Jerome Workman, Jr., and Howard Mark

-axis (photometric axis) is a helpful tool for all spectra when used for both quantitative and qualitative measurements using spectroscopy. The assumptions of regression are that the data channels or 

-axis of data is aligned, and that only amplitude in the 

-axis changes with concentration in some mathematically defined manner. An aligned 

-axis provides an improved method to compare spectra, and to relate amplitude to concentration. The correction of 

-axis (amplitude) variation provides a minimum bias between instruments, and provides a platform for improved calibration or method transfer between instruments. This article provides a discussion for using photometric reference materials to align the 

-axis for ultraviolet, visible, near infrared, infrared, and Raman intensity spectroscopic methods. A general method for correcting each type of absorption-based spectrophotometer is given, as well as the types of reference standards used for calibration or verification of the photometric scale. For Raman spectroscopy, a method is described for intensity correction.

 is a formal term used to describe how well a measuring device (spectrophotometer or illuminance meter) is able to determine the total energy flux transmitted through, reflected from, or scattered from a standard device or standard reference material (SRM). Part I of this series addressed the use of wavelength or wavenumber standards to align the 

-axis for spectroscopic data (1); this Part II article addresses the photometric (or 

-) axis alignment. The various references to National Institute of Standards and Technology (NIST) documents and SRMs are given throughout this article. Note that equations are described as referenced in (2,3) and that SRM data may be found at the NIST website (

	Volume 14.01 of the American Society for Testing and Materials (ASTM) International document (4,5), describes the measurement of the photometric accuracy for an instrument as follows (Sections 20.1 and 20.3):

“Select the appropriate Standard Reference Material and obtain ten successive readings of the apparent absorbance or transmittance at the specified wavelength. Average the ten readings. The photometric accuracy is the difference between the true absorbance or transmittance value and the average observed value.”

	It is further stated that the following apply to the reporting of photometric accuracy:

“Report the photometric accuracy in the following order: reference material, wavelength, true absorbance or transmittance, observed absorbance or transmittance plus or minus the standard deviation.”

Example of Reporting Photometric Accuracy

	As an example the stated specifications for photometric accuracy for a test instrument might be written as:

	±0.002 Au (for range of 0.0 to 0.5 Absorbance units)

	±0.004 Au (for range of 0.5 to 1.0 Absorbance units)

	Here is an evaluation of these specifications in greater detail. A photometric accuracy of "0.002 Au at 0 to 0.5 Au" indicates that, after the instrument has been set to zero, a standard sample that transmits between 30.1% T (0.5 Au) and 100% T (0.0 Au) measures within ±0.002 Au, as compared to the specified value for that reference standard material at any wavelength position (we note that 

	The tolerance for each of the 930d filters has been stated by NIST as ±0.5 percent relative. Thus, a 10% T filter, which corresponds to a measurement of 0.1 T or 1.0 Au for neutral density 930d-type reference standard, would be expected to be within the range of 9.95 to 10.05% T. These values are equivalent to 1.0022 Au to 0.9978 Au. The difference in Au is 1.0022 Au minus 0.9978 Au equals 0.0044 Au. The standard tolerance in Au is 1.0000 ±0.0022 Au, rounded to 1.000 ±0.002 Au. Conservatively, for transmittance, we state 10.00 ±0.03% T; and related to 0.0 Au or 100% T, we specify 100.0 ±0.3% T.

	If the stated tolerance for the NIST 930d filter is ±0.002 Au at 1.0 Au, we are stating that our maximum deviation in photometric accuracy for our instrument at 1.0 Au is also ±0.002 Au. For the worst case measurement for the NIST filter, we would have the filter error of ±0.002 added to the instrument error of ±0.002, to result in a total maximum variation ±0.004 Au error (this is our stated specification for the example given).

	And one can readily calculate the reflectance or transmittance of any measurement given the absorbance of that measurement as follows. For conversion of absorbance (

), we use equations 1–4. (Note that the relationships between transmittance and absorbance are given in Table I.)

