Spectroscopic instrument calibration

• US 5,243,546 A
• Filed: 01/10/1991
• Issued: 09/07/1993
• Est. Priority Date: 01/10/1991
• Status: Expired due to Term

• Current Assignee: Marathon Petroleum Company LLC (Marathon Petroleum Corporation)
• Original Assignee: Ashland Oil, Inc. (Ashland Global Holdings, Inc.)
• Inventors: Maggard, Steven M.
• Primary Examiner(s): Cosimano, Edward R.
• Application Number:
  US07/698,411
• Time in Patent Office: 971 Days
• Field of Search: 250/339, 364/498, 364/571.01, 364/571.02
• US Class Current: 702/90
• CPC Class Codes: G01J 3/28 Investigating the spectrum using colour filters G01J3/51
• CPC Class Codes: G01N 21/274 Calibration, base line adjustment, drift correction
• CPC Class Codes: G01N 21/359 using near infrared light
• CPC Class Codes: G01N 2201/129 Using chemometrical methods

Abstract

A calibrated spectrometer can indirectly determine a physical or chemical property of a sample based upon spectral responses measured by the spectrometer with respect to the particular sample. This invention is directed to a method for calibrating or recalibrating a first spectrometer in light of a second spectrometer, or itself, respectively. The calibration employs a unique selection and manipulation of spectral data obtained from both the first and the second instrument. The recalibration employs a unique selection and manipulation of spectral data from the same first instrument, that is obtained both before and after the need for recalibration arises. Instead of modifying the respective responses of the first and second instrument, or the first instrument before and after the need for calibration arises, this invention modifies the calibration equation of the second, or recalibrated instrument, to yield consistent results to those obtained by the first instrument, or the first instrument before it goes out of calibration. A calibration equation is an equation which transforms spectral data of a particular sample at a variety of wavelengths to a calculated value for a chemical or physical property. Generally, the form of such calibration equations is that of a linear combination of absorbances or mathematical transforms of absorbances measured by the first and second instrument for each sample. The accuracy of the calibrated or recalibrated instrument is maintained and in some instances improved.

Claims

1. In an indirect method for determining a property-p for a sample-u, using spectral data-k measured by an instrument-k, capable of determining such spectral data-k for said sample-u, in combination with a calibration equation-k, having:

at least one constant-k;

a dependent variable-k; and

at least one independent variable-k;

wherein each value for said dependent variable-k is determined specifically by specific values for each said at least one independent variable-k, and wherein each said at least one independent variable-k has a value-k equal to a spectral feature-k, functionally determined from said spectral data-k;

wherein the improvement comprises;

(1) defining said calibration equation-k by recourse to a calibration equation-i for instrument-i, wherein said calibration equation-i has;

at least one constant-i;

at least one independent variable-i; and

a dependent variable-i;

wherein each value for said dependent variable-i is determined specifically by specific values for each said at least one independent variable-i, and wherein each said at least one independent variable-i has a value-i equal to a spectral feature-i, functionally determined from spectral data-i, and wherein calibration equation-i and -k determine, respectively, values for each said dependent variable-k and-i, that are functionally related to one another by a function-f that transforms dependent variable-i into dependent variable-k, said method for defining said calibration equation-k by recourse to said calibration equation-i comprising;

(a) obtaining respectively with instrument-i and instrument-k spectral data-i and spectral data-k, respectively, for each member of a sample-c, comprising at least one member c, thereby functionally determining value-k and value-i, respectively, equal to said at least one spectral feature-k and said at least one spectral feature-i, respectively, for each member of sample-c used in step (b);

(b) determining, in accordance with calibration equation-i, a value for said dependent variable-i for each member of sample-c and in place of each said value of said dependent variable-k in calibration equation-k inserting a value which is functionally related by function-f to said value for said at least one dependent variable-i and selecting an appropriate value for each said at least one constant-k in said calibration equation-k to define a calibration equation-k;

wherein said calibration equation-k in conjunction with said at least one spectral feature-k and the inverse of function-f functionally define each predicted value corresponding to each dependent variable-k for each member c, of sample-c, wherein a sum of each absolute difference between each said predicted value and said dependent variable-i for each member c, of sample-c is minimized at least so that said calibration equation-k predicts a value for property-p of each unknown sample that has a value for property-p within a range of interest with a standard error of prediction not substantially greater than a standard error value selected from the group consisting of a standard error of estimate and a standard error of prediction wherein at least one of said standard errors is necessarily present in a direct method used to determine values for property-p to obtain directly or indirectly calibration equation-i; and

(2) obtaining spectral data-k for said sample-u by means of instrument-k, and determining a value-u for each spectral feature-u, and defining in conjunction with said calibration equation-k a value for dependent variable-u which when transformed by inverse function-f yields a value for property-p of sample-u.

2. The indirect method of claim 1, wherein said value-k and -i for a material at wavelength-j has a value selected from the group consisting of an absorbance at wavelength-j of electromagnetic radiation, and an arithmetic transform of such an absorbance.

3. In the indirect method of claim 1, the further improvement of applying a slope and bias correction, said applying a slope and bias correction comprising the steps of:

determining values for m and b, which substantially satisfy the following expressions;

space="preserve" listing-type="equation">m={(Σ

.sub.c Y.sub.c.sup.a Σ

.sub.c Y.sub.c.sup.k)/I-Σ

.sub.c (Y.sub.c.sup.a Y.sub.c.sup.k)}

space="preserve" listing-type="equation"> /{{Σ

.sub.c Y.sub.c.sup.k Σ

.sub.c Y.sub.c.sup.k }/I-Σ

.sub.c Y.sub.c.sup.k Y.sub.c.sup.k }

space="preserve" listing-type="equation">b={Σ

.sub.c Y.sub.c.sup.a -mΣ

.sub.c Y.sub.c.sup.k }/I

where;

Y_c^a is selected from the group of values consisting of a directly determined value for property-p of a particular member c, of sample-p, a predicted value for property-p of a particular member c, of sample-p by means of a calibration equation for property-p in range of interest, and a combination thereof;

I is the number of members of sample-p;

substitute said values for m and b into the following expression resulting in a further improved calibration equation for the dependent variable-k for instrument-k, which is as follows;

space="preserve" listing-type="equation">Y.sub.c.sup.k ={mA.sup.k +b}+Σ

.sub.j {mB.sub.j.sup.k X.sub.jc.sup.k };

whereby an improved calibration equation over that originally determined without the slope and bias correction is achieved.

4. The indirect method of claim 1, wherein at least one of said calibration equation-i and calibration equation-k is determined by applying a method selected from the group of methods consisting of inverse least squares, classical least squares, partial least squares, principal component regression, and a combination of methods selected from the group consisting of inverse least squares, classical least squares, partial least squares, and principal component regression.

5. In a method of claim 1, wherein said member c is selected from the group consisting of isooctane, n-decane, 1-octene, 2,3,4-trimethylpentane, n-heptane cyclohexane, n-hexane, and toluene.

6. The indirect method of claim 1, wherein:

said at least one constant-i in calibration equation-i corresponds to at least one slope constant-i, B_jⁱ ;

said at least one independent variable-i in calibration equation-i corresponds to X_jsⁱ where j identifies term for wavelength-j and s identifies a member of sample-s; and

said dependent variable-i in calibration equation-i corresponds to Y_sⁱ, wherein;

said at least one constant-k in calibration equation-k corresponds to at least one intercept constant-k, A^k, and at least one slope constant-k, B_j^k ;

said at least one independent variable-k in calibration equation-k for instrument-k corresponds to X_js^k where j identifies term for wavelength-j and s, a member of sample-s; and

said dependent variable-k in calibration equation-k corresponds to Y_s^k ; and

wherein;

said calibration equation-i and calibration equation-k are linear in form and function-f is selected from the group of functions consisting of;

Y_s^k =mY_sⁱ +p, where m is a constant not equal to zero and p is a constant;

Y_s^k =log of Y_sⁱ ; and

a linear combination thereof.

7. The indirect method of claim 6, wherein at least one of said calibration equation-i and calibration equation-k is determined by applying a method selected from the group of methods consisting of inverse least squares, classical least squares, partial least squares, principal component regression, and a combination of methods selected from the group consisting of inverse least squares, classical least squares, partial least squares, and principal component regression.

8. The indirect method of claim 1, wherein said property-p is a property selected from the group consisting of octane and cetane, and said spectral data-k and -i, respectively, each correspond to absorbances for frequencies in a range of about 800 to about 2500 nanometers.

