operator has a causal domain of influence if and only if its associated K is causal.Causality of the inverse Fourier transform of α* follows from causal K, but the converse does not generally hold.Source: §3 · p. 6
For the stated frequency power-law extension, K is causal exactly when γ∈[0, 1); models with γ>1 therefore lack a causal domain of influence.The attenuation law and the cited causality condition do not uniquely determine the attenuation operator.Source: §3 · p. 7

4. Equations for Attenuated Pressure Waves

The section formulates an attenuated pressure-wave equation from causal operators and convolution kernels. It contrasts this framework with standard attenuation models, whose solution operators have a causal domain of influence only for γ in [0, 1).

4. Equations for Attenuated Pressure Waves

The derivation introduces convolution operators and uses an inverse Fourier transform to obtain an equation for pressure from a general source term.The pressure is written as the space–time convolution of the Green function with the source.Source: §4 · pp. 8–9
The section formulates a causal wave equation accounting for attenuation; Equation (29) is the pressure wave equation and obeys attenuation with coefficient (18).Source: §4 · pp. 7, 9
The standard power-law attenuation model is expressed using a fractional time derivative, and the paper relates this formulation to the cited attenuation equations.The fractional derivative is identified with the time-convolution operator in the power-law model.Source: §4 · pp. 9–10
For the standard attenuation models (36) and (37), the solution operator has a causal domain of influence only for γ in [0, 1).The paper contrasts their homogeneous Cauchy formulation with inhomogeneous equations and homogeneous initial conditions, which provide a rigorous framework for defining solution operators.Source: §4 · p. 10

5. The homogeneous Cauchy problem with memory

This section examines when an inhomogeneous wave equation with homogeneous initial conditions and a homogeneous equation with inhomogeneous initial conditions give the same pressure. Their solutions coincide for t > 0 exactly when the initial data and pressure satisfy the stated conditions.

5. The homogeneous Cauchy problem with memory

The section compares an inhomogeneous wave equation with homogeneous initial conditions against a homogeneous equation with inhomogeneous initial conditions.The two formulations are not asserted to be equivalent without the theorem’s conditions.Source: §5 · p. 11
The two formulations have the same solution for t > 0 if and only if q0 and patt satisfy the conditions in Theorem 5.1.The theorem assumes unique solutions for both problems.Source: §5 · p. 11
When the stated conditions hold, the constructed function solves the inhomogeneous problem and equals patt by uniqueness.The proof establishes this equality for t > 0.Source: §5 · p. 12
In the absence of attenuation, the compared solutions are identical for t > 0.In this case the operator B is zero and the condition reduces to the stated unattenuated form.Source: §5 · p. 12

6. The thermo-viscous wave equation

The section shows that the standard thermo-viscous wave equation is not causal, despite its attenuation coefficient satisfying the Kramers–Kronig relation. It then develops a causal variant with the same attenuation law.

6. The thermo-viscous wave equation

The formalism is used to derive a causal variant of the thermo-viscous equation that satisfies the same attenuation law.The section relates the thermo-viscous equation to the operator form used for attenuated waves.Source: §6 · pp. 13–15
The section introduces time-convolution operators T^(1/2) and L^(1/2), which are inverses of one another.Their compositions recover the input function.Source: §6 · pp. 13–14
The thermo-viscous wave equation’s solution operator does not have a causal domain of influence, although its attenuation coefficient satisfies the Kramers–Kronig relation.The paper attributes the failure to the noncausal kernel associated with the standard equation.Source: §6 · p. 15
The real part of the thermo-viscous attenuation law approximates a frequency power law with γ = 2 at small frequencies.This is a low-frequency approximation.Source: §6 · p. 15

7. A causal thermo-viscous wave equation

The section presents a causal variant of the thermo-viscous wave equation. Its solution operator has a causal domain of influence, and the model is described as a potential basis for thermoacoustic tomography in viscous soft tissue.

7. A causal thermo-viscous wave equation

For equation (57), the passage states that the solution operator, denoted A, has a causal domain of influence.Source: §7 · p. 16
The causal model’s attenuation coefficient has the same real part as the standard thermo-viscous coefficient.The causal coefficient’s real and imaginary parts satisfy the Kramers–Kronig relation.Source: §7 · p. 16
The model is a potential basis for thermoacoustic tomography because soft tissue is often modeled as a viscous fluid.The passage also notes that tissue attenuation is frequently modeled by a frequency power law with γ in (1, 2).Source: §7 · p. 16

8. Causal Wave Equations satisfying Frequency Power Laws for small frequencies with γ ∈(1, 2]

The section derives causal wave equations whose attenuation laws approximate frequency power laws at small frequencies for γ∈(1,2]. The solution operator has a causal domain of influence, and γ=2 recovers a causal thermo-viscous variant.

The derived attenuation laws approximate frequency power laws at small frequencies for exponents γ∈(1, 2].Source: §8 · p. 16
For γ=2, the derived equation recovers the causal variant of the thermo-viscous wave equation.Source: §8 · p. 17
The solution operator for the derived equation has a causal domain of influence.This is the section’s stated causality result for equation (61).Source: §8 · p. 17

9. Examples

Simulations of the frequency power law and Szabo’s model show causality only for γ∈[0,1). The section also compares the standard thermo-viscous model with its causal variant.

The solution operator has a causal domain of influence exactly when the inverse Fourier transform of exp(−β∗) is causal.Non-causality is indicated when this inverse transform is nonzero for some t<0.Source: §9 · p. 17
γ∈[0,1): simulations show causality only in this range for the frequency power law and Szabo’s model.Source: §9 · p. 18, Figure 1 · p. 18
The thermo-viscous model and its causal variant differ in attenuation law only by a multiplicative constant α1.The section identifies simulations of the standard equation and of the causal variant with α1=1.Source: §9 · p. 18

10. Conclusions

The conclusions distinguish finite wavefront speed from the Kramers-Kronig condition and summarize the scope of frequency-power-law models. They also report the formulation used for the analysis and its conditional relation to a memory Cauchy problem.

The proposed operator concept guarantees finite wavefront speed, while satisfying the Kramers-Kronig relation alone does not guarantee that speed.Source: §10 · pp. 18–19
Frequency-power-law waves satisfying the Kramers-Kronig relation have finite wavefront speed only when γ∈(0, 1); thermo-viscous waves can have infinite wavefront speed.Source: §10 · p. 19
The authors developed novel equations satisfying their causality requirement and desired attenuation properties because standard models lack causality in the photoacoustic imaging parameter range.Source: §10 · p. 19
The analysis formulates equations as inhomogeneous equations with homogeneous initial conditions; under certain conditions, the problem can instead be formulated as a Cauchy problem with memory.Source: §10 · p. 19

11. Appendix: Nomenclature and elementary facts

The appendix supplies elementary notation and facts used in the paper, including support, convolution, and Fourier-transform conventions.

The support supp(g) is defined as the closure of the points where g does not vanish.Source: §11 · p. 20
The appendix distinguishes time and frequency convolutions from space-time convolution.∗t and ∗ω denote time and frequency convolution, respectively; ∗x,t denotes space-time convolution.Source: §11 · p. 21
F denotes the Fourier transform with respect to t, while F−1 is taken with respect to ω.Source: §11 · p. 21