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The optimal reconstruction

The continuum band light curve is reconstructed from the data using the technique of PRH92. For completeness, this section summarizes their method. The reader who is primarily interested in the application of this method may skip to Section 3.

Suppose that the true flux of the source at time t is s(t), but we measure y(t)=s(t)+n(t), where n(t) is the noise in the measurement. In our case, the noise is Poisson in nature. Our knowledge of s(t) is further impeded by the fact that the measurement is only made at a finite number of times tex2html_wrap_inline971 , where i=1,...,N. We denote tex2html_wrap_inline975 as tex2html_wrap_inline977 and refer to this as the continuum data vector.

We seek an optimal reconstruction of s(t) which is continuous in time, tex2html_wrap_inline981 , such that

equation132

is minimized for all t. As usual, angle brackets denote the expectation value. We impose that tex2html_wrap_inline981 is linear in the data vector in the sense that

equation136

where tex2html_wrap_inline987 are a set of inverse response functions that are also continuous in time.

Assuming that the noise is uncorrelated with both s(t) and itself, PRH92 showed that eqn (3) can be minimized to yield,

equation141

Here,

equation148

is the total covariance matrix. To keep the notation concise, PRH92 define the correlation statistics:

eqnarray152

These functions define what PRH92 call the `covariance model'. The expected variance of the real signal from the optimal reconstruct in eqn (5) is then given by

equation156

Once the covariance model is known, eqns (5) and (10) define the optimal reconstruction of the continuum light curve together with a statistic measuring the expected deviation of the real signal from the reconstruction.



Chris Reynolds
Tue Jan 11 17:27:37 MST 2000