k-Wave
A MATLAB toolbox for the time-domain
simulation of acoustic wave fields
- Getting Started
- Examples
- Initial Value Problems
- Example: Homogenous Propagation Medium
- Example: Using A Binary Sensor Mask
- Example: Defining A Sensor Mask By Opposing Corners
- Example: Loading External Image Maps
- Example: Heterogeneous Propagation Medium
- Example: Saving Movie Files
- Example: Recording The Particle Velocity
- Example: Defining A Gaussian Sensor Frequency Response
- Example: Comparison Of Modelling Functions
- Example: Setting An Initial Pressure Gradient
- Example: Simulations In One Dimension
- Example: Simulations In Three Dimensions
- Example: Photoacoustic Waveforms in 1D, 2D and 3D
- Time Varying Source Problems
- Example: Monopole Point Source In A Homogeneous Propagation Medium
- Example: Dipole Point Source In A Homogeneous Propagation Medium
- Example: Simulating Transducer Field Patterns
- Example: Steering A Linear Array
- Example: Snell's Law And Critical Angle Reflection
- Example: The Doppler Effect
- Example: Diffraction Through A Slit
- Example: Simulations In Three-Dimensions
- Sensor Directivity
- Example: Focussed Detector in 2D
- Example: Focussed Detector in 3D
- Example: Modelling Sensor Directivity in 2D
- Example: Modelling Sensor Directivity in 3D
- Example: Sensor Element Directivity in 2D
- Example: Focussed 2D Array with Directional Elements
- Photoacoustic Image Reconstruction
- Example: 2D FFT Reconstruction For A Line Sensor
- Example: 3D FFT Reconstruction For A Planar Sensor
- Example: 2D Time Reversal For A Line Sensor
- Example: 2D Time Reversal For A Circular Sensor
- Example: 3D Time Reversal For A Planar Sensor
- Example: 3D Time Reversal For A Spherical Sensor
- Example: Image Reconstruction With Directional Sensors
- Example: Image Reconstruction With Bandlimited Sensors
- Example: Iterative Image Improvement Using Time Reversal
- Example: Attenuation Compensation Using Time Reversal
- Example: Attenuation Compensation Using Time Variant Filtering
- Example: Automatic Sound Speed Selection
- Diagnostic Ultrasound Simulation
- Example: Defining An Ultrasound Transducer
- Example: Simulating Ultrasound Beam Patterns
- Example: Using An Ultrasound Transducer As A Sensor
- Example: Simulating B-mode Ultrasound Images
- Example: Simulating B-mode Images Using A Phased Array
- Numerical Analysis
- Example: Controlling The Absorbing Boundary Layer
- Example: Source Smoothing
- Example: Filtering A Delta Function Input Signal
- Example: Modelling Power Law Absorption
- Example: Modelling Nonlinear Wave Propagation
- Example: Optimising k-Wave Performance
- Using The C++ Code
- Elastic Wave Propagation
- Example: Explosive Source In A Layered Medium
- Example: Plane Wave Absorption
- Example: Shear Waves And Critical Angle Reflection
- Example: Simulations In Three Dimensions
- Functions - By Category
- Functions - Alphabetical List
- Release Notes
- License
k-Wave Toolbox |
Modelling Power Law Absorption Example
On this page… |
---|
Overview
This example describes the characteristics of the absorption and dispersion encapsulated by the k-Wave simulation functions.
For a more detailed discussion of the absorption model used in k-Wave, see Treeby, B. E. and Cox, B. T., "Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian," J. Acoust. Soc. Am., vol. 127, no. 5, pp. 2741-2748, 2010.
Absorption and dispersion characteristics
The acoustic absorption within k-Wave is modelled using two phenomenological loss terms. These separately account for absorption that follows a frequency power law in addition to the associated dispersion (dependence of the sound speed on frequency) required by the Kramers-Kronig relations. Under the smallness approximation that the total absorption is much less than the acoustic wavenumber (a condition satisfied for almost all cases of interest in biomedical ultrasound and photoacoustics), these terms allow a wide range of power law absorption characteristics to be accurately modelled. This particular form of absorption is of practical relevance in a number of acoustic applications. For example, the absorption in soft biological tissue over diagnostic ultrasound frequencies has been experimentally shown to follow a frequency power law in which the exponent is between 1 and 2. Similarly, the absorption in marine sediments follows a power law where the exponent is close to 1.
Within the k-Wave simulation functions, the power law absorption is specified by two parameters, medium.alpha_coeff
and medium.alpha_power
. These correspond to the power law pre-factor and exponent, respectively, where the pre-factor is given in units of dB / (MHz^y cm). To illustrate the characteristics of the absorption model, the encapsulated absorption and dispersion for a range of different power absorption parameters are shown below (open circles). These are extracted from the amplitude and phase spectrums of the signals recorded at two different sensor positions in an absorbing medium in 1D using kspaceSecondOrder
(set example_number = 1
within the example m-file). The theoretical curves for absorption and dispersion (calculated using powerLawKramersKronig
) are also shown for comparison (solid line). There is a close agreement between the theoretical and numerical results which demonstrates that the desired absorption characteristics have been correctly modelled.
Numerical errors
If the same example is calculated using kspaceFirstOrder1D
(set example_number = 2
within the example m-file), at higher frequencies there is a small deviation from the theoretical absorption curve and a noticeable deviation from the theoretical dispersion curve.
This is due to the dependence of the loss terms on acoustical quantities in addition to their temporal gradients which are calculated at different stages within the model.
This introduces small phase errors which are accentuated as the absolute absorption level is increased.
These errors can be minimised by reducing the time step used in the simulation (set example_number = 3
within the example m-file).
In the limit, the encapsulated absorption and dispersion properties approach those modelled by kspaceSecondOrder
.
Filtering A Delta Function Input Signal | Modelling Nonlinear Wave Propagation Example |
© 2009-2014 Bradley Treeby and Ben Cox.