9. The method of claim 8, wherein said value-k and value-i are functionally determined from spectral data-k and -i by applying a function to said spectral data-k and -i, which function comprises adjacent derivative orders of spectral data for identical wavelengths.

10. The indirect method of claim 8, wherein said spectral features-k and -i, respectively, determined functionally from spectral data-k and -i, comprises determining differentials, respectively, of said spectral data-k and -i.

11. The indirect method of claim 10, wherein said differentials are at least second order and said at least one frequency is selected from the group consisting of at least one methyne band.

12. In the indirect method of claim 10, the further improvement of applying a slope and bias correction, said applying a slope and bias correction comprising the steps of:

determining values for m and b, which substantially satisfy the following expressions;

space="preserve" listing-type="equation">m={(Σ

.sub.c Y.sub.c.sup.a Σ

Y.sub.c.sup.k)/I-Σ

.sub.c (Y.sub.c.sup.a Y.sub.c.sup.k)}/{{Σ

.sub.c Y.sub.c.sup.k Σ

.sub.c Y.sub.c.sup.k }/I-Σ

.sub.c Y.sub.c.sup.k Y.sub.c.sup.k }

space="preserve" listing-type="equation">b={Σ

.sub.c Y.sub.c.sup.a -mΣ

.sub.c Y.sub.c.sup.k }/I

where;

Y_c^a is selected from the group of values consisting of a directly determined value for property-p of a particular member c, of sample-p, a predicted value for property-p of a particular member c, of sample-p by means of a calibration equation for property-p in range of interest, and a combination thereof;

n is the number of members of sample-p;

substitute said values for m and b into the following expression resulting in a further improved calibration equation for the dependent variable-k for instrument-k, which is as follows;

space="preserve" listing-type="equation">Y.sub.c.sup.k ={mA.sup.k +b}+Σ

.sub.j {mB.sub.j.sup.k X.sub.jc.sup.k };

whereby an improved calibration equation over that originally determined without the slope and bias correction is achieved.

13. In an indirect method for determining a property-p for a sample-u, using spectral data-k measured by an instrument-k, capable of determining such spectral data-k for said sample-u, in combination with a calibration equation-k, having at least one constant-k, a dependent variable-k, and at least one independent variable-k;

wherein each value for said dependent variable-k is determined specifically by particular values for each said at least one independent variable-k, and wherein each said at least one independent variable-k has a value-k equal to a spectral feature-k, functionally determined by said spectral data-k, wherein the improvement comprises;

(1) Defining said calibration equation-k by recourse to a calibration equation-i for instrument-i, wherein said calibration equation-i has at least one constant-i;

at least one independent variable-i; and

a dependent variable-i;

wherein each value for said dependent variable-i is determined specifically by specific values for each said at least one independent variable-i, and wherein each said at least one independent variable-i has a value-i equal to a spectral feature-i, functionally determined from spectral data-i, and wherein calibration equation-i and -k determine, respectively, values for each said dependent variable-k and-i, that are functionally related to one another by a function-f that transforms dependent variable-i into dependent variable-k;

for property-p;

said method for defining said calibration equation-k by recourse to said calibration equation-i comprising;

(a) obtaining a calibration equation for said instrument-i and assuming a calibration equation-k for said instrument-k has the following form;

Y_s^k =A^k +Σ

_j {B_j^k X_js^k } and calibration equation-i has the same or different form;

where;

Y_sⁱ is a dependent variable-i in calibration equation-i, and Y_s^k is functionally related to Y_sⁱ by a function-f;

A^k and Aⁱ are each intercept constants-i and -k in calibration equations-i and -k, respectively;

X_jsⁱ and X_js^k are each one of said at least one independent variable-i and -k in calibration equations-i and -k, respectively, having a value equal to at least one said spectral feature-i and -k, respectively, of a member s, at wavelength-j; and

B_jⁱ and B_j^k are slope constants-i and -k, respectively, corresponding to coefficients for each of said at least one of independent variable-i and -k at wavelength-j, respectively;

(b) obtaining with instrument-i and -k, spectral data-i and -k, respectively, for each member of a sample-c comprising at least one member c, thereby uniquely determining value-k and -i, respectively, equal to said at least one spectral feature-k and-i, respectively, for each member of sample-c used in step (c);

(c) determining said intercept constant-k and said at least one slope constant-k corresponding respectively to A^k and B_j^k which substantially minimizes the following expression;

Σ

_c (Y_cⁱ -A^k -Σ

_j B_j^k X_jc^k)² ;

whereby a calibration equation-k for instrument-k is obtained by reference to said calibration equation for instrument-i; and

(2) obtaining spectral data-u for said sample-u by means of instrument-k, and determining functionally a value-u for each spectral feature-u therein;

wherein each said value-u is used in combination with said calibration equation-k to determine a value for dependent variable-k, Y_u^k, for sample-u which is functionally related by the inverse of function-f to said property-p of said sample-u.

14. In the indirect method of claim 13, the further improvement of applying a slope and bias correction, said applying a slope and bias correction comprising the steps of:

determining values for m and b, which substantially satisfy the following expressions;

space="preserve" listing-type="equation">m={(Σ

.sub.c Y.sub.c.sup.a Σ

Y.sub.c.sup.k)/I-Σ

.sub.c (Y.sub.c.sup.a Y.sub.c.sup.k)}/{{Σ

.sub.c Y.sub.c.sup.k Σ

.sub.c Y.sub.c.sup.k }/I-Σ

.sub.c Y.sub.c.sup.k Y.sub.c.sup.k }

space="preserve" listing-type="equation">b={Σ

.sub.c Y.sub.c.sup.a -mΣ

.sub.c Y.sub.c.sup.k }I

where;

Y_c^a is selected from the group of values consisting of selected from the group of values consisting of a directly determined value for property-p of a particular member c, of sample-p, a predicted value for property-p of a particular member c, of sample-P by means of a calibration equation for property-p in range of interest, and a combination thereof;

I is the number of members of sample-p;

substitute said values for m and b into the following expression resulting in a further improved calibration equation for the dependent variable-k for instrument-k, which is as follows;

space="preserve" listing-type="equation">Y.sub.c.sup.k ={mA.sup.k +b}+Σ

.sub.j {mB.sub.j.sup.k X.sub.jc.sup.k };

whereby an improved calibration equation over that originally determined without the slope and bias correction is achieved.

15. The indirect method of claim 13, wherein at least one of said calibration equation-i and calibration equation-k is determined by applying a method selected from the group of methods consisting of inverse least squares, classical least squares, partial least squares, principal component regression, and a combination of methods selected from the group consisting of inverse least squares, classical least squares, partial least squares, and principal component regression.

16. In a method of claim 13, wherein said member c is selected from the group consisting of isooctane, n-decane, 1-octene, 2,3,4-trimethylpentane, n-heptane cyclohexane, n-hexane, and tolulene.

17. The method of claim 13 wherein said property-p is a property selected from the group consisting of octane and cetane, and said spectral data-k and -i, respectively, each correspond to absorbances for frequencies in a range of about 800 to about 2500 nanometers.

18. In the indirect method of claim 17, the further improvement of applying a slope and bias correction, said applying a slope and bias correction comprising the steps of:

determining values for m and b, which substantially satisfy the following expressions;

space="preserve" listing-type="equation">m={(Σ

.sub.c Y.sub.c.sup.a Σ

Y.sub.c.sup.k)/I-Σ

.sub.c (Y.sub.c.sup.a Y.sub.c.sup.k)}/{{Σ

.sub.c Y.sub.c.sup.k Σ

.sub.c Y.sub.c.sup.k }

space="preserve" listing-type="equation"> /I-Σ

.sub.c Y.sub.c.sup.k Y.sub.c.sup.k }

space="preserve" listing-type="equation">b={Σ

.sub.c Y.sub.c.sup.a -mΣ

.sub.c Y.sub.c.sup.k }/I

where;

Y_c^a is selected from the group of values consisting of a directly determined value for property-p of a particular member c, of sample-p, a predicted value for property-p of a particular member c, of sample-p by means of a calibration equation for property-p in range of interest, and a combination thereof;

I is the number of members of sample-p;

substitute said values for m and b into the following expression resulting in a further improved calibration equation for the dependent variable-k for instrument-k, which is as follows;

space="preserve" listing-type="equation">Y.sub.c.sup.k ={mA.sup.k +b}+Σ

.sub.j {mB.sub.j.sup.k X.sub.jc.sup.k };

Whereby an improved calibration equation over that originally determined without the slope and bias correction is achieved.

