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diff --git a/noao/onedspec/doc/sys/specwcs.ms b/noao/onedspec/doc/sys/specwcs.ms new file mode 100644 index 00000000..a9d90a41 --- /dev/null +++ b/noao/onedspec/doc/sys/specwcs.ms @@ -0,0 +1,612 @@ +.EQ +delim $$ +gsize 10 +.EN +.nr PS 11 +.nr VS 13 +.de V1 +.ft CW +.ps -2 +.nf +.. +.de V2 +.fi +.ft R +.ps +2 +.. +.ND March 1993 +.TL +The IRAF/NOAO Spectral World Coordinate Systems +.AU +Francisco Valdes +.AI +IRAF Group - Central Computer Services +.K2 +.DY + +.AB +The image formats and world coordinate systems for dispersion calibrated +spectra used in the IRAF/NOAO spectroscopy packages are described; in +particular, the image header keywords defining the coordinates are given. +These keywords appear both as part of the IRAF image structure and map +directly to FITS format. The types of spectra include multidimensional +images with one or more spatial axes and a linear or log-linear dispersion +axis and special \fIequispec\fR and \fImultispec\fR formats having multiple +independent one dimensional spectra in a single multidimensional image. +The \fImultispec\fR format also includes general nonlinear dispersion +coordinate systems using polynomial, spline, sampled table, and look-up +table functions. +.AE + +.NH +Types of Spectral Data +.LP +Spectra are stored as one, two, or three dimensional images with one axis +being the dispersion axis. A pixel value is the flux over +some interval of wavelength and position. The simplest example of a +spectrum is a one dimensional image which has pixel values as a +function of wavelength. +.LP +There are two types of higher dimensional spectral image formats. One type +has spatial axes for the other dimensions and the dispersion axis may be +along any of the image axes. Typically this type of format is used for +long slit (two dimensional) and Fabry-Perot (three dimensional) spectra. +This type of spectra is referred to as \fIspatial\fR spectra and the +world coordinate system (WCS) format is called \fIndspec\fR. +The details of the world coordinate systems are discussed later. +.LP +The second type of higher dimensional spectral image consists of multiple, +independent, one dimensional spectra stored in the higher dimensions with +the first image axis being the dispersion axis; i.e. each line is a +spectrum. This format allows associating many spectra and related +parameters in a single data object. This type of spectra is referred to +as \fImultispec\fR and the there are two coordinate system formats, +\fIequispec\fR and \fImultispec\fR. The \fIequispec\fR format applies +to the common case where all spectra have the same linear dispersion +relation. The \fImultispec\fR format applies to the general case of spectra +with differing dispersion relations or non-linear dispersion functions. +These multi-spectrum formats are important since maintaining large numbers +of spectra as individual one dimensional images is very unwieldy for the +user and inefficient for the software. +.LP +Examples of multispec spectral images are spectra extracted from a +multi-fiber or multi-aperture spectrograph or orders from an echelle +spectrum. The second axis is some arbitrary indexing of the spectra, +called \fIapertures\fR, and the third dimension is used for +associated quantities. The IRAF \fBapextract\fR package may produce +multiple spectra from a CCD image in successive image lines with an +optimally weighted spectrum, a simple aperture sum spectrum, a background +spectrum, and sigma spectrum as the associated quantities along the third +dimension of the image. +.LP +Many \fBonedspec\fR package tasks which are designed to operate on +individual one dimensional spectra may operate on spatial spectra by +summing a number of neighboring spectra across the dispersion axis. This +eliminates the need to "extract" one dimensional spectra from the natural +format of this type of data in order to use tasks oriented towards the +display and analysis of one dimensional spectra. The dispersion axis