Photometric Correction for Absorbance-based Spectrophotometers

	For making photometric calibrations using known reference materials, the ratioed replicate spectral data are retained, along with the replicate spectra of a reference sample and dark background, or dark sample. Equation 9 represents Beer's law for taking measurements using a spectrophotometer. To convert the light reflected or transmitted from a sample (

), as a linearized estimate of the spectral response as related to analyte concentration, the following equations 9-13 may be applied.

	If only an internal reference is used for spectral collection, then the absorbance spectrum with respect to wavelength is computed as equation 9:

	If an external standard material is used to calibrate an automated internal reference material, then the absorbance spectrum with respect to wavelength is computed as equations 10 and 11:

 is the dark measurement for the sample, 

 is the dark measurement for the internal reference sample, 

 is the measurement of the external standard reference material for calibration, 

 is the dark measurement for the external standard reference material, and 

 is the ratio between the dark corrected 

. The final spectrum is simplified to the ratio of (internal reference corrected by the external reference minus dark) divided by (sample minus dark). Note that dark is a measurement where no energy from the source is allowed to the detector (the source energy is blocked), or where a totally dark sample or light trap is measured. The dark measurement represents the dark current changes within the instrument during a measurement that relate to electronic noise where the detection electronics, computational electronics, and control electronics perform basic measurement functions with no imputed energy reaching the detector (the source is turned off or blocked using a shutter closure during the dark measurement).

	When a comprehensive method of photometric correction includes a light trap measurement to determine the dark response to correct for optical and window scatter, we use equation 12:

 as the photometric correction calibration measurement of the internal reference sample, 

 is the photometric correction calibration measurement of the dark signal intensity for the internal reference sample, 

 is the intensity measurement of the light trap to determine the window scatter, and 

 is the measurement of the dark signal intensity for the light trap measurement.

	Equation 12 is simplified for understanding by removing the dark correction terms as equation 13:

	There are several available standards used for calibrating or verifying photometric correction when using ultraviolet spectrophotometers, with potassium dichromate demonstrated here. The ultraviolet region is generally considered to include from 190 nm to 380 nm (non-vacuum UV region). The photometric values for UV standards are illustrated in Figures 1-2, and the reference absorbance values are given in the accompanying Table II (reference spectra and data provided with permission by Starna Ltd.).


	Figure 1: Potassium dichromate absorbance and linearity standard traceable to SRM 935a. (Data and spectrum provided with permission by Starna Ltd., Hainault, IG6 3UT, UK. www.starna.com).


	Figure 2: Potassium dichromate absorbance linearity as a photometric standard using twelve concentrations of standard and four specific wavelength measurements (257, 235, 350, and 313 nm) (Table II from 20–200 mg/L). This is a test for photometric linearity throughout the ultraviolet region. (Data and spectrum provided with permission by Starna Ltd., Hainault, IG6 3UT, UK. www.starna.com).

	The visible spectral region is generally considered over the range of 380 nm to 750 nm. Two types of available standards are shown, namely NIST SRM 930d and Russian opal glass, which are used for calibrating or verifying photometric alignment of visible spectrophotometers. Photometric values for these standards are illustrated in Figures 3-4, and the reference absorbance values are given in the accompanying Tables III and IV.


	Figure 3: Transmittance of SRM-930d standard set photometric standards; Filter 10-1225, Filter 20-1255, and Filter 30-1255; data in Table III. (Data and spectrum used with permission from Avian Technologies, New London, NH).


	Figure 4: Russian opal glass glossy reflectance standard measured using 8°/hemispherical spectral reflectance factor geometry. Mean reflectance over the spectral region shown is 0.967. Measured photometric data is given in the Table IV. (Data and spectrum used with permission from Avian Technologies, New London, NH).

Near Infrared (NIR) Reflectance Photometric Standards

	The near infrared spectral region is generally considered to be 780 nm to 2500 nm. Two types of available reflectance standards are shown, namely R50 (~50% reflectance), and sintered Fluorilon (PTFE) (R99 or ~99% reflectance), which are used for calibrating or verifying photometric alignment of NIR spectrophotometers. Photometric values for these standards are illustrated in Figures 5 and 6, and the reference reflectance values are given in the accompanying Tables V and VI. (materials noted provided by courtesy by Avian Technologies). Note that in transmittance measurement mode various solvents or dry air may be used to calibrate or verify the transmittance accuracy to zero (0% T).