19. In an indirect method for determining a property-p for a sample-u, using spectral data-k measured by an instrument-k, capable of determining such spectral data-k for said sample-u, in combination with a calibration equation-k, having at least one constant-k, a dependent variable-k, and at least one independent variable-k;

wherein each value for said dependent variable-k is determined specifically by specific values for each said at least one independent variable-k, and wherein each said at least one independent variable-k has a value-k equal to a spectral feature-k, functionally determined by said spectral data-k;

wherein the improvement comprises;

(1) defining said calibration equation-k by recourse to a calibration equation-i for instrument-i, wherein said calibration equation-i has;

at least one constant-i comprising an intercept constant-i and at least one slope constant-i;

at least one independent variable-i; and

a dependent variable-i;

wherein each value for said dependent variable-i is determined specifically by specific values for each said at least one independent variable-i, and wherein each said at least one independent variable-i has a value-i equal to a spectral feature-i, functionally determined from spectral data-i, and wherein calibration equation-i and -k determine, respectively, values for each said dependent variable-k and -i, that are functionally related to one another by a function-f that transforms dependent variable-i into dependent variable-k;

said method for defining said calibration equation-k by recourse to said calibration equation-i comprising;

(a) obtaining a calibration equation for said instrument-i for property-p, and assuming a calibration equation-k for said instrument-k;

wherein each have the following form;

space="preserve" listing-type="equation">Y.sub.s.sup.i =A.sup.i +Σ

.sub.j {B.sub.j.sup.i X.sub.js.sup.i }

space="preserve" listing-type="equation">and

space="preserve" listing-type="equation">Y.sub.s.sup.k =A.sup.k +Σ

.sub.j {B.sub.j.sup.k X.sub.js.sup.k }

where;

Y_sⁱ is the dependent variable-i for member s in calibration equation-i and Y_s^k is functionally related to Y_sⁱ by a function-f;

A^k and Aⁱ are each intercept constants-i and -k in calibration equations-i and -k, respectively;

X_jsⁱ and X_js^k are each one of said at least one independent variable-i and -k in calibration equations-i and -k, respectively, each having a value equal to at least one said spectral feature-i and -k, respectively, of each member s, of sample-s at wavelength-j; and

B_jⁱ and B_j^k are slope constants-i and -k, respectively, corresponding to coefficients for each of said at least one of independent variables-i and -k at wavelength-j, respectively;

(b) obtaining with instrument-i and -k, spectral data-i and -k, respectively, for each member of a sample-c comprising at least one member c, and functionally determining value-k and -i, respectively, equal to said at least one spectral feature-k and -i, respectively, for each member of sample-c used in step (c);

(c) determining values of B_j^k and A^k so that the following two equations are simultaneously substantially satisfied;

B_j^k =B_jⁱ Σ

_c X_jcⁱ X_jc^k /Σ

_c (X_jc^k)² and A^k =Σ

_c (Y_cⁱ -Σ

_j B_j^k X_jc^k)/n;

where n=number of members to sample-c;

whereby a calibration equation-k for instrument-k is obtained by reference to said calibration equation for instrument-i; and

(2) obtaining spectral data-u by means of instrument-k for said sample-u, and functionally determining a value-u for each spectral features-u therein;

wherein each said value-u is used in combination with said calibration equation-k to determine a value for dependent variable-k, Y_u^k, for sample-u which is functionally related by the inverse of function-f to said property-p of said sample-u.

20. In the indirect method of claim 19, the further improvement of applying a slope and bias correction, said applying a slope and bias correction comprising the steps of:

determining values for m and b, which substantially satisfy the following expressions;

space="preserve" listing-type="equation">m={(Σ

.sub.c Y.sub.c.sup.a Σ

Y.sub.c.sup.k)/I-Σ

.sub.c (Y.sub.c.sup.a Y.sub.c.sup.k)}/{{Σ

.sub.c Y.sub.c.sup.k }/I-Σ

.sub.c Y.sub.c.sup.k Y.sub.c.sup.k }

space="preserve" listing-type="equation">b={Σ

.sub.c Y.sub.c.sup.a -mΣ

.sub.c Y.sub.c.sup.k }/ I

where;

Y_c^a is selected from the group of values consisting of a directly determined value for property-p of a particular member c, of sample-p, a predicted value for property-p of a particular member c, of sample-p by means of a calibration equation for property-p in range of interest, and a combination thereof;

I is the number of members of sample-p;

substitute said values for m and b into the following expression resulting in a further improved calibration equation for the dependent variable-k for instrument-k, which is as follows;

space="preserve" listing-type="equation">Y.sub.c.sup.k ={mA.sup.k +b}+Σ

.sub.j {mB.sub.j.sup.k X.sub.jc.sup.k };

whereby an improved calibration equation over that originally determined without the slope and bias correction is achieved.

21. The indirect method of claim 19, wherein at least one of said calibration equation-i and calibration equation-k is determined by applying a method selected from the group of methods consisting of inverse least squares, classical least squares, partial least squares, principal component regression, and a combination of methods selected from the group consisting of inverse least squares, classical least squares, partial least squares, and principal component regression.

22. In an indirect method for determining a property-p for a sample-u, using spectral data-k measured by an instrument-k, capable of determining such spectral data-k for said sample-u in combination with a calibration equation-k, having at least one constant-k, a dependent variable-k, and at least one independent variable-k;

wherein each value for said dependent variable-k is determined specifically by specific values for each said at least one independent variable-k, and wherein each said at least one independent variable-k has a value-k equal to a spectral feature-k, uniquely determined by reference to said spectral data-k;

wherein the improvement comprises;

(1) defining said calibration equation-k by recourse to a calibration equation-i for instrument-i, wherein said calibration equation-i has;

at least one constant-i;

at least one independent variable-i; and

a dependent variable-i;

wherein each value for said dependent variable-i is determined specifically by specific values for each said at least one independent variable-i, and wherein each said at least one independent variable-i has a value-i equal to a spectral feature-i, functionally determined from spectral data-i, and wherein calibration equation-i and -k determine, respectively, values for each said dependent variable-k and -i, that are functionally related to one another by a function-f that transforms dependent variable-i into dependent variable-k;

said method for defining said calibration equation-k by recourse to said calibration equation-i comprising;

(a) obtaining a calibration equation for said instrument-i for property-p, and assuming a calibration equation-k for said instrument-k wherein each have the following form;

space="preserve" listing-type="equation">Y.sub.s.sup.i =A.sup.i +Σ

.sub.j {B.sub.j.sup.i X.sub.js.sup.i }

space="preserve" listing-type="equation">and

space="preserve" listing-type="equation">Y.sub.s.sup.k =A.sup.k +Σ

.sub.j {B.sub.j.sup.k X.sub.js.sup.k }

where;

Y_sⁱ is the dependent variable-i in calibration equation-i, and Y_s^k is functionally related to Y_sⁱ by a function-f;

A^k and Aⁱ are each intercepts constants-i and -k in calibration equations-i and -k, respectively;

X_jsⁱ and X_js^k are each one of said at least one independent variable-i and -k in calibration equations-i and -k, respectively, each having a value equal to at least one said spectral feature-i and -k, respectively; and

B_jⁱ and B_j^k are slope constants-i and -k, respectively, corresponding to coefficients for each of said at least one of independent variables-i and -k corresponding to spectral features-i and -k at least one wavelength-j, respectively;

(b) obtaining with instrument-i and -k, spectral data-i and -k, respectively, for each member of a sample-s comprising at least one member c, used in step (c), thereby uniquely determining value-k and -i, respectively, equal to said at least one spectral feature-k and -i, respectively, for each member of sample-c having a total of n members;

(c) determining said intercept constant-k and said at least one slope constant-k corresponding, respectively, to values of B_j^k and A^k so that the following are substantially satisfied;

B_j^k =B_jⁱ Σ

_c X_jcⁱ /Σ

_c (X_jc^k) and A^k =Σ

_c (Y_cⁱ -Σ

_j B_j^k X_jc^k)/n;

whereby a calibration equation-k for instrument-k is obtained by reference to said calibration equation for instrument-i;

(2) obtaining spectral data-u for said sample-u by means of instrument-k, and determining functionally a value-u for each spectral feature-u therein;

wherein each said value-u is used in combination with said calibration equation-k to determine a value for dependent variable-k, Y_u^k, for sample-u which is functionally related by the inverse of function-f to said property-p of said sample-u.