is +either given in the image header by the keyword DISPAXIS or the package +\fIdispaxis\fR parameter. The summing factors across the +dispersion are specified by the \fInsum\fR package parameter. +.LP +One dimensional spectra, whether from multispec spatial spectra, have +several associated quantities which may appear in the image header as part +of the coordinate system description. The primary identification of a +spectrum is an integer aperture number. This number must be unique within +a single image. There is also an integer beam number used for various +purposes such as discriminating object, sky, and arc spectra in +multi-fiber/multi-aperture data or to identifying the order number in +echelle data. For spectra summed from spatial spectra the aperture number +is the central line, column, or band. In 3D images the aperture index +wraps around the lowest non-dispersion axis. Since most one dimensional +spectra are derived from an integration over one or more spatial axes, two +additional aperture parameters record the aperture limits. These limits +refer to the original pixel limits along the spatial axis. This +information is primarily for record keeping but in some cases it is used +for spatial interpolation during dispersion calibration. These values are +set either by the \fBapextract\fR tasks or when summing neighboring vectors +in spatial spectra. +.LP +An important task to be aware of for manipulating spectra between image +formats is \fBscopy\fR. This task allows selecting spectra from multispec +images and grouping them in various ways and also "extracts" apertures from +long slit and 3D spectra simply and without resort to the more general +\fBapextract\fR package. +.NH +World Coordinate Systems +.LP +IRAF images have three types of coordinate systems. The pixel array +coordinates of an image or image section, i.e. the lines and +columns, are called the \fIlogical\fR coordinates. The logical coordinates of +individual pixels change as sections of the image are used or extracted. +Pixel coordinates are tied to the data, i.e. are fixed to features +in the image, are called \fIphysical\fR coordinates. Initially the logical +and physical coordinates are the equivalent but differ when image sections +or other tasks which modify the sampling of the pixels are applied. +.LP +The last type of coordinate system is called the \fIworld\fR coordinate +system. Like the physical coordinates, the world coordinates are tied to +the features in the image and remain unchanged when sections of the image +are used or extracted. If a world coordinate system is not defined for an +image, the physical coordinate system is considered to be the world +coordinate system. In spectral images the world coordinate system includes +dispersion coordinates such as wavelengths. In many tasks outside the +spectroscopy packages, for example the \fBplot\fR, \fBtv\fR and +\fBimages\fR packages, one may select the type of coordinate system to be +used. To make plots and get coordinates in dispersion units for spectra +with these tasks one selects the "world" system. The spectral tasks always +use world coordinates. +.LP +The coordinate systems are defined in the image headers using a set of +reserved keywords which are set, changed, and updated by various tasks. +Some of the keywords consist of simple single values following the FITS +convention. Others, the WAT keywords, encode long strings of information, +one for each coordinate axis and one applying to all axes, into a set of +sequential keywords. The values of these keywords must then be pasted +together to recover the string. The long strings contain multiple pieces +called WCS \fIattributes\fR. In general the WCS keywords should be left to +IRAF tasks to modify. However, if one wants modify them directly some +tasks which may be used are \fBhedit\fR, \fBhfix\fR, \fBwcsedit\fR, +\fBwcsreset\fR, \fBspecshift\fR, \fBdopcor\fR, and \fBsapertures\fR. The +first two are useful for the simple keywords, the two "wcs" tasks are +useful for the