### Resolution and aliasing definition according to build_array

Posted:

**Wed Feb 20, 2008 9:49 am**Build_array can calculate the theoretical array response and it automatically picks values for kmin and kmax (resolution and aliasing limits). Here are some explanations how these values are calculated.

kmin is relatively easy to define, the resolution limit is the width of the central peak. kmin/2 is reported (wave number at which the response first drops below 0.5). The wave number map is scanned with a concentric search starting from the origin. The azimuth with the lowest kmin/2 is not reported but you can find it by playing with the azimuth cursor.

For a regular array (let's say a Cartesian grid) the aliasing peaks all reach a value of 1 (the same level as the main lobe). If you take a linear array, the same peaks degenerate into bands (oriented perpendicular to the direction of the array line). These peaks exactly correspond to the the theoretical aliasing defined by the sampling theorem in space (Nyquist, at least two sensors per wavelength). For these arrays, the peaks are strong and the "ground level" of the response function is relatively low. For other shapes, the picture is not that sharp. Theoretical aliasing (peaks reaching 1) may be encountered at very high wave numbers. In between, you can find a lot of intermediate peaks with amplitude between 0 and 1.

To simplify the situation, when several waves travel across the array at the same time, the resulting response is something close to the sum of individual wavelet responses (not exactly there are also cross terms). Hence, the side lobes may stack together, sometimes (or very often) reaching a higher level than the main peaks that correspond to real wavelet wave numbers. By consequence, estimating the aliasing limit for a 2D array (contrary to a simple linear array) requires looking also at the average ground level of the response function and not only at the position of side peaks.

In build_array, the automatic computation of kmax is made by a concentric search (like for kmin) until the amplitude gets over 0.5. I assume that if it goes above 0.5 (with a peak or not) there are more chances to stack sides lobes and aliasing to occur. The azimuth is not reported, but you can easily find it by playing with the azimuth cursor. The precision for the aliasing limit is not so dramatic. Aliasing is obvious in FK results. From experience, it starts slowly from kmax/2 and reaches its maximum at kmax.

In some situations, the kmax deduced from automatic search does not corresponds to your estimation (e.g. just one little peak slightly above 0.5 in one direction, flat elsewhere). Move the kmax cursor and pick another value. For kmin, the choice is much less disputable.

kmin is relatively easy to define, the resolution limit is the width of the central peak. kmin/2 is reported (wave number at which the response first drops below 0.5). The wave number map is scanned with a concentric search starting from the origin. The azimuth with the lowest kmin/2 is not reported but you can find it by playing with the azimuth cursor.

For a regular array (let's say a Cartesian grid) the aliasing peaks all reach a value of 1 (the same level as the main lobe). If you take a linear array, the same peaks degenerate into bands (oriented perpendicular to the direction of the array line). These peaks exactly correspond to the the theoretical aliasing defined by the sampling theorem in space (Nyquist, at least two sensors per wavelength). For these arrays, the peaks are strong and the "ground level" of the response function is relatively low. For other shapes, the picture is not that sharp. Theoretical aliasing (peaks reaching 1) may be encountered at very high wave numbers. In between, you can find a lot of intermediate peaks with amplitude between 0 and 1.

To simplify the situation, when several waves travel across the array at the same time, the resulting response is something close to the sum of individual wavelet responses (not exactly there are also cross terms). Hence, the side lobes may stack together, sometimes (or very often) reaching a higher level than the main peaks that correspond to real wavelet wave numbers. By consequence, estimating the aliasing limit for a 2D array (contrary to a simple linear array) requires looking also at the average ground level of the response function and not only at the position of side peaks.

In build_array, the automatic computation of kmax is made by a concentric search (like for kmin) until the amplitude gets over 0.5. I assume that if it goes above 0.5 (with a peak or not) there are more chances to stack sides lobes and aliasing to occur. The azimuth is not reported, but you can easily find it by playing with the azimuth cursor. The precision for the aliasing limit is not so dramatic. Aliasing is obvious in FK results. From experience, it starts slowly from kmax/2 and reaches its maximum at kmax.

In some situations, the kmax deduced from automatic search does not corresponds to your estimation (e.g. just one little peak slightly above 0.5 in one direction, flat elsewhere). Move the kmax cursor and pick another value. For kmin, the choice is much less disputable.