An overview of the methods of line-by-line and whole-band analysis developed by The Ohio State University spectroscopy group is presented

Shaw, J. H.

English

The development of efficient techniques for extracting the maximum amount of information from the spectra of atmospheric molecules with a minimum of observer bias is discussed. In particular, an overview of the methods of line by line and whole band analysis is presented

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 This document may contain data, which exceeds the sheet parameters. It was 
furnished in this condition by the organizational source and is the best copy 
available. 
 
 This document may contain tone-on-tone or color graphs, charts and/or pictures, 
which have been reproduced in black and white. 
 
 This document is paginated as submitted by the original source. 
 
 Portions of this document are not fully legible due to the historical nature of some 
of the material. However, it is the best reproduction available from the original 
submission. 
 
 
 
 
 
 
 
Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19840008809 2020-03-20T23:44:10+00:00Z
Least Squares Methods of
Analyzing Spectroscopic Data*
(NASA-CR-175335) LEAST SQUARES METHODS OF
ANALYZING SPECTROSCOPIC DATA (Ohio State
Univ., Colusbus.1 23 p HC A02/MF :01
CSCL 12A
N84-16877
Unclas
G3/64 18244
J. H. Shaw
The Ohio State University
Columbus, Ohio 43210
*This work was supported in part by
NASA Grant NSG-7469
J
Ed
0Abstract
An overview of the methods of line-by-line and whole-band analysis developed
by The Ohio State University spectroscopy group is presented.
F
(+Y
Introduction	 OF POOR QUALITY
IT
One of the objectives of our group is the collection and analysis of laboratory
spectra of molecules of atmospheric interest. In recent years we have developed efficient
techniques for extracting the maximum amount of information from these spectra with a
minimum of observer bias. A description of the approach to the analysis of single lines
and entire bands, taken by the members of our group listed in Table 1, is described here.
More detailed descriptions are given elsewhere (see reverences).
Single Line Analysis
A typical problem in spectral analysis is the determination of line positions. One
method is to estimate the line center and to determine its position with respect to other
calibration features by interpolation, as indicated in Fig. 1. This method is often
adequate but it is tedious if there are many lines to be measured. It is also difficult to
estimate the precision of the results since the judgments used to determine the line
center and in the interpolation process cannot be quantified.
Less reliance is placed on the observer if the experimental signals I (u ) are
exp i
compared with the corresponding signals I 	
i
art (u ) of an artificial line centered at v (art) as
o
shown in Figure 2a. The spectrum of the differences
i	 exp i	 art i
shown in Fig. 2b, indicates the sensitivity of the data to changes in the line position. This
sensitivity can also be tested by calculating the partial derivative aI 	 (v )/av . A best
exp i
	
