Observation model

The observation model is based on a physical modeling of dispersion and detection in a turbulent medium proposed in [Vergassola2007] and generalized to an arbitrary number of dimensions in [Loisy2022].

Observations (“hits”) are drawn randomly according to a Poisson distribution

\[\begin{equation} \text{Pr}(h | {\bf x}^a,{\bf x}^s) = \frac{\mu^h \exp(-\mu)}{h!} \qquad \text{with} \, \mu = \mu(\lVert {\bf x}^s - {\bf x}^a \rVert_2) \end{equation}\]

which mean \(\mu\) is a function of the Euclidean distance \(d=\lVert {\bf x}^s - {\bf x}^a \rVert_2\) between the agent and the source (measured in number of grid cells).

The expression of \(\mu(d)\) for an arbitrary number of dimensions \(n\) is

\[\begin{split}\begin{align} & n=1: && \displaystyle \mu(d) = I \frac{2 L}{2 L-1} \exp(-d/L) \\ & n=2: && \displaystyle \mu(d) = I \frac{1}{\ln(2 L)} K_{0} (d/L) \\ & n=3: && \displaystyle \mu(d) = I \frac{1}{2 d} \exp(-d/L) \\ & n=4: && \displaystyle \mu(d) = I \left( \frac{1}{2 L} \right)^{2} \frac{L}{d} K_{1} (d/L) \end{align}\end{split}\]

and more generally for \(n\geqslant 3\)

\[\begin{equation} \mu(d) = I \left( \frac{1}{2 L} \right)^{n-2} \left( \frac{L}{d} \right)^{n/2-1} \frac{(n-2)}{\Gamma(n/2)} \frac{K_{n/2-1} (d/L)}{2^{n/2-1}} \end{equation}\]

where \(L\) is a dimensionless dispersion lengthscale that determines the size of the search domain, \(I\) is a dimensionless source intensity, \(\Gamma\) is the gamma function, and \(K_{\nu}\) is the modified Bessel function of the second kind of order \(\nu\).

In the code

• n is called N_DIMS,

• \(L\) is called LAMBDA_OVER_DX,

• \(I\) is called R_DT.