23. In the indirect method of claim 22, the further improvement of applying a slope and bias correction, said applying a slope and bias correction comprising the steps of:

determining values for m and b, which substantially satisfy the following expressions;

space="preserve" listing-type="equation">m={(Σ

.sub.c Y.sub.c.sup.a Σ

Y.sub.c.sup.k)/I-Σ

.sub.c (Y.sub.c.sup.a Y.sub.c.sup.k)}/{{Σ

.sub.c Y.sub.c.sup.k Σ

.sub.c Y.sub.c.sup.k }/I-Σ

.sub.c Y.sub.c.sup.k Y.sub.c.sup.k }

space="preserve" listing-type="equation">b={Σ

.sub.c Y.sub.c.sup.a -mΣ

.sub.c Y.sub.c.sup.k }/I

where;

Y_c^a is selected from the group of values consisting of a directly determined value for property-p of a particular member c, of sample-p, a predicted value for property-p of a particular member c, of sample-p by means of a calibration equation for property-p in range of interest, and a combination thereof;

I is the number of members of sample-p;

substitute said values for m and b into the following expression resulting in a further improved calibration equation for the dependent variable-k for instrument-k, which is as follows;

space="preserve" listing-type="equation">Y.sub.c.sup.k ={mA.sup.k +b}+Σ

.sub.j {mB.sub.j.sup.k X.sub.jc.sup.k };

whereby an improved calibration equation over that originally determined without the slope and bias correction is achieved.

24. The indirect method of claim 22, wherein at least one of said calibration equation-i and calibration equation-k is determined by applying a method selected from the group of methods consisting of inverse least squares, classical least squares, partial least squares, principal component regression, and a combination of methods selected from the group consisting of inverse least squares, classical least squares, partial least squares, and principal component regression.

25. In a method for recalibrating an instrument-k in light of its former calibrated condition as instrument-i or for calibrating instrument-k, previously uncalibrated, in light of a calibrated instrument-i wherein a calibrated instrument is one having a particular calibration equation, here calibration equation-k, based upon which equation spectraldata of a sample can be used to predict a physical or chemical property of said sample, the improvement which comprises:

determining said calibration equation-k by means of a calibration equation-i, each said calibration equation-k and -i, respectively, having;

at least one constant-k and -i;

a dependent variable-k and -i; and

at least one independent variable-k and -i;

wherein each value, respectively, for said dependent variable-k and -i is determined specifically by particular values for each said at least one independent variable-k and -i and wherein each said at least one independent variable-k and -i has respectively a value-k and -i equal to a spectral feature-k and -i, functionally determined, respectively, by spectral data-k and -i, determined respectively, by an instrument-k and -i capable of determining, respectively, said spectral data-k and -i for a sample, s;

said method comprising;

(a) obtaining respectively with instrument-i and instrument-k spectral data-i and spectral data-k, respectively, for each member of a sample-c, comprising at least one member c, thereby functionally determining value-k and value-i, respectively, equal to said at least one spectral feature-k and said at least one spectral feature-i, respectively, for each member of sample-c used in step (b);

(b) determining in accordance with calibration equation-i, a value for said dependent variable-i for each member of sample-c and in place of each said value of said dependent variable-k in calibration equation-k inserting a value which is related by a function-f to each said value for said dependent variable-i and selecting an appropriate value for each said at least one constant-k in said calibration equation-k to produce a calibration equation-k;

wherein said calibration equation-k in conjunction with said at least one spectral feature-k and the inverse of function-f functionally define each predicted value corresponding to each dependent variable-k for each member c, of sample-c, wherein a sum of absolute differences between each said predicted value and said dependent variable-i for each member c, of sample-c is minimized at least so that said calibration equation-k predicts a value for property-p of each unknown sample that has a value for property-p within a range of interest with a standard error of prediction not substantially greater than a standard error value selected from the group consisting of a standard error of estimate and a standard error of prediction wherein at least one of said standard errors is necessarily present in a direct method used to determine values for property-p to obtain directly or indirectly calibration equation-i.

26. In the method of claim 25, the further improvement of applying a slope and bias correction, said applying a slope and bias correction comprising the steps of:

determining values for m and b, which substantially satisfy the following expressions;

space="preserve" listing-type="equation">m={(Σ

.sub.c Y.sub.c.sup.a Σ

Y.sub.c.sup.k)/I-Σ

.sub.c (Y.sub.c.sup.a Y.sub.c.sup.k)}/{{Σ

.sub.c Y.sub.c.sup.k Σ

.sub.c Y.sub.c.sup.k }/I-Σ

.sub.c Y.sub.c.sup.k Y.sub.c.sup.k }

space="preserve" listing-type="equation">b={Σ

.sub.c Y.sub.c.sup.a -mΣ

.sub.c Y.sub.c.sup.k }I

where;

Y_c^a is selected from the group of values consisting of a directly determined value for property-p of a particular member c, of sample-p, a predicted value for property-p of a particular member c, of sample-p by means of a calibration equation for property-p in range of interest, and a combination thereof;

n is the number of members of sample-p;

substitute said values for m and b into the following expression resulting in a further improved calibration equation for the dependent variable-k for instrument-k, which is as follows;

space="preserve" listing-type="equation">Y.sub.c.sup.k ={mA.sup.k +b}+Σ

.sub.j {mB.sub.j.sup.k X.sub.jc.sup.k };

whereby an improved calibration equation over that originally determined without the slope and bias correction is achieved.

27. The method of claim 25, wherein at least one of said calibration equation-i and calibration equation-k is determined by applying a method selected from the group of methods consisting of inverse least squares, classical least squares, partial least squares, principal component regression, and a combination of methods selected from the group consisting of inverse least squares, classical least squares, partial least squares, and principal component regression.

28. The method of claim 25, wherein said calibration equations for said instrument-i, and said instrument-k, each have the following linear form:

space="preserve" listing-type="equation">Y.sub.s.sup.i =A.sup.i +Σ

.sub.j {B.sub.j.sup.i X.sub.js.sup.i }

space="preserve" listing-type="equation">and

space="preserve" listing-type="equation">Y.sub.s.sup.k =A.sup.k +Σ

.sub.j {B.sub.j.sup.k X.sub.js.sup.k }

where;

Y_sⁱ is the dependent variable-i in calibration equation-i and Y_s^k is functionally related to Y_sⁱ by a function-f;

A^k and Aⁱ are each intercept constants-i and -k in calibration equations-i and -k, respectively;

X_jsⁱ and X_js^k are each one of said at least one independent variable-i and -k in calibration equations-i and -k, respectively, having a value equal to at least one said spectral feature-i and -k, respectively of a member s, at wavelength-j; and

B_jⁱ and B_j^k are slope constants-i and -k, respectively, corresponding to coefficients for each said at least one independent variable-i and -k corresponding to spectral features-i and -k at wavelength-j, respectively;

(a) obtaining with instrument-i and -k, spectral data-i and -k, respectively, for each member of a sample-c comprising at least one member c, thereby functionally determining value-k and -i, respectively, equal to said at least one spectral feature-k and-i, respectively, for each member c, of sample-c used in step (b); and

(b) determining said intercept constant-k, and said at least one slope constant-k corresponding respectively to B_j^k and A^k which substantially minimizes the following expression;

Σ

_c (Y_cⁱ -A^k -Σ

_j B_j^k X_jc^k)²

whereby a method is provided for recalibrating instrument-k in light of its former calibrated condition as instrument-i.

29. The method of claim 28, wherein at least one of said calibration equation-i and calibration equation-k is determined by applying a method selected from the group of methods consisting of inverse least squares, classical least squares, partial least squares, principal component regression, and a combination of methods selected from the group consisting of inverse least squares, classical least squares, partial least squares, and principal component regression.

30. The method of claim 25, wherein said spectral data-k and -i is in the form of electromagnetic radiation selected from a group of electromagnetic radiation consisting of:

ultraviolet, visible, near-infrared, mid-infrared, far-infrared, microwave, and radiowave.