	Figure 5: R50 Fluorilon and carbon black mixture as a ~50% reflectance standard; data in Table V. (Spectrum used with permission from Avian Technologies, New London, NH).


	Figure 6: R99 Fluorilon as a ~99% reflectance standard; data in Table VI. (Spectrum used with permission from Avian Technologies, New London, NH).

Infrared (NIR) Reflectance Photometric Standards

	The infrared spectral region is generally considered to be defined as 2500 nm to 25,000 nm (4000 cm

). Diffuse gold is a primary surface material used for calibrating or verifying reflectance photometric alignment of infrared spectrophotometers. Photometric values for this standard are illustrated in Figures 7-8, and the reference reflectance values are given in the accompanying Table VII. In transmittance measurement mode dry, purged air may be used to calibrate or verify the transmittance accuracy.


	Figure 7: Diffuse gold photometric reflectance standard measured using 8°/hemispherical spectral reflectance factor geometry with a calibrated FT-IR instrument. Mean reflectance over the spectral wavelength region (in micrometers) shown is 0.941. Measured photometric data is given in Table VII. (Data and spectrum used with permission from Avian Technologies, New London, NH).


	Figure 8: Diffuse gold photometric reflectance standard measured using 8°/hemispherical spectral reflectance factor geometry with a calibrated FT-IR instrument. Mean reflectance over the spectral region (in wavenumbers) shown is 0.941. Measured photometric data is given in Table VII. (Data and spectrum used with permission from Avian Technologies, New London, NH).

	The Raman spectrum corresponding to the infrared vibrational region includes 4000 cm

, with Raman spectra referred to as the Stokes shift region defined from 3500 cm

. A variety of materials are used for calibrating or verifying the intensity scale for Raman spectrometers. Materials for this purpose are listed in Table VIII. Materials noted are provided by NIST.

	NIST, previously known as the National Bureau of Standards (NBS), provides materials designed as reference sources to verify the performance characteristics of Raman spectrometers of typical instrument design. The main standards are currently used for intensity standardization. Table VIII lists the typical materials available for Raman intensity calibration. In the case of Raman, there are five active NIST SRMs for measuring Raman intensity, depending upon the excitation laser wavelength used (6–11).

	The Raman intensity standards are certified SRMs useful for the correction of the relative intensity of Raman spectra obtained with instruments employing specific laser excitation sources. Individual intensity standards have been provided for Raman excitation wavelengths of 488 nm, 532 nm, 633 nm, 785 nm, 830 nm, and 1064 nm. For these standards, the relative intensity of the glass luminescence has been calibrated at NIST using a uniform- white-light-source, with an integrating sphere collection design. The shape of the mean luminescence spectrum of the SRM glass is described using a polynomial expression for relative spectral intensity versus Raman shift wavenumber (cm

) based on the specific excitation laser wavelength used (in nanometers).

	The spectral correction for any Raman spectrometer is determined by measuring the luminescence spectrum of the SRM, applying the polynomial model, and computing the spectral intensity-response correction for any Raman instrument. The application of the spectral intensity correction eliminates the instrument induced spectral artifacts to provide a more uniform calibrated Raman spectrum.

	In order for a Raman spectrometer to be calibrated for intensity the Raman wavenumber axis is corrected using the ASTM E1840-96 protocol (12). The laser excitation is aimed at the frosted surface of the glass, to minimize variation in the scattering response. The intensity correction must be completed over the same Raman shift range as that intended for sample measurement.

	The relative intensity of the measured Raman spectrum of the sample can be corrected for the instrument-specific response by a computational procedure that uses a correction curve. This curve is generated using the certified model and the measured luminescence spectrum of the SRM glass. For the spectral range of certification, 

, the elements of the certified relative mean spectral intensity of SRM 2241, 

), are computed according to equation 14:

) is the wavenumber value in units of Raman shift (cm

's are the coefficients listed in Table IX (14–20). The elements of 

) are obtained by evaluating equation 14 at the same data point spacing used for the acquisition of the luminescence spectrum of the SRM and of the Raman spectrum of the sample. 