i
estimate of the line position v is obtained by shifting the position v (art) until the sum of
0	 0
the squares of the differences
Or N 7', , 
SS . EAIs(v)	
GI^1^
OF POOR QJAL,TYi
is a minimum as indicated in Fig. 1c. The rate of change of the SS with respect to the
difference Av = v (exp) - v (art) gives a measure of the precision with which the line
0	 0
center can be found. Unlike the first method, ovice the model for the artificial line and
the criteria for establishing the uncertainty of the line position have been defined, no
other observer judgment is required.
Other parameters can also be determined by this method. By generating
artificial :fines I (vi , S ) as shown in Fig. 1d, corresponding to lines of differentart
intensities, and minimizing
s
SS • EAI 0, S	 ),
i art
where	 IN , S ) = I	 (v) - IN , S ),i art	 exp i	 i art
estimates of the line intensity and its precision can be obtained. In this method
(1) The intensity and position estimates, and also estimates of other variables p j
which affect the signals INN), are obtained from the same set of experimental
data.
(2) Estimates of the goodness of fit to I	
i	
ms
exp 
(v ) are obtained from the su SS. These
minima should not be significantly larger than those due to the noise in the data.
(3) Good fits are only obtained if the models are accurate. The accuracy of the
models as well as the parameter estimates p can be Judged by examination of the
residuals spectra AI(v
i
 ).
(4) All the information about each model and its parameters contained in the spectra
can be extracted since all the data are examined.
(5) The ultimate precision is dependent on the characteristics of the experimental
data. Some of these characteristics are described by the partial derivatives
W )F
,1
) I'l
n•1
^E
OF FOUR QUALt N
BIeXp(v)/8pj.
(6) All of the parameters can be estimated simultaneously by using appropriate
least squares regression techniques.
Niple (8,9) and Tu (10) have considered some of the limitations of the
experimental data by analyzing simulated spectra of single lines. They have shown that
the precisions of the retrieved parameters depend on
(a) the SNR of the spectrum,
(b) the number N of data points analyzed,
0
(c) the spacing dv of the data points,
(d) the number of models required and parameters retrieved,
(e) the accuracy of the models,
(f) the magnitudes and shapes of the partial derivatives &/8p , and
(g) the correlations between the parameters p .i
It is necessary to use a minimum of four models containing at least six adjustable
paratr .eters to create a set of simulated signals Y	 N) suitable for analysis. These are
exp i
briefly described in Table 2. Tu (10) has investigated the information content in spectra
similar to those in Fig. 3, calculated for a Lorentz line, a triangular ISRF wish H - a, a
4background and baseline independent of % and a SNR of 10 . He has estimated the
dependence of the fractional uncertainty AS/S in the retrieved line intercity on S as
shown to Fig. 4 for the case where all six parameters are retrieved. Because of increasing
correlations between some of the parameters there is a limited range of S values for
which satisfactory retrievals can be made. These correlations are related to the
derivatives 8I(v)/ap shown in Fig. S.
OF POOR QUALITY
Some improvement in these fractional uncertainties is obtained by fixing one or
more of the parameters, although this may introduce systematic errors if these
parameters are fixed to incorrect values. It is seen, from Fig. 4, that, at small S values,
the greatest improvement is obtained by fixing the baseline.
From these studies quantitative evaluations of experimental designs can be
made These have yielded some interesting results in addition to confirming the intuitive
conclvzons obtained by less rigorous analyses of experimental spectra. In particular
(1)	 The usable range can be determined. This is the range of experimental
conditions for which all six parameters of these simple models can be estimated.
(a)
	
