Inverse Dispersion Solution Product¶

Metadata¶

Data product name	DpdSirIdsModel
Data product custodian	SIR
Name of the Schema file	euc-sir-IdsModel.xsd
Processing function using the data product	SIR

Proposed for inclusion in EAS/SAS¶

This product is proposed for inclusion in the SAS: no

This is an internal Data Product.

Data product elements¶

Header	object of type sys:genericHeader
Data	object of type sir:sirIdsModel
QualityFlags	object of type dqc:sqfPlaceHolder
Parameters	object of type ppr:genericKeyValueParameters

Detailed description of the data product¶

The sirIdsModel describes the relation between spectra wavelengths and pixels on the detector.

Given a spectrum, this model (combined with Optical Model Product and Curvature Model Product) predicts on which pixel to find a specific \(\lambda\).

The \(\lambda \longleftrightarrow pix\) relationship is not linear and this relationship varies across the field of view; the mathematical description of this model is analogous to the description of the sirCRVModel. Like the sirCRVModel does, the sirIDSModel operates within the SIR rotated layout.

The XML file has the same structure. Each grism/tilt configuration is independently calibrated and the coefficients of each configuration are store the sirIdsModelOrders structure. The sirIdsModel collects all the sirIdsModelOrders available.

Also in this case relevant items of sirIdsModelOrder, are:

GWATilt: the grism nominal tilt value (0, +4, -4);

ExtraTilt: the extra tilt computed during calibration (not implemented yet);

SpectraOrder: the list of sirIdsModelContent, each item contains the geometrical description of a given spectrum order for the current grism (see description below).

SpectraOrders description¶

For each spectral order, one Chebychev mono dimensional polynomial (of order \(N\)) is used to compute the local relation between \(\lambda\) (Angstrom) and pixel.

(2)¶\[\Delta d_{[mm]} = \sum_{k=0}^M \beta_{x,y,k} T_k\left(\lambda \right)\]

\(\Delta d\) is the displacement along the dispersion direction (in mm) between the lambda reference position and the input required \(\lambda\).

Like in the sirCrvModel case the \(\beta\) coefficients variation is described by

a global 2D Chebychev polynomial

\[\beta_{x,y,k} = \sum_{i,j} b^{(k)}_{i,j} T_i(x)T_j(y)\]

These \(\textbf{B}^{(k)} := \{ b^{(k)}_{i,j} \}\) matrices are stored in the sirIdsModelContent structure, which contains:

Order: the order of the spectrum (1 ^st, 0 ^th, 2 ^nd, …) to model;

LocalModelDeg: the degree of the Local ids model (\(M\) in the equation ());

LocalRanges: the validity domain of the local Chebychev polynomial; range extremes are in Angstrom and they contain the blue and the red edges of the spectrum;

GlobalRanges: the validity domain (along the 2 directions of the FOV) of the local 2D Chebychev polynomial. These range extremes are in mm

Model: a list of specificDegreeMatrix, each item of this list contains one \(\textbf{B}^{(k)}\) matrix. If the local polynomial is of degree \(M\), the model will contain \(M+1\) matrices. Each matrix is used to obtain one local coefficient \(\beta_{x,y,k}\) in a given position (\(x,y\)) of the FOV

Usage example¶

This example concatenates all the 3 models (OPT, CRV and IDS) to retrieve the pixel position of a given \(\lambda\) for a given object (RA, Dec coordinates).

1. Given an objects position (RA, Dec) the OpticalModel is used to obtain the mm position the reference lambda:

(3)¶\[x_{ref}, y_{ref}\]

2. This reference lambda position is used to obtain the set of local coefficients: \(\alpha_{0...N}\) for the sirCRVModel and \(\beta_{0...N}\) for the sirIDSModel.

\[\alpha_{x_{ref},y_{ref},k} = \sum_{i,j} a^{(k)}_{i,j} T_i(x_{ref})T_j(y_{ref}) \quad k\in{0...N}\]

\[\beta_{x_{ref},y_{ref},k} = \sum_{i,j} b^{(k)}_{i,j} T_i(x_{ref})T_j(y_{ref}) \quad k\in{0...N}\]

3. The displacement (in mm) wrt to the reference lambda is obtained applying the sirIdsModel

(4)¶\[\Delta d = \sum_{k=0}^M \beta_{x,y,k} T_k\left(\lambda\right)\]

4. The displacement correction (in mm) wrt to the ideal dispersion position is obtained applying the sirCrvModel

(5)¶\[\Delta c = \sum_{k=0}^N \alpha_{x,y,k} T_k\left(\Delta d\right)\]

5. Combining (3), (4) and (5) we obtain the final position for the RGS000 and BGS000:

\[\begin{split}\begin{bmatrix} x_{\lambda} \\ y_{\lambda} \end{bmatrix} = \begin{bmatrix} x_{ref} \\ y_{ref} \end{bmatrix} + \begin{bmatrix} \Delta d \\ \Delta c \end{bmatrix};\end{split}\]

and for RGS180

\[\begin{split}\begin{bmatrix} x_{\lambda} \\ y_{\lambda} \end{bmatrix} = \begin{bmatrix} x_{ref} \\ y_{ref} \end{bmatrix} - \begin{bmatrix} \Delta d \\ \Delta c \end{bmatrix}.\end{split}\]

6. NOT IMPLEMENTED YET. If an extra tilt (\(\theta\)) is computed and the sirPivotModel exist and extra rotation is applied. The pivot position is obtained by sirPivotModel

\[x_{pivot}, y_{pivot}\]

and applied

\[\begin{split}\begin{bmatrix} x_{\lambda} \\ y_{\lambda} \end{bmatrix} = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} \begin{bmatrix} x_{\lambda}-x_{pivot} \\ x_{\lambda}-x_{pivot} \end{bmatrix} + \begin{bmatrix} x_{pivot} \\ x_{pivot} \end{bmatrix}.\end{split}\]

7. The Detector Model, which contains the correct detectors metrology, is used to convert the \(x_{\lambda},y_{\lambda}\) position in mm into a pixel.