) has been normalized to unity and is a relative unit expressed in terms of photons. The data sets that are the measured glass luminescence spectrum, 

, and the measured Raman spectrum of the sample, 

), defined by equation 15, are obtained from ISRM(

) and the elements of the glass luminescence spectrum, 

	The elements of the intensity-corrected Raman spectrum, 

), are derived by multiplication of the elements of the measured Raman spectrum of the sample, 

), by the elements of the correction curve as equation 16 (13).

	The certified model, equation 14, is certified for use between 200 cm

. The certified model is intended as a simple numerical descriptor of the spectral response observed over the wavenumber range studied. It is not intended as a physically meaningful model. The model coefficients listed in Table IX cannot be used to extrapolate the limits of certification without incurring significant error. Extrapolation of the model outside the certification limits of 200 cm

 is not a supported by use of this SRM. This SRM is not intended for use as a standard for the determination of absolute spectral irradiance or radiance.

	The equation describing the mean luminescence spectrum of the glass SRM is given in equation 14, where (

) is the wavenumber in units of Raman shift (cm

-axis is in wavelength with units of nanometers, the same model coefficients can be used to calculate 

) through the following coordinate transformation as equations 17 and 18:

 is the wavelength of the laser in nanometers and 

 is the spectral wavelength in nanometers. The prefactor of 10

 in the first term of equation 17 is needed only if it is desired to preserve the numerical value of spectral areas computed relative to the two 

-axis coordinate systems. Definition of this is from reference (7), with the NIST basic application of the intensity standard.

	A comprehensive paper describing the complete process of calibrating Raman instruments for intensity using various laser sources and instrument types along with the application of NIST SRMs is given in reference 20 as an 

 feature article. This paper is recommended for anyone that would practice Raman intensity correction for analytical measurements, and for understanding the details of this process.

The Concise Handbook of Analytical Spectroscopy: Physical Foundations, Techniques, Instrumentation and Data Analysis, 

J. Workman, Ed. (World Scientific Publishing-Imperial College Press, Singapore, 2016). In 5 Volumes, UV, Vis, NIR, IR, and Raman Volumes 1, pp. 163-214; Volume 2, pp. 217-278; Volume 3, pp. 363-424; Volume 4, pp. 323-360; Volume 5, pp. 229-250. ISBN-13: 978-9814508056, 2016). (Permission from NIST for data and spectra related to SRM materials).

	(3) Each table refers to specific NIST SRM datasheets and published references and as referred to in reference (2).

Standard Practice for Describing and Measuring Performance of Ultraviolet, Visible, and Near-Infrared Spectrophotometers. American Society for Testing and Materials, 

ASTM International designation E 275-08(2013), Philadelphia, PA (2013).

Standard Terminology Relating to molecular Spectroscopy; American Society for Testing and Material

s, ASTM International designation E 131-10, 2010, Philadelphia, PA (2010).

http://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.260-176.pdf

, pp. 162–163 (accessed Sept. 9, 2017).

	(7) Standard Reference Material (SRM) 2241 Certificate, 

Relative Intensity Correction Standard for Raman Spectroscopy: 785 nm Excitation

, National Institute of Standards and Technology, Gaithersburg, MD 20899, 6 pages (28 February 2014).

	(8) Standard Reference Material (SRM) 2242 Certificate, 

Relative Intensity Correction Standard for Raman Spectroscopy: 532 nm Excitation

, National Institute of Standards and Technology, Gaithersburg, MD 20899, 6 pages (22 October 2013).

Articles

Alignment of the instrument y-axis is a critical step for quantitative and qualitative measurements using spectroscopy. Here, we explain in detail how to use photometric standards for ultraviolet, visible, near infrared, infrared, and Raman spectroscopy.