For the cases considered here this range is primarily limited by correlations
between S, B, a, and H.
(3) The usable range can be extended by fixing one or more of the adjustable
parameters.
(4) The systematic errors introduced by fixing the parameters may be larger than
the uncertainties of the parameters retrieved by the regression methods used
here. This is especially likely if the fixed parameter was highly correlated with
free parameter.
Whole Band Analysis
For many atmospheric problems it is sufficient to know the intensit-1, width, and
position of a few lines. However, these data also serve as the raw material from which
12^
2. the number of parameters retrieved is relatively small and this allows their
precision to be increased,
3. the usable range is larger than that for single lines, because many of the highly
correlated parameters previously measured are no longer estimated directly, and
4. the accuracy of the band models can be rigorously tested.
When the band systems in Fig. 6 were analyzed (5) by this method the residuals
spectrum shown in the lower part of the figure was obtained. The weak Q branch near
2615 c,n and associated P and R branch lines are just above the noise level and are too
weak to be modelled satisfactorily. Even so, the sum of the squares of the differences is
within 50% of that expected from the noise and this is considered to be a good retrieval.
This agreement confirms that the line shape, the band models, the ISRF, and
other instrumental effects have been modelled sufficiently well to describe the
experimental data. These data have an estimated SNR of 200-300 and were obtained with
-i
a Fourier Transform Spectrometer (FTS) having a spectral resolution of about 0.06 cm
The results of the analysis of this band system have been described by Hoke (5, 6). The
-i
center of the principal band was estimated to occur at 2614.241124 t 0.000055 cm , this
precision is at least an order of magnitude better than is usually claimed from
measurements made with instruments of similar resolution and performance.
If the background, baseline, and ISRF parameters are retrieved, together with
the band intensity and line width parameters, then the line intensities and widths can be
estimated to about 2%. In all other reported measurements of similar quantities these
instrumental parameters have been assumed known. When our analyses were repeated
with these same assumptions the uncertainties in the line intensities and widths were
nV16 (:;y
more fundamental molecular properties can be derived, provided a sufficiently large
number of lines in a vibration-rotation band can be analyzed.
The strongest band of CO
2 
shown in Fig. 6 contains over 100 lines. If a
line-by-line analysis on this band were performed by the methods just described up to 600
parameter values would be required. However, the spectrum in Fig. 6 contains three CO
2
bands with lines of measurable intensity and thus more than 1000 parameters may be
required to describe this spectrum. Many of these lines lie outside the usable range
describe? ;above and most are overlapped and blended with other lines. Unles= the
in- ►estigator makes arbitrary decisions concerning some of the adjustable parameters in
these line by lisle models, only a small fraction of the lines can be analyzed satisfactorily.
However, the parameters describing these lines are not independent and. the
models for the background and baseline can be assurnt-d to be smoothly varying functions
over the band. Thus the spectrum in Fig. 6 can be described by models similar to those in
Table 2 and a series of additiona, models which relate the parameters associated with the
individual lines (S, 6). The models describing these additional relationships and the typical
numbers of adjustable parameters in these models are shown in Table 3. By extending the
analysis of the previous section to the retrieval of the parameters in this table, the
problem is reduced to estimating of the order of 40 parameters rather than over 1000. We
have used this technique to analyze overlapping bands of CO, CO , and N O which contain
2	 2
over 3000 data values and several hundred lines (see Refs. 3-7). The advantages of this
approach are
1.	 the parameters of primary interest are found directly from the experimental
data,
f
e
reduced to less than 0.1%.
Portions of the experimental spectrum, the corresponding calculated spectrum,
and the residuals spectrum obtained during the course of this investigation are shown in
Fig. 7. Although it is difficult to detect significant differences between the observed and
calculated spectra the residuals spectrum shows systematic variations attributed to poor
modelling of the ISRF. These systematic differences are not seen in the final residuals
spectrum shown in Fig. 8.
In our analysis of CO, CO 2 and N2 O bands we have consistently obtained stable,
reproducible solutions, but the size of the data sets which can be handled and the number
of parameters retrieved is limited by the computer resources available. As these
resources expand it is expected that several spectra of the same band and/or several band
systems will be analyzable simultaneously. The band systems of most molecules are
