The following is based on the report from Peter Swerling: Probability of Detection for Fluctuating Targets.
Case II is defined as fluctuations that are independent from pulse to pulse (page 2), that is, the RCS is pulse to pulse uncorrelated.
On page 8 of the report the expression for Case II probability of detection, $P_{D}$, is given as:
$$P_{D}=1 - I\left[\frac{Y_{b}}{(1+\bar{x})\sqrt{N}},N-1\right]$$
where $I$ is the incomplete gamma function, $\bar{x}$ is the expected value of the signal-to-noise ratio, $N$ is the number of pulses integrated, and $Y_{b}$ is the designed detection threshold.
For partial correlated pulses I will refer to Exact Detection Probability for Partially Correlated Rayleigh Targets. (My apologies for the pay wall.). While an exact formula for partial correlated pulses is presented in this paper it is too involved a computation to easily transcribe here. I am reproducing the abstract below.
The probability of detection of the sum of N square-law-detected
pulses is derived for the case where the signal fluctuation obeys
chi-square statistics with four degrees of freedom. P. Swerling's
(1960) case III and IV represent the cases where the signal is
completely correlated and completely decorrelated, respectively, from
pulse to pulse. An exact expression for probability of detection is
derived for the condition of partial signal correlation. The results
given are compared with the approximate technique commonly used to
handle partial signal correlation.
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