Using Reference Materials, Part II: Photometric Standards

Calibration transfer involves multiple strategies and mathematical techniques for applying a single calibration database consisting of samples, reference data, and calibration equations to two or more instruments. The instruments used for initial calibration development and transfer may be of like or meaningfully different optical designs. In a two-part series of articles, we take a broad overview of the chemometric and tactical strategies used for the calibration transfer process. Here in part I, the emphasis is on methods that modify the spectra or regression vectors to correct differences between instruments involved in the calibration transfer process. Part II will review calibration transfer mathematics and approaches.

	Calibration transfer in its purest form allows a calibration to be moved from one instrument to another with statistically retained accuracy and precision (1,2). Ideally, a model developed on one instrument would provide a statistically identical analysis when used on other instruments, without bias or slope adjustments, without additional product samples for standardization, and without product recalibration.

	A variety of textbooks and papers exist describing the general methods and history of calibration transfer. Among these are references describing the details of the background, chemometrics, types of instruments, and differences in sample chemistry (3,4). One comprehensive review of calibration transfer presents an overview of the different mathematical methods used for calibration transfer as well as a critical assessment of their validity and applicability (5). The focus of this review was on a description of methods used for transfer of calibrations for near-infrared (NIR) spectra (5).

	We note there are at least four basic strategies that can be used before the calibration modeling step for calibration transfer:

		instrument matching or making instruments truly alike,

		global modeling or developing models that include varying instrumental parameters, sample preparation, and environmental factors in the data before computing the calibration method,

		model updating, which adds samples from each transfer instrument before computing a revised calibration, and

		sensor selection or finding regression variables that are somewhat insensitive to variation in measurement conditions, yet remain sensitive to the analyte signal.

	Most of the methods discussed in this column refer to making one set of spectral responses from one instrument similar to a second set of responses from a second instrument for transfer; these methods would generally be categorized as instrument matching or standardization methods. 

	Another prominent review of calibration transfer methods describes multiple chemometric techniques used (6). In this review paper, calibration transfer methods are classified as being represented by three main technical approaches:

		by developing more robust calibrations, derived by combining data from multiple instruments and by using spectral pretreatments of that data,

		by modifying calibrations using bias and slope adjustments (the traditional method), and

		by adjusting spectra, using such methods as piecewise direct standardization (PDS) combined with a patented method that requires spectra from transfer sample sets measured on each instrument involved in the transfer experiment (7).

	Again, most of the methods described within this column refer to adjusting spectra or coefficients to match instruments.

	A comprehensive book chapter presents a review of calibration transfer methods and a critical assessment of the more common methods in terms of applicability and capabilities (8). This chapter focuses on methods for calibration transfer using NIR spectra. It is noted that the methodology described is widely applicable to other analytical measurement techniques (8). Comprehensive reviews of chemometric methods, including calibration transfer, have been published since 1980 as part of the 

Fundamental Reviews in Analytical Chemistry

 (9). One of the most cited of those fundamental chemometrics reviews delves into multiple topics related to calibration methods and calibration transfer techniques (10).

	A popular and patented technique for transferring calibration models from a reference analytical instrument to a target analytical instrument is demonstrated as a U.S. patent (11). For this technique, a set of diverse transfer samples are measured on a reference instrument, producing a reference instrument response for each sample. These measurements are repeated for the target instrument, to produce a target instrument response for each transfer sample. One then generates transfer coefficients capable of performing a multivariate estimation of the reference instrument responses for the transfer samples from the target instrument responses for those samples. The transfer coefficients may then be used to convert a target instrument response for an unknown sample into the equivalent response for the reference instrument. The patent describes piecewise, classical, and inverse transfer techniques. The target instrument responses for the transfer samples are combined with reference instrument responses for the full set of calibration samples, to derive a multivariate prediction model for the target instrument (11).

	A research paper describes the transfer of calibration models between NIR spectrometric instruments using three different standardization sets representing different sample chemistry types to be used as transfer standards. The first set contained samples that are very similar to the agricultural samples from three different sets to be analyzed, the second set contained generic biological and chemical standards, and the third transfer sample set contained pure organic and inorganic chemicals (12). Another research paper describes several standardization strategies for calibration transfer methods as well as an analysis of problems associated with the choice of the standardization samples (13).