V I	 described by more complicated models than those required in these investigations. In
these cases the whole band methods of analysis can be powerful tools for testing the
	 t
accuracies of these models and for identifying the significant parameters.
These analytical methods involve the application of standard statistical
techniques to the analysis of typical experimental data. However they have advantages£,
over the more conventional methods of spectral analysis which are similar to the
advantages claimed for FTS over dispersive spectroscopic techniques. These are usually
described as the throughput and multiplex advantages.
By analogy with these comparisons the multiplex advantage of whole band
analysis over line-by-line analysis is the direct, simultaneous retrieval of the parameters
of interest. There is also a corresponding throughput advantage in that the entire data set
is searched for information about each parameter.
a^P111i;FW90^ 1.
ab
Selected References
1. Y. S. Chang Ph.D. diss. The Ohio State Univ. (1976).
2. Y. S. Chang and J. H. Shaw, Applied Spectroscopy, 31, 218 ( 1977).
3. R. L. Hawkins Ph.D. diss. The Ohio State Univ. (1982).
4. R. L. Hawkins and J. H. Shaw JQSRT, 29, 543 (1983).
5. M. L. Hoke Ph.D. diss. The Ohio State Univ. (1982).
6. ri:. . Hoke & J. H. Shaw, Applied Optics 21, 929, 935 (1983).
7. C. L. Lin, J. H. Shaw and J. G. Calvert JQSRT 13, 387 (1980).
8. E. Niple Ph. D. diss. The Ohio Stste Univ. ( 1980).
9. E. Niple and J. H. Shaw Appl. Spectroscopy 3" 569 (1979).
10. N. Tu and J. H. Shaw, paper CEi, Fall Meeting, Ohio Section, American Physical
Society (1983).
I r.
Table 1 Contributors w these studies
R. L. Hawkins	 M. L. Hoke
E. Niple	 N. Tu
Y. S. Chang
C. L. Lin
I 
o 
N = 1Q
v 
1
Io
H
4	 Baseline to line
(polynomial in W
Synthetic spectrum
i-
r
BNi) = B
H
B
Z/
v
Table 2 Single Line Models and Parameters 	 C)r,^t a .
OF POOR QJA .li'Y
Minimum
Model	 Parameters
1 Line Shape	 I	 vo
(Lorentz, Voigt, ...)	 Y	 I -a
L —),
^	
a
: 
w	 S=/kdu	 I
0
2 Instrument Spectral	 CResponse Function	 0
(ISR F)	 a
(sinc, triangle
Gaussian, ... )
3 Background to
iine (polynomial
in u and other
terms)
H
v
Quantity Model Number of Parameters
Line Intensities Band Intensity —3 if interaction terms
S 	 = ES(lines) are included(the sample
temperature and E" Me
are also required)
Line Posi l iui_s u = E' - E" For CO 2 the upper and
lover state energies can
each be modelled by
expressions containing
3 or 4 parameters
<8
The vari stion in the Lorentz
Line widths widths o• +er the band is '10
modelled 1 ^v empirical relations.
The models contain 3-4
ISRF parameters which are assumed <4
constant over the band.
Several parameters may be
IoNI ) required if channel spectra <10
are present.
B(vi ) Usually 2 terms are adequate. <3
no.
F
c
Table 3 Models for Describing Entire Bands
Total number of parameters retrieved 30 - 40.
[E
aP4
(+'Y
Figure Captions
Fig. 1. Line positions by interpolation. The posit'on of the )th line is
-1
(2622 + Ax/Axl ] cm
Fig. 2. a) The spectrum I
art i
(v ) of an artificial line is compared with the experimental
dais I	
iI
exp (u ). b) the differences AI(u ) are similar in shape to the partial
derivative aI
exp i
(u )/6v. c) the best estimate of the line position occurs at the
2
minimum value of the sum EAI (vi ); d), e), f) corresponding steps required to
determine the line intensity.
Fig. 3. A set of lines generated by using a Lorentz line shape, a triangular ISRF with
-1
H = a = 0.1 cm , and constant I an,' B functions.
0
Fig. 4. The dependence of the fractional uncertainty in the retrieved line intensity on
the line intensity for the case of 6 retrieved parameters and for the cases where
B, I , and H are fixed to their correct values.
0
Fig. 5. The derivatives (3I(v )/Zp as functions of v for the parameters p = S, a, u , H, I
i	 j	 j	 o	 0
and B and the dependence of these derivatives on the line intensity S.
-1
Fig. 6. Upper curve: a spectrum of CO near 2600 cm . Lower curve: the residuals
2
spectrum after modelling the entire spectrum and retrieving more than 30
adjustable parameters.
n^
Fig. 7. The upper portion of this figure shows a comparison between a portion of the
experimental spectrum in Fig. 6 and a calculated spectrum. The residuals
spectrum in the lower part of this figure has systematic variations attributed to
incorrect modelling.
Fig. 8. The upper part of this figure is the same portion of the experimental spectrum as
that in Fig. 7. The lower part of this figure is the residuals spectrum obtained by
using the best estimates for the parameters.
VC4
o'
v
c
v+
U)
ORIU&NA:. -er;(;`C 13
OF POOR QUALITY
2620	 2621	 2622	 2623
Wavenumber (cm-1)
I(a)
vo(art)
ORIGINAL PAGE IS
OF POOR QUALITY
vo(art)
Iexp
P
I
c
Lp (art)
1^sI
Lo(art)
%(art)
ai
as
S (art)
8	 999	 1000
Frequency (cm-1)
1001	 1002
A
C
D
E
A S=00625
B S=0.1
C S=0.25
D S= 1.0
E S=4.0
QE
4--
O
Cl
E
(f)
D
0.4
0.2
OF POOR QUALrry
'n^ 0
U.1	 0.5	 1.0	 5	 10	 5C
S=Sou/v
104
—.a
0+4
V)
Cnd
OF POOR QUAL'Ty
q None Fixed
p	 o B Fixed
0 Io Fixed
0 H Fixed
O
10-1
X10
c
c
c
D
c
D
v 103
cO
Li
'LIS
q O
0 O
°o
o° o
o 0O0 q0 q
0 00
°O q q
0 0q
D
I Z^
---
OT
0 01-
OFtPOOR QJr..9T``T
n
0
0
o
T-
11 p
0
C\j
(DO
T
-
p
t^ 0p
o
a N
0 0 — 0H (/7
0-
NN
> 0
0
O
u 0
0
0
° C
IO
q VV
0-V) I 
v 0
0- p 0
N
Ac-- 0 p
u
vo 	 H
	