linear ndspec and equispec formats, the next two are for the +common cases of shifting the coordinate zero point or applying a doppler +correction, and the last one is the one to use for the more complex +multispec format attributes. +.NH +Physical Coordinate System +.LP +The physical coordinate system is used by the spectral tasks when there is +no dispersion coordinate information (such as before dispersion +calibration), to map the physical dispersion axis to the logical dispersion +axis, and in the multispec world coordinate system dispersion functions +which are defined in terms of physical coordinates. +.LP +The transformation between logical and physical coordinates is defined by +the header keywords LTVi, LTMi_j (where i and j are axis numbers) through +the vector equation + +.EQ I + l vec~=~|m| cdot p vec + v vec +.EN + +where $l vec$ is a logical coordinate vector, $p vec$ is a physical +coordinate vector, $v vec$ is the origin translation vector specified by +the LTV keywords and $|m|$ is the scale/rotation matrix +specified by the LTM keywords. For spectra rotation terms (nondiagonal +matrix elements) generally do not make sense (in fact many tasks will not +work if there is a rotation) so the transformations along each axis are +given by the linear equation + +.EQ I + l sub i~=~LTMi_i cdot p sub i + LTVi. +.EN + +If all the LTM/LTV keywords are missing they are assumed to have zero +values except that the diagonal matrix terms, LTMi_i, are assumed to be 1. +Note that if some of the keywords are present then a missing LTMi_i will +take the value zero which generally causes an arithmetic or matrix +inversion error in the IRAF tasks. +.LP +The dimensional mapping between logical and physical axes is given by the +keywords WCSDIM and WAXMAP01. The WCSDIM keyword gives the dimensionality +of the physical and world coordinate system. There must be coordinate +information for that many axes in the header (though some may be missing +and take their default values). If the WCSDIM keyword is missing it is +assumed to be the same as the logical image dimensionality. +.LP +The syntax of the WAXMAP keyword are pairs of integer values, +one for each physical axis. The first number of each pair indicates which +current \fIlogical\fR axis corresponds to the original \fIphysical\fR axis +(in order) or zero if that axis is missing. When the first number is zero +the second number gives the offset to the element of the original axis +which is missing. As an example consider a three dimensional image in +which the second plane is extracted (an IRAF image section of [*,2,*]). +The keyword would then appear as WAXMAP01 = '1 0 0 1 2 0'. If this keyword +is missing the mapping is 1:1; i.e. the dimensionality and order of the +axes are the same. +.LP +The dimensional mapping is important because the dispersion axis for +the nspec spatial spectra as specified by the DISPAXIS keyword or task +parameter, or the axis definitions for the equispec and or multispec +formats are always in terms of the original physical axes. +.NH +Linear Spectral World Coordinate Systems +.LP +When there is a linear or logarithmic relation between pixels and +dispersion coordinates which is the same for all spectra the WCS header +format is simple and uses the FITS convention (with the CD matrix keywords +proposed by Hanisch and Wells 1992) for the logical pixel to world +coordinate transformation. This format applies to one, two, and three +dimensional data. The higher dimensional data may have either linear +spatial axes or the equispec format where each one dimensional spectrum +stored along the lines of the image has the same dispersion. +.LP +The FITS image header keywords describing the spectral world coordinates +are CTYPEi, CRPIXi, CRVALi, and CDi_j where i and j are axis numbers. As +with the physical coordinate transformation the nondiagonal or rotation +terms are not expected in the spectral WCS and may cause problems if they +are not zero. The CTYPEi keywords will have the value LINEAR to