31. In the method claim 30 the further improvement of applying a slope and bias correction, said applying a slope and bias correction comprising the steps of:

determining values for m and b, which substantially satisfy the following expressions;

space="preserve" listing-type="equation">m={(Σ

.sub.c Y.sub.c.sup.a Σ

Y.sub.c.sup.k)/I-Σ

.sub.c (Y.sub.c.sup.a Y.sub.c.sup.k)}/{{Σ

.sub.c Y.sub.c.sup.k Σ

.sub.c Y.sub.c.sup.k }/I-Σ

.sub.c Y.sub.c.sup.k Y.sub.c.sup.k }

space="preserve" listing-type="equation">b={Σ

.sub.c Y.sub.c.sup.a -mΣ

.sub.c Y.sub.c.sup.k }/I

where;

Y_c^a is selected from the group of values consisting of a directly determined value for property-p of a particular member c, of sample-p, a predicted value for property-p of a particular member c, of sample-p by means of a calibration equation for property-p in range of interest, and a combination thereof;

I is the number of members of sample-p;

substitute said values for m and b into the following expression resulting in a further improved calibration equation for the dependent variable-k for instrument-k, which is as follows;

space="preserve" listing-type="equation">Y.sub.c.sup.k ={mA.sup.k +b}+Σ

.sub.j {mB.sub.j.sup.k X.sub.jc.sup.k };

whereby an improved calibration equation over that originally determined without the slope and bias correction is achieved.

32. In the method of claim 25, wherein said value-k and value-i are functionally determined from spectral data-k and -i by applying a function to said spectral data-k and -i, which function is selected from the group consisting of:

first, second, third, forth, fifth, and higher derivatives of spectral data.

34. The method of claim 25, wherein:

said calibration equation-i and calibration equation-k are linear in form.

35. In the method of claim 34, the further improvement of applying a slope and bias correction, said applying a slope and bias correction comprising the steps of:

determining values for m and b, which substantially satisfy the following expressions;

space="preserve" listing-type="equation">m={(Σ

.sub.c Y.sub.c.sup.a Σ

Y.sub.c.sup.k)/I-Σ

.sub.c (Y.sub.c.sup.a Y.sub.c.sup.k)}/{{Σ

.sub.c Y.sub.c.sup.k Σ

.sub.c Y.sub.c.sup.k }/I-Σ

.sub.c Y.sub.c.sup.k Y.sub.c.sup.k }

space="preserve" listing-type="equation">b={Σ

.sub.c Y.sub.c.sup.a -mΣ

.sub.c Y.sub.c.sup.k }/I

where;

Y_c^a is selected from the group of values consisting of a directly determined value for property-p of a particular member c, of sample-p, a predicted value for property-p of a particular member c, of sample-p by means of a calibration equation for property-p in range of interest, and a combination thereof;

I is the number of members of sample-p;

substitute said values for m and b into the following expression resulting in a further improved calibration equation for the dependent variable-k for instrument-k, which is as follows;

space="preserve" listing-type="equation">Y.sub.c.sup.k ={mA.sup.k +b}+Σ

.sub.j {mB.sub.j.sup.k X.sub.jc.sup.k };

whereby an improved calibration equation over that originally determined without the slope and bias correction is achieved.

36. In a method of claim 34, wherein said member c is selected from the group consisting of isooctane, n-decane, 1-octene, 2,3,4, -trimethylpentane, n-heptane, cyclohexane, n-hexane, and toluene.

37. The method of claim 25, wherein said calibration equations for said instrument-i, and said instrument-k, each have the following linear form:

space="preserve" listing-type="equation">Y.sub.s.sup.i =A.sup.i +Σ

.sub.j {B.sub.j.sup.i X.sub.js.sup.i }

space="preserve" listing-type="equation">and

space="preserve" listing-type="equation">Y.sub.s.sup.k =A.sup.k +Σ

.sub.j {B.sub.j.sup.k X.sub.js.sup.k }

where;

Y_sⁱ is the dependent variable-i in calibration equation-i and Y_s^k is functionally related to Y_sⁱ by a function-f;

A^k and Aⁱ are each intercept constants-i and -k in calibration equations-i and -k, respectively;

X_jsⁱ and X_js^k are each one of said at least one independent variable-i and -k in calibration equations-i and -k, respectively, having a value equal to at least one said spectral feature-i and -k, respectively of a member s, at wavelength-j; and

B_jⁱ and B_j^k are slope constants=i and -k, respectively, corresponding to coefficients for each said at least one independent variable-i and -k corresponding to spectral features-i and -k at wavelength-j, respectively;

(a) determining by measuring respectively with instrument-i and -k, spectral data-i and -k, respectively, for each member of a sample-c comprising at least one member c, thereby functionally determining value-k and -i, respectively, equal to said at least one spectral feature-k and-i, respectively, for each member c, of sample-c used in step (b); and

(b) determining said intercept constant-k and said at least one slope constant-k corresponding, respectively, to B_j^k and A^k which substantially satisfy the following two equations simultaneously;

B_j^k =B_jⁱ Σ

_c X_jcⁱ /Σ

_c (X_jc^k) and A^k =Σ

_c (Y_c^k -Σ

_j B_j^k X_jc^k)/n;

where n=number of members of sample-c;

whereby a method is provided for recalibrating instrument-k in light of its former calibrated condition as instrument-i or for calibrating instrument-k, previously uncalibrated, in light of a calibrated instrument-i.

38. The method of claim 37, wherein at least one of said calibration equation-i and calibration equation-k is determined by applying a method selected from the group of methods consisting of inverse least squares, classical least squares, partial least squares, principal component regression, and a combination of methods selected from the group consisting of inverse least squares, classical least squares, partial least squares, and principal component regression.

39. The method of claim 37, wherein said property-p is a property selected from the group consisting of octane and cetane, and said spectral data-k and -i, respectively, each correspond to absorbances for frequencies in a range of about 800 to about 2500 nanometers.

40. The method of claim 25, wherein said calibration equations for said instrument-i and said instrument-k each have the following linear form:

space="preserve" listing-type="equation">Y.sub.s.sup.i =A.sup.i +Σ

.sub.j {B.sub.j.sup.i X.sub.js.sup.i }

space="preserve" listing-type="equation">and

space="preserve" listing-type="equation">Y.sub.s.sup.k =A.sup.k +Σ

.sub.j {B.sub.j.sup.k X.sub.js.sup.k }

where;

Y_sⁱ is the dependent variable-i in calibration equation-i and Y_s^k is functionally related to Y_sⁱ by a function-f;

A^k and Aⁱ are each intercept constants-i and -k in calibration equations-i and -k, respectively;

X_jsⁱ and X_js^k are each one of said at least one independent variable-i and -k in calibration equations-i and -k, respectively, having a value equal to at least one said spectral feature-i and -k, respectively of a member s, at wavelength-j; and

B_jⁱ and B_j^k are slope constants-i and -k, respectively, corresponding to coefficients for each said at least one independent variable-i and -k corresponding to spectral features-i and -k at wavelength-j, respectively;

(a) obtaining with instrument-i and -k, spectral data-i and -k, respectively, for each member of a sample-c comprising at least one member c, thereby functionally determining value-k and -i, respectively, equal to said at least one spectral feature-k and-i, respectively, for each member c, of sample-c used in step (b); and

(b) determining values of B_j^k and A^k which substantially satisfy the following two equations simultaneously;

B_j^k =B_jⁱ Σ

_c X_jcⁱ X_jc^k /Σ

_c (X_jc^k)² and A^k =Σ

_c (Y_c^k -Σ

_j B_j^k X_jc^k)/n;

where n=number of members of sample-c;

whereby a method is provided for recalibrating instrument-k in light of its former calibrated condition as instrument-i or for calibrating instrument-k, previously uncalibrated, in light of a calibrated instrument-i.

41. In the method of claim 40, the further improvement of applying a slope and bias correction, said applying a slope and bias correction comprising the steps of:

determining values for m and b, which substantially satisfy the following expressions;

space="preserve" listing-type="equation">m={(Σ

.sub.c Y.sub.c.sup.a Σ

Y.sub.c.sup.k)/I-Σ

.sub.c (Y.sub.c.sup.a Y.sub.c.sup.k)}/{{Σ

.sub.c Y.sub.c.sup.k Σ

.sub.c Y.sub.c.sup.k }/I-Σ

.sub.c Y.sub.c.sup.k Y.sub.c.sup.k }

space="preserve" listing-type="equation">b={Σ

.sub.c Y.sub.c.sup.a -mΣ

.sub.c Y.sub.c.sup.k }/I

where;

Y_c^a is selected from the group of values consisting of a directly determined value for property-p of a particular member c, of sample-p, a predicted value for property-p of a particular member c, of sample-p by means of a calibration equation for property-p in range of interest, and a combination thereof;

n is the number of members of sample-p;

substitute said values for m and b into the following expression resulting in a further improved calibration equation for the dependent variable-k for instrument-k, which is as follows;

space="preserve" listing-type="equation">Y.sub.c.sup.k ={mA.sup.k +b}+Σ

.sub.j {mB.sub.j.sup.k X.sub.jc.sup.k };

whereby an improved calibration equation over that originally determined without the slope and bias correction is achieved.

