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## Sensor Element Directivity in 2D Example

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Overview Defining directional detectors Running the simulation (initial value problem) Running the simulation (time-varying source)

### Overview

This example shows how to attribute a directional response to a single-element sensor, or individual elements of a multi-element sensor array. Directionality can be included without a separate function through explicit averaging, as shown in the examples Modelling Sensor Directivity in 2D and Focussed Detector in 2D, but the functionality described here allows greater flexibility. Note that directivity defined in this way is currently only supported in 2D. This example builds on the Homogeneous Propagation Medium and Using A Binary Sensor Mask examples.

- open the file in the MATLAB Editor
- run the file in MATLAB

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### Defining directional detectors

When using a binary sensor mask, the directionality of each sensor element (defined by the 1's within `sensor.mask`) can be set by assigning a directivity angle to the corresponding elements within an Nz by Nx matrix assigned to `sensor.directivity_angle`. This angle defines the direction of greatest sensitivity for each element. For waves incident from other angles, the sensitivity will be reduced. In this example, a line of spaced single element sensors is assigned to a binary sensor mask with each given a different directionality. A directivity angle of 0 (or pi) results in the element being most sensitive to waves travelling in the z (up/down) direction. A directivity of pi/2 (or -pi/2) results in the element being most sensitive to waves travelling in the x (left/right) direction. The magnitude of the directionality can be adjusted by defining the `sensor.directivity_size` (this value is assigned to all elements). For a plane wave incindent on a parallel linear detector, this value is equivalent to the size of sensor that each element averages across. If this field is not set, it defaults to `10*kgrid.dx` (i.e., 10 times the width of a single pixel).

    % define a line of single-element sensors
    sensor.mask = zeros(Nz,Nx);
    sensor.mask(24,2:2:63) = 1;

    % define the angle of max directivity for each single-element sensor
    % 0 = max sensitivity in z direction (up/down)
    % pi/2 or -pi/2 = max sensitivity in x direction (left/right)
    sensor.directivity_angle = zeros(Nz,Nx);
    sensor.directivity_angle(24,2:2:63) = (-1:1/15:1)*pi/2;

    % define the directivity size
    sensor.directivity_size = 16*kgrid.dx;

A visualisation of the sensor mask showing the directivity of each element is shown below, superimposed over an image of the source mask (or initial pressure distribution). Note that having many different directivity angles in a single simulation can increase the computation time. However, arrays in which all the elements have the same directionality will run almost as fast as the default omni-directional case.

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### Running the simulation (initial value problem)

The initial pressure distribution is set up in the usual way and the computation is invoked by running `kspaceFirstOrder2D` with the inputs defined above. To ensure the plane wave source is not distorted by the perfectly matched layer, the part of the PML perpendicular to the propagation direction is turned off by setting the optional input `'PMLAlpha'` to `[0 4]` (see Controlling The Absorbing Boundary Layer).

    % define the initial pressure distribution
    source.p0 = zeros(Nz,Nx);
    source.p0(39:41 ,:) = 2;

    % run the simulation
    [sensor_data p_final] = kspaceFirstOrder2D(kgrid, medium, source, sensor, 'PMLAlpha', [0 4]);

The maxima of the time series recorded by each sensor element are plotted below. It is clear that the elements with their directivity aligned to the plane wave exhibit greater sensitivity.

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### Running the simulation (time-varying source)

It is interesting to run this example with a single frequency source, rather than the broadband source due to the initial pressure distribution. A single frequency time-varying source plane wave source can defined as:

    % define a time varying sinusoidal source instead of an initial pressure
    source.p_mask = source.p0;
    source.p0 = [];
    source_freq = 12e6;
    source_mag = 1;
    source.p = source_mag*sin(2*pi*source_freq*kgrid.t_array);

The computation is invoked by running `kspaceFirstOrder2D` with the inputs defined above. As above, it is ensure the source remains as a plane wave, is is necessary to turn off the PML along the sides perpendicular to the wavefront by setting the optional input `'PMLAlpha'` to `[0 4]` (see Controlling The Absorbing Boundary Layer).

    % run the simulation
    sensor_data = kspaceFirstOrder2D(kgrid, medium, source, sensor, 'PMLAlpha', [0 4]);

The maxima of the time series recorded by each sensor element are plotted below on Cartesian and polar plots. Because this example used a single frequency, the characteristic side lobes associated with spatial averaging detectors can be seen.

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© 2009, 2010, 2011 Bradley Treeby and Ben Cox.