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diff --git a/noao/onedspec/doc/specwcs.hlp b/noao/onedspec/doc/specwcs.hlp new file mode 100644 index 00000000..ed8852e3 --- /dev/null +++ b/noao/onedspec/doc/specwcs.hlp @@ -0,0 +1,586 @@ +.help specwcs Mar93 noao.onedspec + +.ce +\fBThe IRAF/NOAO Spectral World Coordinate Systems\fR + + +.sh +1. Types of Spectral Data + +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. + +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. + +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. + +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. + +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. +See "help onedspec.package" for information on these parmaeters. + +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. + +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. +.sh +2. World Coordinate Systems + +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. + +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. + +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. +.sh +3. Physical Coordinate System + +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. + +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 + +.nf + l = |m| * p + v +.fi + +where l is a logical coordinate vector, p is a physical +coordinate vector, v 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 + +where l is a logical coordinate vector, p is a physical coordinate vector, +v is the origin translation vector specified by the LTV keywords and |m| is +the scale/rotation matrix specified by the LTM keywords. For spectra a +rotation term (nondiagonal matrix elements) generally does 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 + +.nf + li = LTMi_i * pi + LTVi. +.fi + +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. + +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. + +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. + +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. +.sh +4. Linear Spectral World Coordinate Systems + +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. + +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 coordinate system. The transformation between dispersion +coordinate, wi, and logical pixel coordinate, li, along axis i is given by + +.nf + wi = CRVALi + CDi_i * (li - CRPIXi) +.fi + +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". + +If the pixel sampling is logarithmic in the dispersion coordinate, as +required for radial velocity cross-correlations, the WCS coordinate values +are logarithmic and wi (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. + +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. + +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. + +.ce +Figure 1: Long Slit Spectrum + +.nf + 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 +.fi + +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. + +.ce +Figure 2: Equispec Spectrum + +.nf + 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. +.fi +.sh +5. Multispec Spectral World Coordinate System + +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. + +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. + +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. + +.nf + specN = ap beam dtype w1 dw nw z aplow aphigh [functions_i] +.fi + +where there are zero or more functions having the following fields, + +.nf + function_i = wt_i w0_i ftype_i [parameters] [coefficients] +.fi + +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. + +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. + +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_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. + +The equation below shows how the final wavelength is computed based on +the nfunc individual dispersion functions W_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. + +.nf + w = sum from i=1 to nfunc {wt_i * (w0_i + W_i(p)) / (1 + z)} +.fi + +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, pmin and pmax 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. + +.nf + p = (l - LTV1) / LTM1_1 + n = (p - pmiddle) / (prange / 2) + = (p - (pmax+pmin)/2) / ((pmax-pmin) / 2) + s = (p - pmin) / (pmax - pmin) * npieces + j = int(s) +.fi +.sh +5.1 Linear and Log Linear Dispersion Function + +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. + +.nf + w = (w1 + dw * (p - 1)) / (1 + z) + w = 10 ** {(w1 + dw * (p - 1)) / (1 + z)} +.fi + +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. + +.ce +Figure 3: Echelle Spectrum with Linear Dispersion Function + +.nf + 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. +.fi +.sh +5.2 Chebyshev Polynomial Dispersion Function + +The parameters for the chebyshev 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, ci. The equation below shows how to +evaluate the function using an iterative definition where x_1 = 1, +x_2 = n, and x_i = 2 * n * x_{i-1} - x_{i-2}. + +The parameters for the chebyshev 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_i. The equation below shows how to evaluate the +function using an iterative definition +where x_1 = 1, x_2 = n, and x_i = 2 * n * x_{i-1} - x_{i-2}. + +.nf + W = sum from i=1 to order {c_i * x_i} +.fi +.sh +5.3 Legendre Polynomial Dispersion Function + +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_i. The equation below shows how to evaluate the +function using an iterative definition where x_1 = 1, x_2 = n, and +x_i = ((2i-3)*n*x_{i-1}-(i-2)*x_{i-2})/(i-1). + +.nf + W = sum from i=1 to order {c_i * x_i} +.fi + +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. + +.ce +Figure 4: Echelle Spectrum with Legendre Polynomial Function + +.nf + 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. +.fi +.sh +5.4 Linear Spline Dispersion Function + +The parameters for the linear spline dispersion function are the number of +spline pieces, npieces, and the range of physical coordinates, pmin +and pmax, 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_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_0=a, and x_1=b. + +.nf + W = sum from i=0 to 1 {c_(i+j) * x_i} +.fi +.sh +5.5 Cubic Spline Dispersion Function + +The parameters for the cubic spline dispersion function are the number of +spline pieces, npieces, and the range of physical coordinates, pmin +and pmax, 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_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_0 =a sup 3, x_1 =(1+3*a*(1+a*b)), +x_2 =(1+3*b*(1+a*b)), and x_3 =b**3. + +The parameters for the cubic spline dispersion function are the number of +spline pieces, npieces, and the range of physical coordinates, pmin +and pmax, 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_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_0=a**3, x_1=(1+3*a*(1+a*b)), x_2=(1+3*b*(1+a*b)), and x_3=b**3. + +.nf + W = sum from i=0 to 3 {c_(i+j) * x_i} +.fi +.sh +5.6 Pixel Array Dispersion Function + +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=int(p), a=(k+1)-p, b=p-k, +and x_0=a, and x_1=b. + +.nf + W = sum from i=0 to 1 {c_(i+j) * x_i} +.fi +.sh +5.7 Sampled Array Dispersion Function + +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. +.endhelp |