42. The method of claim 40, wherein at least one of said calibration equation-i and calibration equation-k is determined by applying a method selected from the group of methods consisting of inverse least squares, classical least squares, partial least squares, principal component regression, and a combination of methods selected from the group consisting of inverse least squares, classical least squares, partial least squares, and principal component regression.

43. The method of claim 40, wherein said property-p is a property selected from the group consisting of octane and cetane, and said spectral data-k and -i, respectively, each correspond to absorbances for frequencies in a range of about 800 to about 2500 nanometers.

44. In a method of claim 25, wherein said member c is selected from the group consisting of isooctane, n-decane, 1-octene, 2,3,4-trimethylpentane, n-heptane cyclohexane, n-hexane, and toluene.

45. The method of claim 25, wherein said property-p is a property selected from the group consisting of octane and cetane, and said spectral data-k and -i, respectively, each correspond to absorbances for frequencies in a range of about 800 to about 2500 nanometers.

46. In a method of claim 45, wherein said member c is selected from the group consisting of pure components of a hydrocarbon fuel.

47. In a method of claim 45, wherein said member c is selected from the group consisting of isooctane, n-decane, 1-octene, 2,3,4-trimethylpentane, cyclohexane, n-hexane, and toluene.

48. In a method of claim 45, wherein said member c is selected from the group consisting of paraffin, isoparaffin, aromatic, naphthene, and olefin.

49. In a method of claim 48, wherein said paraffin is selected from the group consisting of hexane and hepthane;

said isoparaffin is selected from the group consisting of 2,3-dimethylbutane, 2,3,4-trimethylpentane, and isooctane;

said aromatic is selected from the group consisting of ethyl benzene, S-butyl benzene, and t-butyl benzene;

said naphthene is 1,2-dimethylclyclohexene, as either a cis- or trans-isomer; and

said olefin is 1-octene as either a cis- or trans isomer.

33. In the method of 32, the further improvement of applying a slope and bias correction, said applying a slope and bias correction comprising the steps of:

determining values for m and b, which substantially satisfy the following expressions;

space="preserve" listing-type="equation">m={(Σ

.sub.c Y.sub.c.sup.a Σ

Y.sub.c.sup.k)/IΣ

.sub.c (Y.sub.c.sup.a Y.sub.c.sup.k)}/{{Σ

.sub.c Y.sub.c.sup.k Σ

.sub.c Y.sub.c.sup.k }

space="preserve" listing-type="equation"> /I-Σ

.sub.c Y.sub.c.sup.k Y.sub.c.sup.k }

space="preserve" listing-type="equation">b={Σ

.sub.c Y.sub.c.sup.a -mΣ

.sub.c Y.sub.c.sup.k }/I

where;

Y_c^a is selected from the group of values consisting of a directly determined value for property-p of a particular member c, of sample-p, a predicted value for property-p of a particular member c, of sample-p by means of a calibration equation for property-p in range of interest, and a combination thereof;

I is the number of members of sample-p;

substitute said values for m and b into the following expression resulting in a further improved calibration equation for the dependent variable-k for instrument-k, which is as follows;

space="preserve" listing-type="equation">Y.sub.c.sup.k ={mA.sup.k +b}+Σ

.sub.j {mB.sub.j.sup.k X.sub.jc.sup.k };

whereby an improved calibration equation over that originally determined without the slope and bias correction is achieved.

50. In an indirect method for determining a property-p for a sample-u by measuring and using spectral data-(ku) measured by an instrument-k capable of measuring such spectral data-(ku) for said sample-u in conjunction with a calibration equation-k,

wherein said calibration equation-k defines a dependent-variable-(ku), in terms of at least one constant-k and at least one independent-variable-(kju),

wherein said at least one independent-variable-(kju) has a value equal to at least one spectral-feature-(kju) determined from said spectral data-(ku) for said sample-u by a first functions that involves wavelength-j,

and wherein said dependent-variable-(ku) has a value substantially equal to said property-p for said sample-u when values for each said at least one constant-k and each said at least one independent-variable-(kju) are substituted appropriately into said calibration equation-k;

wherein the improvement comprises;

determining a value for each said at least one constant-k by recourse to a calibration equation-i,

wherein said calibration equation-i defines dependent-variable-(ic) as a function of an independent-variable-(ijc) which independent-variable-(ijc) is equal to a spectral feature-(ijc) functionally determined from spectral data-(ic) corresponding both to member c of a calibration sample-c having a total of n members and to each wavelength-j, and said dependent-variable-(ic) has a value substantially equal to said property-p of said member c, of said sample-c;

said determining a value for each said at least one constant-k comprises the steps of;

(1) using respectively spectral data-(ic) and spectral data-(kc) for each member c, of said calibration sample-c measured respectively by instrument-i and instrument-k, to determine each said at least one independent-variable-(kjc) corresponding to each member c, and to determine from said calibration equation-i a value for said dependent-variable-(ic) corresponding to each member c;

(2) maintaining consistent a set of equations that results from substituting in place of the value for dependent-variable-(kc) that value for dependent-variable-(ic), while selecting a value for each said at least one constant-k so that a sum of absolute differences in value between each new-dependent-variable-(kc) and dependent-variable-(ic) corresponding to each said member c, is substantially minimized;

wherein said new-dependent-variable-(kc) is determined by calibration equation-k after substituting each selected value for each said at least one constant-k along with each said at least one independent-variable-(kjc) in said calibration equation-k.

51. The improved method of claim 50, wherein said selecting a value for each said at least one constant-k is continued at least until standard error of estimation for property-p is not significantly greater than that standard error of estimation for property-p used to establish calibration equation-i.

52. The improved method of claim 50, wherein said calibration equation-k has the following form which defines and at least one constant-k that comprises an intercept-constant-k and at least one slope-constant-(kj):

space="preserve" listing-type="equation">Y.sup.k.sub.c =A.sup.k +Σ

.sub.j B.sup.k.sub.j I.sup.k.sub.jc

wherein;

Y^k_c =dependent-variable-(kc)

A^k =intercept-constant-k

B^k_j =slope-constant-(kj); and

I^k_jc =independent-variable-(kjc);

wherein said selecting each value for each said at least one constant-k comprises selecting at least one intercept-constant-k, A^k, and at least one slope-constant-(kj), B^k_j, so that the expression immediately following is reduced substantially to its minimum value;

space="preserve" listing-type="equation">Σ

.sub.c {Y.sup.i.sub.c -(A.sup.k -Σ

.sub.j B.sup.k.sub.j I.sup.k.sub.jc)}.sup.2.

53. The improved method of claim 50, wherein said calibration equation-i is in the same linear form as calibration equation-k and wherein said calibration equation-i defines dependent-variable-(ic) in terms of at least one constant-i and at least one independent-variable-(ijc), wherein said at least one constant-i comprises:

an intercept-constant-i, and at least one slope-constant-(ij) involving each wavelength-j used in calibration equation-i; and

wherein each said at least one independent-variable-(ijc) has a value substantially equal to each at least one spectral feature-(ijc) determined by said first function from spectral data-(ic) measured by said instrument-i that involves each said wavelength-j for member c; and

wherein said dependent-variable-(ic) has a value substantially equal to said property-p for said sample-u when values for each said at least one constant-i and each said at least one independent-variable-(ijc) are appropriately substituted into said calibration equation-i and sample-u is member c, wherein calibration equation-k and calibration equation-i respectively, have the linear forms;

space="preserve" listing-type="equation">Y.sup.i.sub.c =A.sup.i +Σ

.sub.j B.sup.i.sub.j I.sup.i.sub.jc

space="preserve" listing-type="equation">and

space="preserve" listing-type="equation">Y.sup.k.sub.c =A.sup.k +Σ

.sub.j B.sup.k.sub.j I.sup.k.sub.jc ;

wherein;

Yⁱ_c =dependent-variable-(ic)

Aⁱ =intercept-constant-i

Bⁱ_j =slope-constant-(ij); and

Iⁱ_jc =independent-variable-(ijc)

A^k =intercept-constant-k

B^k_j =slope-constant-(kj); and

I^k_jc =independent-variable-(kjc)

and wherein the following two equations are simultaneously substantially satisfied;

space="preserve" listing-type="equation">B.sup.k.sub.j ={B.sup.i.sub.j ×

Σ

.sub.c (I.sup.i.sub.jc ×

I.sup.kj.sub.c)}/Σ

.sub.c (I.sup.k.sub.jc).sup.2

space="preserve" listing-type="equation">and

space="preserve" listing-type="equation">A.sup.k =Σ

.sub.c {Y.sup.i.sub.c -Σ

.sub.j B.sup.k.sub.j I.sup.k.sub.jc }/n

where n=number of members in calibration sample-c;

whereby said calibration equation-k for instrument-k is obtained by reference to said calibration equation-i.

