Establishing the model matrix

For numeric calculations we approximate the density $ \rho(r)$ by a piecewise constant function:

$\displaystyle \rho(r) = \sum_{j=1}^{N_0} \rho_j \chi_j(r),$ (3)

where $ \chi_j(x)$ designates the characteristic function of the interval $ (j-1)R/N_0 \leq x < jR/N_0$. Using this simple ansatz, we can substitute (3) into (2) and perform the integration analytically. Assuming that $ N_d$ measurements are made at positions

$\displaystyle y_i = \left(i - \frac{1}{2}\right) \frac{R}{N_d},$ (4)

we obtain the matrix equation:

$\displaystyle A_i \equiv A(y_i) = \sum_{j=1}^{N_0} \tilde{M}_{ij} \rho_j.$ (5)

Setting

$\displaystyle r_j = j\frac{R}{N_0},$ (6)

the model matrix $ \tilde{M}$ is given by:

\begin{equation*}\tilde{M}_{ij} = \left\{ \begin{aligned}& 2\left(\sqrt{r_j^2-y_...
... < y_i \leq r_{j} \\ & 0 & \mbox{otherwise} \end{aligned} \right.\end{equation*}

In order to ensure smoothness of the plasma density $ \rho(r)$ we represent it by a natural cubic spline passing through $ N_k$ control points $ (\hat{r}_k,
\hat{\rho}_k)$ and calculate $ \rho_j$ by interpolation at $ r_j$. Fixing $ \hat{r}_k$ and $ r_j$ while varying only $ \hat{\rho}_k$, we get $ \rho_j$ by a simple matrix multiplication: $ \rho_j = \sum_k S_{jk} \hat{\rho}_k$. In order to obtain the spline matrix $ S_{jk}$ we calculate the cubic spline $ S_k(r)$ passing through $ (\hat{r}_l,\delta_{kl})$ and set $ S_{jk} = S_k(r_j)$.

We finally arrive at a matrix equation relating $ \hat{\rho}_k$ and the absorbance $ A_i$:

$\displaystyle A_i = \sum_{j,k} \tilde{M}_{ij} S_{jk} \hat{\rho}_k \equiv \sum_{k=1}^{N_k} M_{ik} \hat{\rho}_k$ (8)

Note that the Maximum Entropy method ensures that the ordinates of the control points $ \hat{\rho}_k$ will be always positive but interpolated values may be negative.
Danilo Neuber 2003-10-03