identify +the type of of coordinate system. The transformation between dispersion +coordinate, $w sub i$, and logical pixel coordinate, $l sub i$, along axis i is given by + +.EQ I + w sub i~=~CRVALi + CDi_i cdot (l sub i - CRPIXi) +.EN + +If the keywords are missing then the values are assumed to be zero except +for the diagonal elements of the scale/rotation matrix, the CDi_i, which +are assumed to be 1. If only some of the keywords are present then any +missing CDi_i keywords will take the value 0 which will cause IRAF tasks to +fail with arithmetic or matrix inversion errors. If the CTYPEi keyword is +missing it is assumed to be "LINEAR". +.LP +If the pixel sampling is logarithmic in the dispersion coordinate, as +required for radial velocity cross-correlations, the WCS coordinate values +are logarithmic and $w sub i$ (above) is the logarithm of the dispersion +coordinate. The spectral tasks (though not other tasks) will recognize +this case and automatically apply the anti-log. The two types of pixel +sampling are identified by the value of the keyword DC-FLAG. A value of 0 +defines a linear sampling of the dispersion and a value of 1 defines a +logarithmic sampling of the dispersion. Thus, in all cases the spectral +tasks will display and analyze the spectra in the same dispersion units +regardless of the pixel sampling. +.LP +Other keywords which may be present are DISPAXIS for 2 and 3 dimensional +spatial spectra, and the WCS attributes "system", "wtype", "label", and +"units". The system attribute will usually have the value "world" for +spatial spectra and "equispec" for equispec spectra. The wtype attribute +will have the value "linear". Currently the label will be either "Pixel" +or "Wavelength" and the units will be "Angstroms" for dispersion corrected +spectra. In the future there will be more generality in the units +for dispersion calibrated spectra. +.LP +Figure 1 shows the WCS keywords for a two dimensional long slit spectrum. +The coordinate system is defined to be a generic "world" system and the +wtype attributes and CTYPE keywords define the axes to be linear. The +other attributes define a label and unit for the second axis, which is the +dispersion axis as indicated by the DISPAXIS keyword. The LTM/LTV keywords +in this example show that a subsection of the original image has been +extracted with a factor of 2 block averaging along the dispersion axis. +The dispersion coordinates are given in terms of the \fIlogical\fR pixel +coordinates by the FITS keywords as defined previously. + +.DS +.ce +Figure 1: Long Slit Spectrum + +.V1 +WAT0_001= 'system=world' +WAT1_001= 'wtype=linear' +WAT2_001= 'wtype=linear label=Wavelength units=Angstroms' +WCSDIM = 2 +DISPAXIS= 2 +DC-FLAG = 0 + +CTYPE1 = 'LINEAR ' +LTV1 = -10. +LTM1_1 = 1. +CRPIX1 = -9. +CRVAL1 = 19.5743865966797 +CD1_1 = 1.01503419876099 + +CTYPE2 = 'LINEAR ' +LTV2 = -49.5 +LTM2_2 = 0.5 +CRPIX2 = -49. +CRVAL2 = 4204.462890625 +CD2_2 = 12.3337936401367 +.V2 +.DE + +Figure 2 shows the WCS keywords for a three dimensional image where each +line is an independent spectrum or associated data but where all spectra +have the same linear dispersion. This type of coordinate system has the +system name "equispec". The ancillary information about each aperture is +found in the APNUM keywords. These give the aperture number, beam number, +and extraction limits. In this example the LTM/LTV keywords have their +default values; i.e. the logical and physical coordinates are the same. + +.DS +.ce +Figure 2: Equispec Spectrum + +.V1 +WAT0_001= 'system=equispec' +WAT1_001= 'wtype=linear label=Wavelength units=Angstroms' +WAT2_001= 'wtype=linear' +WAT3_001= 'wtype=linear' +WCSDIM = 3 +DC-FLAG = 0 +APNUM1 = '41 3 7.37 13.48' +APNUM2 = '15 1 28.04 34.15' +APNUM3 = '33 2 43.20 49.32' + +CTYPE1 = 'LINEAR ' +LTM1_1 = 1. +CRPIX1 = 1. +CRVAL1 = 4204.463 +CD1_1 = 6.16689700000001 + +CTYPE2 = 'LINEAR ' +LTM2_2 = 1. +CD2_2 = 1. + +CTYPE3 = 'LINEAR ' +LTM3_3 = 1. +CD3_3 = 1. +.V2 +.DE +.NH +Multispec Spectral World Coordinate