	There are a number of research publications that describe strategies and techniques for instrument alignment and correction (14–18). In these approaches, the instrument wavelength or frequency (

) axes are carefully maintained as compared to physical standards so that instrument drift and the differences between instrument wavelength and photometric axes scales are minimized. Full spectrum instruments and research associated with basic instrument measurement technology, instrumentation alignment, mathematical calculations, measuring agreement between instruments, and applying wavelength and photometric standards are described in a series of papers (19–25). The current state of commercial NIR spectrophotometers and their comparative accuracies are presented in this series. The performance of different instrument types, such as monochromator, Fourier transform, and diode-array–based instruments are compared in detail with data comparisons shown in tabular and graphical forms (19–25). An example of the practical issues associated with multivariate calibration transfer (or calibration transfer) for NIR spectroscopic instruments using reflection spectroscopy for flour analysis was discussed in a recent paper (26). A trademarked and patent pending method that aligns instrument parameters to first principles physics has been described, namely True Alignment Spectroscopy (TAS, Unity Scientific) (27). In this technique, all instrument parameters are aligned to the "true" physical values of first principles standards, such as emission lamps and pure solids, gases, or liquid materials. This method does not rely on a single instrument response function as the standard like what is applied using a "master" instrument concept, but rather relies on basic physical science and the measurement of constant physical standards. Figure 1 illustrates the differences between the concept of first principles alignment and master instrument alignment.


	Figure 1: The use of a first physical principles alignment technique provides constant and accurate alignment of all instruments to unchanging physics at any time. The use of a master instrument provides uncertain similarity alignment to the master instrument response that will vary over time.

	The utilization of NIR spectroscopy by the forage and feed industry was initially limited by the requirement to individually calibrate each instrument. Calibration transfer became a necessity for practical application and implementation of NIR over time because of the collection of large, comprehensive databases of more than 1000 or more samples. In the initial application of the "master instrument" concept, 60 sealed samples were used for measurements on primary ("master") and target ("slave") instruments for the transfer process. Note that master and slave terms are anachronisms, and have been changed to parent and child instruments, or primary and secondary instruments, respectively. In this method, one monochromator is selected to be the master instrument. Calibration equations are developed on it and transferred to one or more slave instruments. In one publication, six transfer instruments were evaluated (28). The standard errors of difference (SED) including bias among the six transfer instruments for crude protein, acid detergent fiber, neutral detergent fiber, lignin, and in vitro dry matter disappearance were less than the comparative SED among laboratories for reference chemical analysis. It was concluded that satisfactory calibration transfer could be accomplished with 10 samples. This was the most successful and practical method for calibration transfer in the mid-1980s.

	There have been many approaches to correcting instruments for calibration transfer. In one instance, filter instruments were used and a method was demonstrated for computing calibrations that are robust against filter differences and bandwidths of filters. Common methods for wavelength selection involve using absorbance maxima for analyte spectra, using correlation plots relating the change in absorbance with respect to a change in concentration, and using computerized wavelength search algorithms relating to specific wavelengths and their correlation to changes in concentration.

	The technique of selecting isonumeric wavelengths allows the development of general wavelength positions and regression coefficients that are robust against small variations in center wavelength positions for filters because of manufacturing variations. This technique also applies to the wavelength variations that exist in full scanning instruments (29). In the isonumeric method, small changes in wavelength center positions do not affect regression coefficients or the predicted results. To illustrate, given the standard regression equations for every multivariate model expressed in algebraic notation as follows:

 is the measured absorbance at wavelength 

 are the concentrations for samples 1 and 2, and 

 represents the regression coefficients for sample 

. The bias is the offset value for the regression.

	For the regression coefficients to be immune to wavelength shifts, the following conditions must hold for the regression coefficients 

	An equation derived through a proof in the reference paper leaves the final equation as follows:

 is the predicted value for any sample. The end result of a wavelength search under the conditions that the change in wavelength does not change the regression coefficients, allows the derivation of a calibration equation that is robust against small wavelength shifts within and between instruments.

