Establishing the model matrix

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

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

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

we obtain the matrix equation:

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

Setting

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

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

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

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 $A_i$:

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