Sampling in , , and

The present version, DDSCAT.5a, chooses the angles , , and to sample the intervals (BETAMI,BETAMX), (THETMI,THETMX), (PHIMIN,PHIMAX), where BETAMI, BETAMX, THETMI, THETMX, PHIMIN, PHIMAX are specified in ddscat.par . The prescription for choosing the angles is to:

• uniformly sample in ;
• uniformly sample in ;
• uniformly sample in .

This prescription is appropriate for random orientation of the target, within the specified limits of , , and .

Note that when DDSCAT.5a chooses angles it handles and differently from . The range for is divided into NBETA intervals, and the midpoint of each interval is taken. Thus, if you take BETAMI=0, BETAMX=90, NBETA=2 you will get and . Similarly, if you take PHIMIN=0, PHIMAX=180, NPHI=2 you will get and .

Sampling in is done quite differently from sampling in and . First, as already mentioned above, DDSCAT.5a samples uniformly in , not . Secondly, the sampling depends on whether NTHETA is even or odd.

• If NTHETA is odd, then the values of selected include the extreme values THETMI and THETMX; thus, THETMI=0, THETMX=90, NTHETA=3 will give you .
• If NTHETA is even, then the range of will be divided into NTHETA intervals, and the midpoint of each interval will be taken; thus, THETMI=0, THETMX=90, NTHETA=2 will give you and [ and 0.75].

The reason for this is that if odd NTHETA is specified, then the ``integration'' over is performed using Simpson's rule for greater accuracy. If even NTHETA is specified, then the integration over is performed by simply taking the average of the results for the different values.

If averaging over orientations is desired, it is recommended that the user specify an odd value of NTHETA so that Simpson's rule will be employed.