54. The improved method of claim 50, wherein said calibration equation-i is in the same linear form as calibration equation-k and wherein said calibration equation-i defines dependent-variable-(ic) in terms of at least one constant-i and at least one independent-variable-(ijc), wherein said at least one constant-i of calibration equation-i comprises:

an intercept-constant-i, and at least one slope-constant-(ij) involving each wavelength-j used in calibration equation-i; and

wherein each said at least one independent-variable-(ijc) has a value substantially equal to each at least one spectral feature-(ijc) determined by said first function from spectral data-(ic) measured by instrument-i that involves each said wavelength-j for member c, and wherein said dependent-variable-(ic) has a value substantially equal to said property-p for said member c when values for each said at least one constant-i and each said at least one independent-variable-(ijc) are appropriately substituted into said calibration equation-i;

wherein calibration equation-i and calibration equation-k have, respectively, the linear forms;

space="preserve" listing-type="equation">Y.sup.i.sub.c =A.sup.i +Σ

.sub.j {B.sup.i.sub.j I.sup.i.sub.jC }

space="preserve" listing-type="equation">Y.sup.k.sub.c =A.sup.k +Σ

.sub.j {B.sup.k.sub.j I.sup.k.sub.jc }

and wherein the following two equations are simultaneously substantially satisfied;

space="preserve" listing-type="equation">B.sup.k.sub.j =B.sup.i.sub.j Σ

.sub.c I.sup.i.sub.jc /Σ

.sub.c I.sup.k.sub.jc

space="preserve" listing-type="equation">and

space="preserve" listing-type="equation">A.sup.k ={Σ

.sub.c (Y.sup.i.sub.c -Σ

.sub.j B.sup.k.sub.i I.sup.k.sub.jc)}/n;

whereby said calibration equation-k for instrument-k is obtained by reference to said calibration equation-i.

55. The improved method of claim 50, wherein there is a further improvement of applying a slope and bias correction comprising the steps of:

determining values for m and b, which substantially satisfy the following expressions simultaneously;

space="preserve" listing-type="equation">m={Σ

.sub.c {Y.sup.a.sub.c Σ

.sub.c Y.sup.k.sub.c }/I-{Σ

.sub.c Y.sup.a.sub.c Y.sup.k.sub.c }}

space="preserve" listing-type="equation"> /{{Σ

.sub.c Y.sup.k.sub.c Σ

.sub.c Y.sup.k.sub.c }/I-Σ

.sub.c Y.sup.k.sub.c Y.sup.k.sub.c }

space="preserve" listing-type="equation">b={Σ

.sub.c Y.sup.a.sub.c -mΣ

.sub.c Y.sup.k.sub.c }/I

where;

Y^a_c =Dependent-variable-(ac) is selected from the group of values consisting of;

a directly determined value for property-p of a particular member c, of sample-p;

a predicted value for property-p of said member c, of sample-p by means of a calibration equation for property-p in a range of interest; and

a combination thereof;

I is the number of members of sample-p; and

to apply said slope and bias correction, substitute said values for m and b into the following expression;

space="preserve" listing-type="equation">Y.sup.k.sub.c ={mA.sup.k +b}+Σ

.sub.j {mB.sup.k.sub.j I.sup.k.sub.jc }

whereby an improved calibration equation-k for dependent-variable-(kc) for instrument-k is obtained.

56. The improved method of claim 50, wherein said property-p is selected from the group consisting of octane and cetane and said spectral data-k and -i, respectively, each correspond to at least one absorbance for light radiation with a frequency for each member c, in the range of about 800 to 2500 nanometers.

57. In a method of claim 50, wherein said member c is selected from the group consisting of isooctane, n-decane, 1-octene, 2,3,4-trimethylpentane, n-heptane cyclohexane, n-hexane, and toluene.

58. The improved method of claim 50, wherein instead of a value equal to dependent-variable-(ic), Yⁱ_c, being substituted for that value for dependent-variable-(kc), Y^k_c, a mathematically transformed dependent-variable-(ic), Y^t_c, which is transformed by means of a function is used to yield a value that is substituted for dependent-variable-(kc) in calibration equation-k.

59. The improved method of claim 58, wherein said mathematically transformed dependent-variable-(ic) value, Yⁱ_c, is determined by a function selected from the group consisting of:

Y^t_c =u×

Yⁱ_c +v;

Y^t_c =u×

log Yⁱ_c +v;

where u is any real number not equal to zero and v is any real number including zero.

60. The improved method of claim 50, wherein said first function to obtain spectral feature-(ic) and spectral feature-(kc), respectively from spectral data-(ic) and spectral data-(kc) for each member c, comprises differentials.

61. The improved method of claim 60, wherein said differentials comprise at least second order.

62. The improved method of claim 61, wherein said wavelength-j includes at least one frequency selected from the group consisting of at least one methyne band.

63. In a method for recalibrating an instrument-k in light of its former calibrated condition as instrument-i or for calibrating instrument-k, previously uncalibrated, in light of a calibrated instrument-i wherein a calibrated instrument is one having a particular calibration equation, here calibration equation-k, based upon which equation spectral data of a sample can be used to predict a physical or chemical property of said sample, the improvement which comprises:

determining said calibration equation-k for instrument-k by means of said calibration equation-i for instrument-i, wherein said calibration equation-k defines a dependent-variable-(ku), in terms of at least one constant-k and at least one independent-variable-(kju), wherein said at least one independent-variable-(kju) has a value equal to at least one spectral feature-(kju) determined from said spectral data-(ku) measured by said instrument-k for said sample-u by a first function that involves wavelength-j, and wherein said dependent-variable-(ku) has a value substantially equal to a property-p for said sample-u when values for each said at least one constant-k and each said at least one independent-variable-(kju) are substituted appropriately into said calibration equation-k;

wherein said calibration equation-i defines dependent-variable-(iu) as a function of an independent-variable-(iju) which independent-variable-(iju) is equal to a spectral feature-(iju) determined from spectral data-(iu) measured by said instrument-i for said sample-u corresponding to each wavelength-j, and has a value substantially equal to said property-p of said sample-u;

said method for determining said calibration equation-k by means of said calibration equation-i comprising;

(1) using respectively spectral data-(ic) and spectral data-(kc) for each member c, of a group of samples having n members that are each measured respectively by said instrument-i and instrument-k, to determine each said at least one independent-variable-(ijc) and independent-variable-(kjc), respectively, involving each wavelength-j corresponding to each member c, and to determine from calibration equation-i a value for said dependent-variable-(ic) corresponding to each member c;

(2) maintaining consistent a set of equations that results from substituting in place of the value for dependent-variable-(kc) that value for dependent-variable-(ic), while selecting a value for each said at least one constant-k so that a sum of absolute differences between each new-dependent-variable-(kc) and dependent-variable-(ic) corresponding to each said member c, is minimized;

wherein said each new-dependent-variable-(kc) is determined by calibration equation-k after substituting each selected value for each said at least one constant-k along with each said at least one independent-variable-(kjc) in said calibration equation-k.

64. The method of claim 63, wherein said selecting a value for each said at least one constant-k is continued at least until standard error of estimation for property-p is not significantly greater than that standard of estimation for property-p used to establish calibration equation-i.

65. The method of claim 63, wherein said calibration equation-k has the following form which defines an intercept-constant-k, A^k, and an at least one slope-constant-(kj), B^k_j :

space="preserve" listing-type="equation">Y.sup.k.sub.c =A.sup.k +Σ

.sub.j B.sup.k.sub.j I.sup.k.sub.jc

and wherein said selection of values for said at least one constant-k comprises selecting at least one intercept-constant-k and at least one slope-constant-(kj) so that the immediately following expression is reduced substantially to its minimum value;

space="preserve" listing-type="equation">Σ

.sub.c (Y.sup.i.sub.c -A.sup.k -Σ

.sub.j {B.sup.k.sub.j I.sup.k.sub.jc }).sup.2.

