Note that this formulation ignores the reflected atmospheric downwelling radiance. This assumption is reasonable in the Longwave Infrared band (LWIR) because the reflected radiance contribution to observed signal is negligible [7].The Beer-Bourger-Lambert Law [8] gives an explicit expression for the transmissivity pathway signaling of a plume in terms of the chemical effluent’s concentration path-length, c (with c measured in parts-per-million-meter, denoted ppm-m), as follows:��p=exp(�\��j=1NcAj(��)cj)(3)where Aj(��) is the absorbance coefficient of chemical Inhibitors,Modulators,Libraries j in (ppm-m)?1 [8] and Nc denotes the number of chemicals in the plume. For optically thin plumes, this term is well approximated by the first two terms in a Taylor Series expansion Inhibitors,Modulators,Libraries [1]. This gives:��p(��)?1�\��j=1NcAj(��)cj(4)for Inhibitors,Modulators,Libraries small c.

Substituting Equation (4) into Equation (1) yields the working gas-plume linearized model:Lobs(��)=��a(��)[B(Tp;��)�\Lg(��)]��j=1NcAj(��)cj+��a(��)Lg(��)+Lu(��)n(��)(5)where the noise term n(��) now includes the approximation Inhibitors,Modulators,Libraries error due to application of Equation (4). Equation (5) shows that the sensor incident radiance Lobs(��) can be represented as an additive layering of the chemical signal ��a(��)[B(Tp;��)�\Lg(��)]��j=1NcAj(��)cj, ground radiance transmitted through the atmosphere, ��a(��)Lg(��), atmospheric upwelling radiance, Lu(��), and noise, n(��). This formulation motivates scene whitening i.e. background radiance subtraction and decorrelation. In the next section we discuss scene whitening in the context of the detection methods.3.

?Detection Method FormulationsIn this section we discuss the matched filter Entinostat or generalized least squares (GLS) approach to gas detection and the Basis Vector Detection (BVD) method.3.1. Matched FilterAs a hyperspectral sensing instrument records radiance at a number of channels, we will present the physics-based model and data processing in vector-matrix notation. We will also restrict our exploration to the single chemical case.Let N�� denote the number of spectral channels recorded by the instrument. The physics-based model in Equation (5) can be written in vector form as:Lobs=��a��(B(Tp)�\Lg)��Ac+��a��Lg+Lu+n(6)where bold terms represent N�� �� 1 vectors and denotes the Hadamard product (element-wise multiplication).An initial step is to remove the background radiance. To do this we compute the scene-wide mean radiance = ��a g + Lu + while assuming constant atmospheric terms ��a and Lu.

This provides a reasonable approximation to the background (non-plume) radiance provided the plume(s) are small (up to a few tens of pixels out of tens of thousands) and weak (in concentration and temperature contrast with the background) [1]. We subtract from both sides of Equation (6) to arrive at:r=Lobs�\L��=��a��(B(Tp)�\Lg)��Ac+z(7)where from r is the mean subtracted pixel radiance and we assume z = ��a (Lg ? g) + (n?) is zero mean with covariance matrix ��.