TD
Wavenumber (cm-')
G
02585.00	 2605.00	 2625.00	 2645.00	 2665.00
WAVENUMBERS (CM- 1)
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O
O
2565.00
4
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O
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>- o
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0--. N
(r) O
Ln0
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JQo
^o
O o
W
cc
Ln00
2565.00 2585.00
	 2605.00
	 2625.00
WAVFN(1Ml3ERS tCM- 1 l
2645.00
	 2865.00
=i
OBSERVED
0
m
^ o
F-
Z
0
m
m o
Q
m
h--
m o
mQ o
Q
Z
m oru
Le) o
0
00	 CALCULATED
OF POOR ^,.I i Y
0Co
0-
2620.00	 2621.00	 2622.00
WRVENUMBERS
2623.00
	
2624.00	 2625
(CM-1)
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0
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--19
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(+m"'Y
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	 2621.00	 2622.00
	
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	 2625
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vw


Least squares methods of analyzing spectroscopic data

General Disclaimer 
One or more of the Following Statements may affect this Document 
 
 This document has been reproduced from the best copy furnished by the 
organizational source. It is being released in the interest of making available as 
much information as possible. 
 
 This document may contain data, which exceeds the sheet parameters. It was 
furnished in this condition by the organizational source and is the best copy 
available. 
 
 This document may contain tone-on-tone or color graphs, charts and/or pictures, 
which have been reproduced in black and white. 
 
 This document is paginated as submitted by the original source. 
 
 Portions of this document are not fully legible due to the historical nature of some 
of the material. However, it is the best reproduction available from the original 
submission. 
 
 
 
 
 
 
 
Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19840008790 2020-03-20T23:44:08+00:00Z
Least Squares Methods of
Analyzing Spectroscopic Data*
L; I
ego
Q.^r
c
J. H. Shaw
JAN 1984
The Ohio State University
	 RECEIVED
Columbus, Ohio 43110
	 - j0M DEPT.Wasnumm
*This work was supported in part by
NASA Grant NSG-7469
( NASA-C8-173241) LEAST SQUARES METHODS OF
	 N84-16858ANALYZING SPECTROSCOPIC DATA (Ohio State
Univ., Columbus.) 23 p HC A02/MF A01
CSCL 12A
	 Unclas
G3/64 00604
O
^TJ^
Abstract.
An overview of the methods of line-by-line and whole-band analysis developed
by The Ohio State University spectroscopy group is presented.
ROF POOH QUAL11-Y
Introduction
One of the objectives of our group is the collection and analysis of laboratory
spectra of molecules of atmospheric interest. In recent years we have developed efficient
techniques for extracting the maximum amount of information from these spectra with a
minimum of observer bias. A description of the approach to the analysis of single lines
and entire bands, taken by the members of our group listed in Table 1, is described here.
More detailed descriptions are given elsewhere (see references).
Single Line Analysis
A typical problem in spectral analysis is the determination of line positions. One
method is to estimate the line center and to determine its position with respect to other
calibration features by interpolation, as indicated in Fig. 1. This method is often
adequate but it is teWous if there are many lines to be measured. It is also difficult to
estimate the precision of the results since the Judgments used to determine the line
center and in the interpolation process cannot be quantified.
Less reliance is placed on the observer if the experimental signals 1 (v) are
exp i
	
compared with the corresponding signals I 	
i
art (u ) of an artificial line centered at v (art) as
o
shover in Figure aa. The spectrum of the differences
	
AN = I	 (u) - I	 (u ),i	 exp i	 art i
shown in Fig. ab, indicates the sensitivity of the data to changes in the line position. This
sensitivity can also be tested by calculating the partial derivative 
aIexp N IV&oI . A best
estimate of the line position u is obtained by shifting the position v (art) until the sum of
0	 0
the squares of the differences
i
tT/•
	