Direct Standardization and Piecewise Direct Standardization 

	The second and third most used approaches to multivariate calibration transfer involve the application of direct standardization (DS) and piecewise direct standardization (PDS) (30–33). These approaches are described in detail with application descriptions, examples, and equations within the cited references. The DS and PDS approaches are also often combined with small adjustments in bias or slope of the predicted values to compensate for small differences not accounted for by using standardization algorithms. Note that the frequency with which standardization approaches must be applied to child instruments is dependent on the frequency of the calibration updates required and the drift of the child instruments with respect to the parent (or original calibration instrument).

	PDS was applied to a set of gasoline samples measured on two different NIR spectrometers (34). PDS was applied to a set of two-component samples measured on the same ultraviolet–visible (UV–vis) spectrometer with the use of a cuvette cell with a 10-mm pathlength and a fiber-optic probe with a 2-mm pathlength. Piecewise direct standardization proceeds by determining a structured transformation matrix using the spectra of a few samples measured with both devices. This transformation matrix can then be used to transform any spectrum measured on one device to that obtained on another device, thereby making the calibration model transferable between devices (35). More detailed chemometric improvements to the basic technique of PDS for NIR spectrometric instruments are further described for multivariate calibration transfer in a 1996 paper (36).

	For the PDS computation, we apply the following equations:

 is the response of the standardization samples measured on the parent instrument at wavelength 

 is the response for the standardization samples measured on the child instrument at wavelength 

 is the transformation or correction vector of coefficients for the 

	The transformation matrix for PDS is computed using equation 6:

 is the total number of wavelengths in the measured standardization samples.

	Finally, the response prediction equation for any unknown sample measured on the child instrument is estimated using equation 7:

	For the DS method, the test sample set is measured on the parent and child instruments as typically absorbance (

 individual wavelengths. A transformation matrix (T) is used to match the child instrument data (

). And so equation 8 demonstrates the matrix notation. Note that for DS a linear relationship is assumed between the parent and child measurement values.

 is the parent data for the test sample set as an 

 is the child instrument data for the test samples as an 

 is the pseudoinverse approximated using singular value decomposition of the 

 spectral data matrix for a set of transfer or standardization samples measured on the child instrument, and 

 spectral data matrix for the same set of transfer or standardization samples measured on the parent instrument. The transform matrix is used to convert a single spectrum measured on the child instrument to be converted to "look" like a parent instrument spectrum.

	The response vector for a new measured sample designated as 

 is predicted using the original model equation as follows:

	where superscript T is the matrix transpose and 

 represents the estimated or predicted value for the measured sample 

	For the PDS method, the DS method is used piecewise or with a spectral windowing method to more closely match the spectral nuances and varying resolution and lineshape of spectra across the full spectral region, and there is no assumption of linearity between the parent and child prediction results. The transformation matrix is formed in an iterative manner across multiple windows of the spectral data in a piecewise fashion. Many other approaches have been published and compared, but for many users these are not practicable or have not been adopted for various reasons; basic method comparisons are demonstrated in the literature (37).

	An example of orthogonal signal correction (OSC) was applied to NIR spectra that were used in a calibration for the water content in a pharmaceutical product (38). Partial least squares (PLS) calibrations were then compared to other calibration models with uncorrected spectra, models with spectra subjected to multiplicative signal correction, and a number of other transfer methods. The performance of OSC was on the same level as for piecewise direct standardization and spectral offset correction for each individual instrument and PLS models with both instruments included.

	The goal of OSC is to remove variation from the spectral data, 

. This orthogonal variation is modeled by additional components for 

 and results in the decomposition as shown in equation 11:

 represent the scores and loadings for the orthogonal component and 

', OSC maximizes correlation and covariance between the 

 scores to achieve both good prediction and interpretation. As for OSC, there are several algorithms that have been reported in the literature (40). These algorithms differ in the approach used to obtain 

, a new score vector is calculated and multiplied with the loading vector (

') computed previously. In the final step, the product of the two is subtracted from 

	In analytical chemistry, it is necessary to form instrument-dependent calibration models. Problems such as instrument drift, repair, or use of a new instrument create a need for recalibration. Since recalibration can require considerable costs and cause time delays, methods for calibration transfer have been developed. One paper shows that many of these approaches are based on the statistical procedure known as 

) (41). Transfer by PA methods is known to involve translation (mean-centering), rotation, and stretching of instrument responses (41). The standard PA steps include translation, uniform scaling, rotation, and fine adjustments to superimposing the signals or spectra. An excellent tutorial on the general use of the procrustes technique in chemistry is found in the literature (42).

