For numeric calculations we approximate the density by a piecewise constant function:

where designates the characteristic function of the interval . Using this simple ansatz, we can substitute (3) into (2) and perform the integration analytically. Assuming that measurements are made at positions

(4) |

we obtain the matrix equation:

(5) |

Setting

(6) |

the model matrix is given by:

In order to ensure smoothness of the plasma density we represent it by a natural cubic spline passing through control points and calculate by interpolation at . Fixing and while varying only , we get by a simple matrix multiplication: . In order to obtain the spline matrix we calculate the cubic spline passing through and set .

We finally arrive at a matrix equation relating and the absorbance :

Note that the Maximum Entropy method ensures that the ordinates of the control points will be always positive but interpolated values may be negative. Danilo Neuber 2003-10-03