System +.LP +The \fImultispec\fR spectral world coordinate system applies only to one +dimensional spectra; i.e. there is no analog for the spatial type spectra. +It is used either when there are multiple 1D spectra with differing +dispersion functions in a single image or when the dispersion functions are +nonlinear. +.LP +The multispec coordinate system is always two dimensional though there may +be an independent third axis. The two axes are coupled and they both have +axis type "multispec". When the image is one dimensional the physical line +is given by the dimensional reduction keyword WAXMAP. The second, line +axis, has world coordinates of aperture number. The aperture numbers are +integer values and need not be in any particular order but do need to be +unique. This aspect of the WCS is not of particular user interest but +applications use the inverse world to physical transformation to select a +spectrum line given a specified aperture. +.LP +The dispersion functions are specified by attribute strings with the +identifier \fIspecN\fR where N is the \fIphysical\fR image line. The +attribute strings contain a series of numeric fields. The fields are +indicated symbolically as follows. + +.EQ I + specN~=~ap~beam~dtype~w1~dw~nw~z~aplow~aphigh~[functions sub i ] +.EN + +where there are zero or more functions having the following fields, + +.EQ I + function sub i~=~ wt sub i~w0 sub i~ftype sub i~[parameters]~[coefficients] +.EN + +The first nine fields in the attribute are common to all the dispersion +functions. The first field of the WCS attribute is the aperture number, +the second field is the beam number, and the third field is the dispersion +type with the same function as DC-FLAG in the \fInspec\fR and +\fIequispec\fR formats. A value of -1 indicates the coordinates are not +dispersion coordinates (the spectrum is not dispersion calibrated), a value +of 0 indicates linear dispersion sampling, a value of 1 indicates +log-linear dispersion sampling, and a value of 2 indicates a nonlinear +dispersion. +.LP +The next two fields are the dispersion coordinate of the first +\fIphysical\fR pixel and the average dispersion interval per \fIphysical\fR +pixel. For linear and log-linear dispersion types the dispersion +parameters are exact while for the nonlinear dispersion functions they are +approximate. The next field is the number of valid pixels, hence it is +possible to have spectra with varying lengths in the same image. In that +case the image is as big as the biggest spectrum and the number of pixels +selects the actual data in each image line. The next (seventh) field is a +doppler factor. This doppler factor is applied to all dispersion +coordinates by multiplying by $1/(1+z)$ (assuming wavelength dispersion +units). Thus a value of 0 is no doppler correction. The last two fields +are extraction aperture limits as discussed previously. +.LP +Following these fields are zero or more function descriptions. For linear +or log-linear dispersion coordinate systems there are no function fields. +For the nonlinear dispersion systems the function fields specify a weight, +a zero point offset, the type of dispersion function, and the parameters +and coefficients describing it. The function type codes, $ftype sub i$, +are 1 for a chebyshev polynomial, 2 for a legendre polynomial, 3 for a +cubic spline, 4 for a linear spline, 5 for a pixel coordinate array, and 6 +for a sampled coordinate array. The number of fields before the next +function and the number of functions are determined from the parameters of +the preceding function until the end of the attribute is reached. +.LP +The equation below shows how the final wavelength is computed based on +the $nfunc$ individual dispersion functions $W sub i (p)$. Note that this +is completely general in that different function types may be combined. +However, in practice when multiple functions are used they are generally of +the same type and represent a calibration before and after the actual +object observation with the weights