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Produced by the NASA Center for Aerospace Information (CASI)
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
Abstract

An overview of the methods of line-by-line and whole-band analysis developed

by The Ohio State University spectroscopy group is presented.

F
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,TY i

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( v

i

 , S ) as shown in Fig. 1d, corresponding to lines of different

art

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
^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

4

background 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
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
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 

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.
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).
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

ir

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 C Response 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
P4

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

I i

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.
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
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

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Cnd

OF POOR QUAL'Ty

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p o B Fixed

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c

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Li

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> 0

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u 0

0

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° C

I O

q VV

0- V) I 

v

0

0- p

0

N

- 0 p Acu

vo H TD

Wavenumber (cm-')

G
0

2585.00 2605.00 2625.00 2645.00 2665.00

WAVENUMBERS (CM- 1)

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2565.00

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WAVFN(1Ml3ERS tCM- 1 l

2645.00 2865.00
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OF POOR ^,.I i Y

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2620.00 2621.00 2622.00

WRVENUMBERS

2623.00 2624.00 2625

(CM-1)

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2620.00 2621.00 2622.00 2623.00 2624.00 2625

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