66. The method of claim 63, wherein said calibration equation-i is in the same linear form as calibration equation-k and wherein said calibration equation-i defines dependent-variable-(ic), Yⁱ_c, in terms of at least one constant-i and at least one independent-variable-(ijc), Iⁱ_jc, wherein said at least one constant-i comprises:

an intercept-constant-i, Aⁱ, and at least one slope-constant-(ij), Bⁱ_j, involving each wavelength-j used in calibration equation-i; and

wherein each said at least one independent-variable-(ijc), Iⁱ_jc, has a value substantially equal to each at least one spectral feature-(ijc) determined by said first function from spectral data-(ic) measured by said instrument-i that involves each said wavelength-j for member c; and

wherein said dependent-variable-(ic), Yⁱ_c, has a value substantially equal to said property-p for said sample-u when values for each said at least one constant-i and each said at least one independent-variable-(ijc) are appropriately substituted into said calibration equation-i and sample-u is member c, whereby calibration equation-i and equation-k, respectively, have the linear forms;

space="preserve" listing-type="equation">Y.sup.i.sub.c =A.sup.i +Σ

.sub.j {B.sup.i.sub.j I.sup.i.sub.jc }

space="preserve" listing-type="equation">and

space="preserve" listing-type="equation">Y.sup.k.sub.c =A.sup.i +Σ

.sub.j {B.sup.i.sub.j I.sup.i.sub.jc };

and wherein the following two equations are simultaneously substantially satisfied;

space="preserve" listing-type="equation">B.sup.k.sub.j =B.sup.i.sub.j {Σ

.sub.c (I.sup.i.sub.jc I.sup.k.sub.jc)}/Σ

.sub.c (I.sup.k.sub.jc).sup.2

space="preserve" listing-type="equation">and

space="preserve" listing-type="equation">A.sup.k =Σ

.sub.c {Y.sup.i.sub.c -Σ

.sub.j (B.sup.k.sub.j I.sup.k.sub.jc)}/n

whereby said calibration equation-k for instrument-k is obtained by reference to said calibration equation-i.

67. The method of claim 63, wherein said calibration equation-i is in the same linear form as calibration equation-k and wherein said calibration equation-i defines dependent-variable-(ic) in terms of at least one constant-i and at least one independent-variable-(ijc), wherein said at least one constant-i of calibration equation-i comprises:

an intercept-constant-i, and at least one slope-constant-(ij) involving each wavelength-j used in calibration equation-i; and

wherein each said at least one independent-variable-(ijc) has a value substantially equal to each at least one spectral feature-(ijc) determined by said first function from spectral data-(ic) measured by instrument-i that involves each said wavelength-j for member c, and wherein said dependent-variable-(ic) has a value substantially equal to said property-p for said sample-u when values for each said at least one constant-i and each said at least one independent-variable-(ijc) are appropriately substituted into said calibration equation-i and sample-u is member c;

whereby calibration equation-i and equation-k, respectively have the linear forms;

space="preserve" listing-type="equation">Y.sup.i.sub.c =A.sup.i +Σ

.sub.j {B.sup.i.sub.j I.sup.i.sub.jc }

space="preserve" listing-type="equation">and

space="preserve" listing-type="equation">Y.sup.k.sub.c =A.sup.k +Σ

.sub.j {B.sup.k.sub.j I.sup.k.sub.jc };

and wherein the following two equations are simultaneously substantially satisfied;

space="preserve" listing-type="equation">B.sup.k.sub.j =B.sup.i.sub.j Σ

.sub.c I.sup.i.sub.jc /Σ

.sub.c I.sup.k.sub.jc

space="preserve" listing-type="equation">and

space="preserve" listing-type="equation">A.sup.k =Σ

.sub.c {Y.sup.i.sub.c -Σ

.sub.j (B.sup.k.sub.j I.sup.k.sub.jc)}/n

whereby said calibration equation-k for instrument-k is obtained by reference to said calibration equation-i.

68. The method of claim 63, wherein there is a further improvement of applying a slope and bias correction comprising the steps of:

determining values for m and b, which substantially satisfy the following expressions simultaneously;

space="preserve" listing-type="equation">m={Σ

.sub.c Y.sup.a.sub.c Σ

.sub.c Y.sup.k.sub.c }/I-Σ

.sub.c (Y.sup.a.sub.c Y.sup.k.sub.c)}/{(Σ

.sub.c Y.sup.k.sub.c Σ

.sub.c Y.sup.k.sub.c)/I-Σ

.sub.c Y.sup.k.sub.c Y.sup.k.sub.c }

space="preserve" listing-type="equation">b={Σ

.sub.c Y.sup.a.sub.c -mΣ

.sub.c Y.sup.k.sub.c }/I

where;

dependent-variable-(ac), Y^a_c, is selected from the group of values consisting of;

a directly determined value for property-p of a particular member c, of sample-u;

a predicted value for property-p of a particular member c, of sample-u by means of a calibration equation for property-p in a range of interest; and

a combination thereof;

I is the number of members of sample-u; and

to apply said slope and bias correction, substitute said values for m and b into the following expression;

space="preserve" listing-type="equation">Y.sup.k.sub.c ={mA.sup.k +b}+Σ

.sub.j {mB.sup.k.sub.j I.sup.k.sub.jc }

whereby an improved calibration equation-k for a slope and bias corrected dependent-variable-(kc), Y^k_c, for instrument-k is obtained.

69. The method of claim 63, wherein said property-p is selected from the group consisting of octane and cetane and said spectral data-k and -i, respectively, each correspond to at least one absorbance for light radiation with a frequency for each member c, in the range of about 800 to 2500 nanometers.

70. In a method of claim 63, wherein said member c is selected from the group consisting of isooctane, n-decane, 1-octene, 2,3,4-trimethylpentane, n-heptane cyclohexane, n-hexane, and toluene.

71. The method of claim 63, wherein instead of a value equal to dependent-variable-(ic) being substituted for that value for dependent-variable-(kc), a mathematically transformed dependent-variable-(ic), Y^t_c, by means of a function is used to yield Y^t_c that is substituted for dependent-variable-(kc), Y^k_c, in calibration equation-k.

72. The method of claim 71, wherein said Y^t_c is determined by a function selected from the group consisting of:

Y^t_c =u×

Yⁱ_c +v;

Y^t_c =u×

log Yⁱ_c +v;

where u is any real number not equal to zero and v is any real number including zero.

73. The method of claim 63, wherein said first function to obtain spectral feature-(ic) and spectral feature-(kc), respectively, from spectral data-(ic) and spectral data-(kc) for each member c, comprises differentials.

74. The method of claim 73, wherein said differentials comprise at least second order.

75. The method of claim 74, wherein said wavelength-j includes at least one frequency selected from the group consisting of at least one methyne band.

76. A method for calibrating or recalibrating a first instrument, instrument-k, by reference to a second instrument, instrument-i, comprising in any order:

(a) obtaining spectral data by means of instrument-k and instrument-i, corresponding respectively to data-k and data-i for each member of a calibration set;

(b) determining by means of a calibration equation, calibration equation-i, of said instrument-i, a value, value-i, for a dependent variable, dependent variable-i, corresponding to each member of said calibration set;

(c) substituting a value, value-f, related by function-f to said value-i for said dependent variable-i, for each member of the calibration set in place of that value for said dependent variable-k in a calibration equation-k which contains constants-k and a dependent variable, dependent variable-k;

(d) selecting values for said constants-k in calibration equation-k so that the sum of absolute differences between;

(1) each dependent variable-k calculated based on said calibration-k with said selected constants-k and data-k for each member of said calibration set; and

(2) corresponding values for identical members of said calibration set determined by function-f and corresponding value-i for dependent variable-i of each said member;

is substantially minimized;

whereby said constants-k can be inversely transformed by an inverse to function-f to define calibration equation-k.

77. A method for determining constants in a calibration equation, calibration equation-1, for a first instrument by recourse to a second instrument having a calibration equation, calibration equation-2, comprising in any order:

(a) determining from calibration equation-2 a value for each dependent variable-2 corresponding to each member of a calibration set based on spectral data-2 for each member of said calibration set measured by means of instrument-2; and

(b) determining constants-1 in said calibration equation-1 so that the sum of absolute differences between;

(1) said value for each said dependent variable-2 for each member of said calibration set; and

(2) that corresponding value of dependent variable-1 calculated for each said member based upon calibration equation-1;

is substantially minimized.

78. The method of claim 77, wherein said calibration set has at least one member that has spectral data-2 which in conjunction with calibration equation-2 determines a value for said dependent variable-2 for said at least one member which is significantly different from that value determined by a direct test method for which said calibration equation-1 and calibration equation-2 were established to model.

79. The method of claim 77 wherein a slope and bias adjustment is made to instrument-k'"'"'s calibration equation to minimize the errors on subsequent samples.