2	
ORIGINAL ^=eta.: E ;'
	
SS . Edi(1)	
of POOR QUALITY
is it minimum as indicated in Fig. 1c. The rate of change of the SS with respect to the
difference Au = v (exp) - u (art) gives a measure of the precision with which the line
0	 0
center can be found. Unlike the first method, once the model for the artificial line and
the criteria for establishing the uncertainty of the line position have been defined, no
other observer judgment is required.
Other parameters can also be determined by this method. By generating
artificial lines IN
i , 
S 
art ) 
as shown in Fig. ad, corresponding to lines of different
intensities, and minimizing
2
Ss - ELI (v , S ),
i art
where	 I(v, S ) = I (v) - IN
i art	 exp i	 i 'Sart
^i	 estimates of the line intensity and its precision can be obtained. In this method
(1) The intensity and position estimates, and also estimates of other variables p
which affect the signals Ilv i), are obtained from the same set of experimental
	 y
data.
(a) Estimates of the goodness of fit to I	
iexp 
(v ) are obtained from the sums SS. These
minima should not be significantly larger than those due to the noise in the data.
(3) Good fits are only obtained if the models are accurate. The accuracy of the
models as well as the parameter estimates pj can be judged by examination of the
residuals spectra AI(vi).
(4) All the information about each model and its parameters contained in the spectra
can be extracted since all the data are examined.
(5) The ultimate precision is dependent on the characteristics of the experimental
data. Some of these characteristics are described by the partial derivatives
o'
01^^1a:._
aieXp(v)/apj. 	 OF POOR QUALM,
(6) All of the parameters can be estimated simultaneously by using appropriate
least squares regression techniques.
Niple (8,9) and Tu (10) have consiiered some of the limitations of the
experimental data by analyzing simulated spectra of single lines. They have shown that
the precisions of the retrieved parameters depend on
(a) the SNR of the spectrum,
(b) the number N of data points analyzed,
0
(c) the spacing 6v of the data points,
(d) the number of models required and parameters retrieved,
(e) the accuracy of the models,
(f) the magnitudes and shapes of the partial derivatives Wittj , and
(g) the correlations between the parameters p .j
It is necessary to use a minimum of four models containing at least six adjustable
parameters to create a set of simulated signals I  (v i) suitable for analysis. These are
xp
briefly described in Table 2. Tu (10) has investigated the information content in spectra
similar to those in Fig. 3, calculated for a Lorentz line, a triangular ISRF with H = a, a
4background and baseline independent of v, and a SNR of 10 . He has estimated the
dependence of the fractional uncertainty AS/S in the retrieved line intensity on S as
shown in Pig. 4 for the case where all six parameters are retrieved. Because of increasing
correlations between some of the parameters there is a limited range of S values for
which satisfactory retrievals can be made. These correlations are related to the
derivatives aI(v)/apj shown in Fig. 5.
U^
ORIGINAL PAGE 1S
OF POOR QUALITY
Some improvement in these fractional uncertainties is obtained by fixing one or
more of the parameters, although this may introduce systematic errors if these
parameters are fixed to incorrect values. It is seen, from Fig. 4, that, at small S values,
the greatest improvement is obtained by fixing the baseline.
From these studies quantitative evaluations of experimental designs can be
made. These have yielded some interesting results in addition to confirming the intuitive
conclusions obtained by less rigorous analyses of experimental spectra. In particular
(1)	 The usable range can be determined. This is the range of experimental
conditions for which all six parameters of these simple models can be estimated.
(1)	 For the cases considered here this range is primarily limited by correlations
between S, B, a, and H.
(3) The usable range can be extended by fixing one or more of the adjustable
parameters.
(4) The systematic errors introduced by fixing the parameters may be larger than
the uncertainties of the parameters retrieved by the regression methods used
here. This is especially likely if the fixed parameter was highly correlated with
	