	In an example paper, four different calibration transfer techniques are compared (43). Three established techniques, finite impulse response (FIR) filtering, generalized least squares weighting (GLSW), and PDS, were evaluated. A fourth technique, baseline subtraction, was reported to be the most effective for calibration transfer. Using as few as 15 transfer samples, the predictive capability of the analytical method was maintained across multiple instruments and major instrument maintenance (43). Previously, a standard-free method using the FIR filter was successfully used to transfer the NIR spectra of caustic brines, analgesics, and terpolymer resins. This example paper carries the FIR transfer method one step further, leading to an improved algorithm that makes the transfer more robust and general by avoiding transfer artifacts in the filtered spectra (44).

	In a second example paper, FIR filtering was used for a set of spectra to be transferred, using a spectrum on the target instrument to direct the filtering process (45). Often, the target spectrum is the mean of a calibration set. The method is compared against direct transfer and piecewise direct transfer on NIR reflectance spectra in two representative data sets. Results from these studies suggest that FIR transfer compares favorably with piecewise direct transfer in terms of accuracy and precision of the match of transferred spectra to the predictive calibration models developed on the target instrument. Unlike piecewise direct transfer, FIR transfer requires no measurement of standard samples on both the source and target spectrometers. Details and limitations of the FIR transfer method are presented in the literature (45).

Maximum Likelihood Principal Component Analysis 

maximum likelihood principal component analysis

), is analogous to conventional principal component analysis (PCA), but incorporates measurement error variance information in the decomposition of multivariate data (46). A very detailed description of the derivation, computations, and results discussion is given in the literature (47).

Using Wavelength Standards for FT-NIR Alignment

	A series of papers describe the use of a powdered mixture of Er

, and talc measured at a constant resolution of 2 cm

 on four combinations of spectrometers and sampling accessories. The wavenumber scale of each mixture spectrum was corrected with lines in the vibration–rotation spectrum of water vapor to an accuracy better than 0.02 cm

. The claim for this work is that the reported values for the DR standard are improved by an order of magnitude compared to the values reported by the National Institute of Standards and Technology (NIST). These values for the standard are used for testing or calibrating alignment for the wavelength axis of a spectrometer. A discussion of the various methods, and results, is provided in the literature (48–50) and will be further described in greater detail in subsequent columns on the subject of reference standards.

	There are multiple mathematical techniques being applied for transfer of multivariate calibrations. Some of these techniques were described within this column with appropriate references. In the next installment in this series, we will continue to provide a basic top-level review of calibration transfer methods.

40 Years of Chemometrics–From Bruce Kowalski to the Future

 (American Chemical Society, Washington, D.C., 2015), pp. 257–282.

1st Edition (Elsevier/Academic Press, Boston, Massachusetts, 2010).

The Concise Handbook of Analytical Spectroscopy: Physical Foundations, Techniques, Instrumentation and Data Analysis

, J. Workman, Ed. (World Scientific Publishing-Imperial College Press, in 5 Volumes, UV, Vis, NIR, IR, and Raman, New Jersey and Singapore, 2016), volume 3, pp. 185–188, volume 4, pp. 159–162 (2016).

	(5) R.N. Feudale, N.A. Woody, H. Tan, A.J. Myles, S.D. Brown, and J. Ferré, 

	(7) J. Shenk and M. Westerhaus, U.S. Patent 4,866,644 September 12, 1989.

Calibration transfer involves multiple strategies and mathematical techniques for applying a single calibration database to two or more instruments. Here, we explain the methods to modify the spectra or regression vectors to correct differences between instruments.

Jerome Workman, Jr., and Howard Mark

Articles

Using Reference Materials, Part II: Photometric Standards

Calibration Transfer Chemometrics, Part I: Review of the Subject