based on the relative time difference +between the calibration dispersion functions and the object observation. + +.EQ I +w~=~sum from i=1 to nfunc {wt sub i cdot (w0 sub i + W sub i (p)) / (1 + z)} +.EN + +The multispec coordinate systems define a transformation between physical +pixel, $p$, and world coordinates, $w$. Generally there is an intermediate +coordinate system used. The following equations define these coordinates. +The first one shows the transformation between logical, $l$, and physical, +$p$, coordinates based on the LTM/LTV keywords. The polynomial functions +are defined in terms of a normalized coordinate, $n$, as shown in the +second equation. The normalized coordinates run between -1 and 1 over the +range of physical coordinates, $p sub min$ and $p sub max$ which are +parameters of the function, upon which the coefficients were defined. The +spline functions map the physical range into an index over the number of +evenly divided spline pieces, $npieces$, which is a parameter of the +function. This mapping is shown in the third and fourth equations where +$s$ is the continuous spline coordinate and $j$ is the nearest integer less +than or equal to $s$. + +.EQ I + p mark~=~(l - LTV1) / LTM1_1 +.EN +.EQ I + n lineup~=~(p - p sub middle ) / (2 cdot p sub range ) +.EN +.EQ I + lineup~=~(p - (p sub max + p sub min )/2) / (2 cdot (p sub max - p sub min )) +.EN +.EQ I + s lineup~=~(p - p sub min ) / (p sub max - p sub min ) cdot npieces +.EN +.EQ I + j lineup~=~roman "int" (s) +.EN +.NH 2 +Linear and Log Linear Dispersion Function +.LP +The linear and log-linear dispersion functions are described by a +wavelength at the first \fIphysical\fR pixel and a wavelength increment per +\fIphysical\fR pixel. A doppler correction may also be applied. The +equations below show the two forms. Note that the coordinates returned are +always wavelength even though the pixel sampling and the dispersion +parameters may be log-linear. + +.EQ I + w mark~=~(w1 + dw cdot (p - 1)) / (1 + z) +.EN +.EQ I + w lineup~=~10 sup {(w1 + dw cdot (p - 1)) / (1 + z)} +.EN + +Figure 3 shows an example from a multispec image with +independent linear dispersion coordinates. This is a linearized echelle +spectrum where each order (identified by the beam number) is stored as a +separate image line. + +.DS +.ce +Figure 3: Echelle Spectrum with Linear Dispersion Function + +.V1 +WAT0_001= 'system=multispec' +WAT1_001= 'wtype=multispec label=Wavelength units=Angstroms' +WAT2_001= 'wtype=multispec spec1 = "1 113 0 4955.44287109375 0.05... +WAT2_002= '5 256 0. 23.22 31.27" spec2 = "2 112 0 4999.0810546875... +WAT2_003= '58854293 256 0. 46.09 58.44" spec3 = "3 111 0 5043.505... +WAT2_004= '928358078002 256 0. 69.28 77.89" +WCSDIM = 2 + +CTYPE1 = 'MULTISPE' +LTM1_1 = 1. +CD1_1 = 1. + +CTYPE2 = 'MULTISPE' +LTM2_2 = 1. +CD2_2 = 1. +.V2 +.DE +.NH 2 +Chebyshev Polynomial Dispersion Function +.LP +The parameters for the chebyshev polynomial dispersion function are the +$order$ (number of coefficients) and the normalizing range of physical +coordinates, $p sub min$ and $p sub max$, over which the function is +defined and which are used to compute $n$. Following the parameters are +the $order$ coefficients, $c sub i$. The equation below shows how to +evaluate the function using an iterative definition where $x sub 1 = 1$, +$x sub 2 = n$, and $x sub i = 2 cdot n cdot x sub {i-1} - x sub {i-2}$. + +.EQ I + W~=~sum from i=1 to order {c sub i cdot x sub i} +.EN +.NH 2 +Legendre Polynomial Dispersion Function +.LP +The parameters for the legendre polynomial dispersion function are the +order (number of coefficients) and the normalizing range of physical +coordinates, pmin and pmax, over which the function is defined +and which are used to compute n. Following the parameters are the +order coefficients, c sub i. The equation below shows how to evaluate the +function using an iterative definition where $x sub 1 = 1$, $x sub 2 = n$, and +$x sub i = ((2i-3) cdot n cdot x sub {i-1} - (i-2) cdot x sub {i-2}) / (i-1)$. + +.EQ I + W~=~sum from i=1 to order {c sub i cdot x sub i} +.EN +.LP +Figure 4 shows an example from a multispec image with independent nonlinear +dispersion coordinates. This is again from an echelle spectrum. Note that +the IRAF \fBechelle\fR package determines a two dimensional dispersion +function, in this case a bidimensional legendre polynomial, with the +independent variables being the order number and the extracted pixel +coordinate. To assign and store this function in the image is simply a +matter of collapsing the two dimensional dispersion function by fixing the +order number and combining all the terms with the same order. + +.DS +.ce +Figure 4: Echelle Spectrum with Legendre Polynomial Function + +.V1 +WAT0_001= 'system=multispec' +WAT1_001= 'wtype=multispec label=Wavelength units=Angstroms' +WAT2_001= 'wtype=multispec spec1 = "1 113 2 4955.442888635351 0.05... +WAT2_002= '83 256 0. 23.22 31.27 1. 0. 2 4 1. 256. 4963.0163112090... +WAT2_003= '976664 -0.3191636898579552 -0.8169352858733255" spec2 =... +WAT2_004= '9.081188912082 0.06387049476832223 256 0. 46.09 58.44 1... +WAT2_005= '56. 5007.401409453303 8.555959076467951 -0.176732458267... +WAT2_006= '09935064388" spec3 = "3 111 2 5043.505764869474 0.07097... +WAT2_007= '256 0. 69.28 77.89 1. 0. 2 4 1. 256. 5052.586239197408 ... +WAT2_008= '271 -0.03173489817897474 -7.190562320405975E-4" +WCSDIM = 2 + +CTYPE1 = 'MULTISPE' +LTM1_1 = 1. +CD1_1 = 1. + +CTYPE2 = 'MULTISPE' +LTM2_2 = 1. +CD2_2 = 1. +.V2 +.DE +.NH 2 +Linear Spline Dispersion Function +.LP +The parameters for the linear spline dispersion function are the number of +spline pieces, $npieces$, and the range of physical coordinates, $p sub min$ +and $p sub max$, over which the function is defined and which are used to +compute the spline coordinate $s$. Following the parameters are the +$npieces+1$ coefficients, $c sub i$. The two coefficients used in a linear +combination are selected based on the spline coordinate, where $a$ and $b$ +are the fractions of the interval in the spline piece between the spline +knots, $a=(j+1)-s$, $b=s-j$, and $x sub 0 =a$, and $x sub 1 =b$. + +.EQ I + W~=~sum from i=0 to 1 {c sub (i+j) cdot x sub i} +.EN +.NH 2 +Cubic Spline Dispersion Function +.LP +The parameters for the cubic spline dispersion function are the number of +spline pieces, $npieces$, and the range of physical coordinates, $p sub min$ +and $p sub max$, over which the function is defined and which are used +to compute the spline coordinate $s$. Following the parameters are the +$npieces+3$ coefficients, $c sub i$. The four coefficients used are +selected based on the spline coordinate. The fractions of the interval +between the integer spline knots are given by $a$ and $b$, $a=(j+1)-s$, +b=$s-j$, and $x sub 0 =a sup 3$, $x sub 1 =(1+3 cdot a cdot (1+a cdot b))$, +$x sub 2 =(1+3 cdot b cdot (1+a cdot b))$, and $x sub 3 =b sup 3$. + +.EQ I + W~=~sum from i=0 to 3 {c sub (i+j) cdot x sub i} +.EN +.NH 2 +Pixel Array Dispersion Function +.LP +The parameters for the pixel array dispersion function consists of just the +number of coordinates $ncoords$. Following this are the wavelengths at +integer physical pixel coordinates starting with 1. To evaluate a +wavelength at some physical coordinate, not necessarily an integer, a +linear interpolation is used between the nearest integer physical coordinates +and the desired physical coordinate where $a$ and $b$ are the usual +fractional intervals $k= roman "int" (p)$, $a=(k+1)-p$, $b=p-k$, +and $x sub 0 =a$, and $x sub 1 =b$. + +.EQ I + W~=~sum from i=0 to 1 {c sub (i+j) cdot x sub i} +.EN +.NH 2 +Sampled Array Dispersion Function +.LP +The parameters for the sampled array dispersion function consists of +the number of coordinate pairs, $ncoords$, and a dummy field. +Following these are the physical coordinate and wavelength pairs +which are in increasing order. The nearest physical coordinates to the +desired physical coordinate are located and a linear interpolation +is computed between the two sample points. |