V	 free parameter.
1.
	1	 Whole Band Analysis
For many atmospheric problems it is sufficient to lmow the intensity, width, and
position of a few lines. however, these data also serve as the raw material from which
2.	 the number of parameters retrieved is relatively small and this allows their
precision to be increased,
3	 the usable range is larger than that for single lines, because many of the highly
correlated parameters previously measured are no longer estimated directly, and
4.	 the accuracy of the band models can be rigorously testel.
When the band systems in Fig. 6 were analyzed (5) by this method the residuals
spectrum shown in the lower part of the figure was obtained. The weak Q branch near
--12615 c w and associated P and R branch lines are Just above the noise level and are too
weak to be modelled satisfactorily. Even so, the sum of the squares of the differences is
within 50% of that expected from the noise and this is considered to be a good retrieval.
This agreement confirms that the line shape, the band models, the ISRF, and
other instrumental effects have been modelled sufficiently well to describe the
experimental data. These data have an e-timated SNR of 200-300 and were obtained with
-i
s Fourier Transform Spectrometer (FTS) having a spectral resolution of about 0.06 cm
The results of the analysis of this band system have been described by Hoke (5, 6). The
-i
center of the principal band was estimated to occur at 2614.241224 t 0.000055 cm , this
precision is at least an order of magnitude better than is usually claimed from
measurements made with instruments of similar resolution and performance.
If the background, baseline, and ISRF parameters are retrieved, together with
the band intensity and line width parameters, then the line intensities and widths can be
estimated to about 2%. In all other reported measurements of similar quantities these
instrumental parameters have been assumed known. When our analyses were repeated
with these same assumptions the uncertainties in the line intensities and widths were
OF POOR QUALITY
was)
ORIGINAL PACE FS
OF POOR QUALITY
more fundamental molecular properties can be derived, provided a sufficiently large
number of lines in a vibration-rotation band can be analyzed.
The strongest band of CO 2 shown in Fig. 6 contains over 100 lines. If a
line-by-line analysis on this band were performed by the methods just described up to 600
parameter values would be required. However, the spectrum in Fig. 6 contains three CO2
bands with ?t,;es of measurable intensity and thus more than 1000 parameters may be
required to describe this spectrum. Many of these lines lie outside the usable range
describe(' above and most are overlapped and blended with other .ines. Unless the
inv!stigator makes arbitrary decisions concerning some of the adjustable parameters in
these line by line models, only a small fraction of the lines can be analyzed satisfactorily.
However, the parameters describing these lines are not independent and the
models for the background and baseline can be assum:d to be smoothly varying functions
over the band. Thus the spectrum in Fig. 6 can be described by models similar to those in
Table 2 and a series of additional models which relate the parameters associated with the
individu.1. lines (5, 6). The models describing these additional relationships and the typical
numbers of adjustable parameters in these models are shown in Table 3. By extending the
analysis of the previous section to the retrieval of the parameters in this table, the
problem is reduced to estimating of the order of 40 parameters rather than over 1000. We
have used this technique to analyze overlapping bands of CO, CO , and N
2 
0 which contain
2 
over 3000 data values and several hundred lines (see Refs. 3- 7). The advantages of this
approach are
1.	 the parameters of primary interest are found directly from the experimental
data,
r+^
reduced to less than 0.1%.	 OF POOR QUALITY
Portions of the experimental spectrum, the corresponding calculated spectrum,
and the residuals spectrum obtained during the course of this investigation are shown in
Fig. 7. Although it is difficult to detect significant differences between the observed and
calculated spectra the residuals spectrum shows systematic variations attributed to poor
modelling of the ISRF. These systematic differences are not seen in the final residuals
spectrum shown in Fig. 8.
In our analysis of CO, CO 2 and N2 0 bands we have consistently obtained stable,
reproducible solutions, but the size of the data rats which can be handled and the number
of parameters retrieved is limited by the computer resources available. As these
resources expand it is expected that several spectra of the same band and/or several band
systems will be analyzable simultaneously. The band systems of most molecules are
