### Compound form of the K distribution

The clutter intensity has an exponential distribution of mean power \( x \)\[ P(z \vert x ) = \frac {1} {x} \exp(-\frac{z}{x}) \] where x itself fluctuates with a Gamma distribution

\[ P(x) = \frac{b^\nu}{\Gamma(\nu)} x^{\nu - 1}\exp(-bx) \]

Here, \( \nu \) is the shape parameter, and \( \langle x \rangle = \frac{\nu}{b} \) is the sea clutter mean power.

### Shape parameter dependence on sensor and environmental parameters

Empirical models for the K-distribution shape parameter have been developed using data collected using an I-band radar which provide an initial guide to the range of values to be expected.\[ \log_{10} ( \nu ) = \frac{2}{3}\log_{10}(\phi^{0}_{gr}) + \frac{5}{8} \log_{10}(A_{c})-k_{pol}-\frac{\cos(2 \theta_{SW})}{3} \]

- \( \phi^{0}_{gr} \) is the grazing angle in degrees
- \( A_{c} \) is the radar resolved area
- \(k_{pol} \) is a polarisation dependent parameter (1.39 for VV and 2.09 for HH)
- \( \theta_{SW} \) is the aspect angle with respect to the swell direction (omitted if there is no swell)

### Probability of false alarm

Include the thermal noise in the mean power\[ P(z \vert x ) = \frac {1} {x + p_n} \exp(-\frac{z}{x+ p_n}) \] If we assume that speckle decorrelates then the sum of N pulses \( \mu = \sum\limits_{i=1}^{N} z_{i} \) has the PDF

\[ P(\mu \vert x) = \frac{\mu^{N - 1}}{(x+p_{n})^{N}(N-1)!} \exp (\frac{-\mu}{x+ p_{n}}) \] The probability of false alarm, given \( x \), for a threshold \( Y \) is

\[ P_{FA} (Y \vert x) = \int_Y^\infty \frac{\mu^{N - 1}}{(x+p_{n})^{N}(N-1)!} \exp (\frac{-\mu}{x+ p_{n}}) d \mu = \\ \frac{1} {(N - 1)!} \Gamma(N, \frac{Y}{x + p_n}) \] The overall probability of false alarm after pulse-to-pulse integration is therefore

\[ P_{FA} (Y) = \frac{1} {(N - 1)!} \int_0^\infty \Gamma(N, \frac{Y}{x + p_n}) P(x) dx \] where \[ P(x) = \frac{b^\nu}{\Gamma(\nu)} x^{\nu - 1}\exp(-bx) \]

This is evaluated using numerical integration to produce radar detection performance predictions.

### Probability of detection for a single pulse

The target plus noise and speckle is described by the Rice distribution\[ P( z \vert A,x) = \frac{1}{x + p_n} \exp(- \frac{z + A^2}{x+p_n}) I_0( \frac {2A \sqrt{z}}{x + p_n}) \] Integrate for probability of detection given local mean

\[ P_D(Y \vert x) = \int_Y^\infty \frac{1}{x + p_n} \exp(-\frac{z + A^2}{x + p_n}) I_0(\frac{2A\sqrt{z}}{x+p_n}) dz \] Integrate result over the distribution of the local mean

\[ P_D(Y) = \int_0^\infty P_D(Y \vert x) P(x) dx \]

### Probability of detection, pulse to pulse integration

The PDF of target plus noise and speckle has a multi-look Rice distribution given by\[ P(\mu \vert s, N) = (\frac{\mu}{s})^{\frac{N-1}{2}} \exp(-(\mu + s)) I_{N-1}(2 \sqrt{\mu s}) \] where

- \( \mu = \frac{1}{x + p_n} \sum \limits_{i=1}^{N} z_i \)
- \( s = \frac{1}{x + p_n} \sum \limits_{i=1}^{N} A_i^2 \)

\[ P_D(Y) = \int_0^\infty P_D(Y \vert x) P(x) dx \]

### Target fluctuations

Target plus noise and speckle, multi-look Rice distribution, only depends upon the sum of target powers, not the individual values. Therefore a Gamma distribution model for target fluctuations\[ P(s \vert S, k) = \frac{s^{k-1}}{\Gamma(k)} (\frac{k}{S})^k e^{-\frac{ks}{S}} \] may be used to model correlated and uncorrelated targets including all Swerling models

- Swerling 1: \( k = 1 \)
- Swerling 2: \( k = N \)
- Swerling 3: \( k = 2 \)
- Swerling 4: \( k = 2N \)

### Speckle correlation

There may be pulse to pulse correlation of speckle due to fixed frequency operation or limited frequency agility. If the noise is very low a reasonable approximation may be obtained by reducing the number of integrated pulses. However this is often not the case and the calculations become more complicated. < br> Put the clutter speckle (L independent pulses) with the target to form a Rice distribution \[ P(\beta \vert s, \alpha, L) = \frac{1}{\alpha} (\frac{\beta}{s})^{\frac{L-1}{2}} \exp(-\frac{s+\beta}{\alpha})I_{L-1}(\frac{2 \sqrt{s \beta}}{\alpha}) \] where- \( \alpha = \frac{xN}{L p_n} \)
- \( s = \frac{1}{p_n} \sum \limits_{i=1}^{N} A_i^2 \)
- \( S=\frac{N \langle A^2 \rangle} {p_n} \)

\[ (L+n) \Psi(n+1, L, k, z) =\\ (2n + L + z(n+k))\Psi(n, L, k, z) - n(z+1) \Psi(n - 1, L, k, z) \] where \( \Psi(0, L, k, z) = 1 \) and \( \Psi(1, L, k, z) = 1 + \frac{kz}{L} \)