Example forward-difference

Consider the example of an ADCP with a beam angle of [math]\displaystyle{ 20^{\circ} }[/math], configured with a vertical bin size of 10 cm, recording profiles at 1 second intervals with a data segment length of 300 seconds. The Level 1 QC of the data identified that good data was typically returned from bins 1 to 30.

The velocity data from a single beam for a single data segment can therefore be visualised as:

with [math]\displaystyle{ N=300 }[/math]. The square of the velocity difference between bins separated by [math]\displaystyle{ \delta }[/math] bins is then evaluated for each [math]\displaystyle{ t }[/math]. So for [math]\displaystyle{ t_1 }[/math], bin 1 and [math]\displaystyle{ \delta=1 }[/math], we get:

[math]\displaystyle{ D(1,1,1) = \left[v^\prime(1,1) - v^\prime(2,1)\right]^2 }[/math]

The forward-difference evaluatedfor bin 1 and is then can be evaluated for [math]\displaystyle{ 1\leqslant\delta\leqslant29 }[/math], whilst that for bin 2 is restricted