described by more complica'ed models than those required in these investigations. hn
these cases the whole band methods of analysis car. be powerful tools for testing the
accuracies of these models and for identifying the significant parameters.
These analytical methods involve the application of standard statistical i
techniques to the analysis of typical experimental data. However they have advantages
over the more conventional methods of spectral analysis which are similar to the
advantages claimed for FTS over dispersive sptctroscopic techniques. These are usually
described as the throughput and multiplex advantages.
By analogy with these comparisons the multiplex advantage of whole band
analysis over line-by-line analysis is the direct, simultaneous retrieval of the parameters
of interest. There is also a corresponding throughput advantage in that the entire data set
is searched for information about each parameter.
r_-
Selected Preferences
1. Y. S. Chang Ph.D. diss. The Ohio State Univ. (1976).
2. Y. S. Chang and J. If. Shaw, Applied Spectroscopy, 31, 218 (1977).
3. R. L. Hawkins Ph.D. diss. The Ohio State Univ. (1982).
4. R. L. Hawkins and J. H. Shaw JQSRT, 29, 543 (1983).
5. M. L. Hoke Ph.D. diss. The Ohio State Univ. (1982).
6. Y1. L. Hoke & J. H. Shaw, Applied Optics 21, 919, 935 (1983).
7. C. L. Lin, J. H. Shaw and J. G. Calvert JQSRT 23, 387 (1980).
8. E. Niple Ph. D. diss. The Ohio State Univ. (1980).
9. E. Niple and J. H. Shaw Appl. Spectroscopy 33, 569 (1979).
10. N. Tu and J. H. Shaw, paper CEI, Fall Meeting, Ohio Section, American Physical
r=	 Society (1983).
r_ .
F
1
S
W.
Table ) Contributors to these studies
Y. S. Chang	 R. L. Hawkins	 M. L. Hoke
C. L. Lin	 B. Niple	 N. Tu
Oy
F'0"" ^'^
AL
ORIGINAL PAGE W
+ POOR QUALITY Table 2 Single line Models and Parameters
Minimum
Model Parameters
1 Line Shape v
(Lorentz, Voigt, ...) Y 1	 a o
a
w S = f kdv
!
Y
2 Instrument Spectral N
Response Function p
H
H
(IS RF) N(sinc, triangle 4)
Gaussian, ...)
v'
3 Background to T lo (vi) 3 1 
line (polynomial °
In v and other H
terms)
v
4 Baseline to line
i
B(vi) o B
(polynomial in W
H
E3
Y
Synthetic spectrum
L
O
VH
1
Table 3 Models for Describing Entire Bands
Quantity Model Number of Parameters
Line Intensities Band Intensity —3 if interaction terms
S A = ES(lines) are included(the sample
temperature and E" line
are also required)
Line PoOtions u = E' - B"
I
For CO 2
 the upper and
lower state energies can
each be modelled by
expressions containing
3 or 4 parameters
<8
The variation in the Lorentz
Line widths widths over the band is <10
modelled by empirical relations.
The models contain 3- 4
ISRF parameters which are assumed <4
constant over the band.
Several parameters may Le
IoNI ) required if channel spectra <10
are present.
B0.)i ) Usually 2 terms are adequate. <3
Total number of parameters re t rieved 30 - 40.
nUNIGINAL PAGE 5;:
Figure Captions	 OF POOR QUALITY
Fig. 1. Line positions by interpolation. The position of the ith line is
-1(2622 + Ax 
i 
/Ax 1]cm
Fig. 2. a) The spectrum I (v ) of an artificial lira is compared with the experimental
art i
s data I	
iexp i
(u ). b) the differences ^I(v ) are similar ir. shape to the partial
^.	 derivative X
	 (u )/bu. c) the best estimate of the line position occurs at the
exp i
^^	 a
minimum value of the sum EDI (vi ); d), e), f) corresponding steps required to
determine the line intensity.
Fig. 3. A set of lines generated by using a Lorentz line shape, a triangular ISRF with
-1
H = a = 0.1 cm , and constant I and B functions.0
Fig. 4. The dependence of the fractional uncertainty in the retrieved line intensity on
the line intensity for the case of 6 retrieved parameters and for the csses where
1.	
B, I , and H are fixed to their correct values.0
Fig. 5. The derivatives WN )/ap as functions of x. for the parameters p = S, (i, v , H, Ii	 1	 j	 o	 0
and B and the dependence of these derivatives on the line intensity S.
-1
Fig. 6. Upper curve: a spectrum of CO near 2600 cm
	 Lower curve: the residuals
2
spectrum after modelling the entire spectrum and retrieving more than 30
ad)ustable. parameters.
Fig. 7. The upper portion of this figure shows a comparison between a portion of the
experimental spectrum in Fig. 6 and a calculated spectrum. The residuals
spectrum in the lower part of this figure has systematic variations attributed to
incorrect modelling.
Fig. 3. The upper part of this figure is the same portion of tae experimental spectrum as
that in Fig. 7. The lower part of this figure is the residuals spectrum obtained by
i using the best estimates for the parameters.
c
v+
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ORIGINAL PAGE
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Least squares methods of analyzing spectroscopic data

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