Getting the Dispersion Solution

The command CALIBRATE/LONG approximates the dispersion relation  for each searched row of the arc spectrum. The algorithm can be activated in different modes, controlled by the parameter WLCMTD:

• The mode IDENT must be used if the lines have been identified interactively using IDENTIFY/LONG.

• The mode GUESS allows a previously saved session to be used (command SAVE/LONG) for the determination of the dispersion relation. The two observation sets must in principle correspond to the same instrumental set-up. Limited shifts (up to 5 pixels) are taken into account by a cross-correlation algorithm. The name of the reference session must be indicated by the parameter GUESS.

• It is also noteworthy to indicate the presence of a mode LINEAR introduced in the package for the purpose of on-line calibration. This mode is still in evaluation phase and allows the results of the command ESTIMATE/DISPERSION to be taken into account in the calibration process.

The command CALIBRATE/LONG provides the following results:

• The coefficients of the dispersion relation for each searched row of the two-dimensional spectrum are computed as a series of polynomials $\lambda = p_y (x)$. The results are written in the table coerbr.tbl.

• The starting, final and average step wavelength  of the spectrum are estimated and written in the keywords REBSTRT, REBEND, REBSTP used by the commands REBIN/LONG and RECTIFY/LONG.

• A two-dimensional dispersion relation of the form $\lambda = \lambda(X, Y)$ is estimated if the parameter TWODOPT is set to YES. This bivariate dispersion relation is necessary to use the command RECTIFY/LONG. The resulting dispersion coefficients are stored in the keyword KEYLONGD.

The command CALIBRATE/TWICE performs a two-pass determination of the dispersion relation. In a first pass, the lines are identified by a standard CALIBRATE/LONG. Only the lines which have consistently identified at all rows are selected for the second pass, which then performs a new calibration on a stable set of arc lines. If after selection a good spectral coverage of the arc spectrum is secured, this method provides very stable estimates of the dispersion relation.

The command PLOT/CALIBRATE visualizes the lines found by the calibration process. The dispersion curve and the lines that were used to determine it are presented by PLOT/DELTA. Residuals to the dispersion curve are plotted by PLOT/RESIDUAL. For two-dimensional spectra, the command PLOT/DISTORTION can be used to check the stability of the dispersion relation along the slit.

The iterative identification loop consists of estimating the wavelength of all lines in the arc spectrum and associate them to laboratory wavelengths to refine the estimates of the dispersion relation. The line identification criterion will associate a computed wavelength $\lambda_c$ to the nearest catalog wavelength $\lambda_{cat}$ if the residual:

$$\delta \lambda = \vert \lambda_c - \lambda_{cat} \vert$$

is small compared to the distance of the next neighbours in both the arc spectrum and the catalog:

$$\delta \lambda < min(\delta \lambda_{cat}, \delta \lambda_c)*\alpha$$

where $\delta \lambda_{cat}$ is the distance to the next neighbour in the line catalog, $\delta \lambda_c$ the distance to the next neighbour in the arc spectrum and alpha the tolerance parameter. Optimal values of $\alpha$ are in the range $0 < \alpha < 0.5$. The tolerance value is controlled by the parameter ALPHA.

Lines are identified in a first pass without consideration of the rms of the residual values by an iterative loop controlled by the parameter WLCNITER. The residuals for each line are then checked in order to reject outliers which residual is above the value specified by the final tolerance parameter TOL. The degree of the polynomials is controlled by the parameter DCX and the iterative loop is stopped if residuals are found to be larger than MAXDEV.