Estimation of continuous solar spectral distributions from discrete filter measurements

Solar Energy Vol. 33, No. 1, pp. 57-64, 1984

0038-092X/84 $3.00+ .00 © 1984 Pergamon Press Ltd.

Printed in the U.S.A.

ESTIMATION OF C O N T I N U O U S SOLAR SPECTRAL DISTRIBUTIONS FROM DISCRETE FILTER MEASUREMENTS J. J. MICHALSKYt a n d E. W. KLECKNER Pacific Northwest Laboratory, Richland, WA 99352, U.S.A.

(Received 18 May 1983; accepted 23 January 1984) Abstract--Techniques exist for estimating a well-behaved continuous function given a relatively small number of carefully chosen samples of that function. Two of these methods are applied to the problem of obtaining a low-resolution spectral distribution of direct solar radiation from a few filter samples. The minimum variance estimation and cubic spline fitting techniques are different approaches which yield integrated irradiances within one percent of each other and the test spectrum. They both produce reasonable averaged spectra over most of the spectral interval tested.

1. BACKGROUND

operating and maintaining such a device, except as a standard, is difficult to justify. As part of the National Insolation Resource Assessment Program, the Office of Conservation and Solar Energy of the U.S. Department of Energy (DOE) sponsored a solar spectral resource assessment program in the 1970s at the Pacific Northwest Laboratory (PNL). The instrument used in this program was a filter radiometer under computer control. The device, described in Kleckner et al.[4], has operated at as many as nine sites in the U.S. and Canada for periods of up to five years. To promote the effective use of these data, techniques were sought to produce low resolution spectra from the discrete data obtained from the radiometer. The problem of estimating continuous distributions from discrete data, in general, and the solar spectral distribution from filter data, in particular, is introduced in Section 2. Sections 3 and 4 contain the mathematical procedures for obtaining solar spectra using a minimum variance and a cubic spline technique, respectively. Assessments of their ability to accurately reproduce test spectra are also made. The last section of the paper draws conclusions about the applicability of the methods, and it outlines the attempts which will be made to validate estimated solar spectra based on actual measurements.

Solar spectral distribution data are important to research both outside and within the field of solar energy. The medical, materials, and photochemistry research disciplines are generally interested in the UV portion of the spectrum. Agronomists are concerned with the photosynthetic spectrum between 400 and 700 nm. Solar technologies which are sensitive to the spectral distribution include biomass and photovoltaics. Silicon, the prime photovoltaic candidate in the near term, is particularly sensitive to the nearinfrared portion of the spectrum and obtains its peak sensitivity in the vicinity of strong water vapor bands. Water vapor variability and, to a lesser extent, aerosol variability make it essential that the spectral distribution of solar radiation be measured at candidate sites for this technology. Insolation measurements to assess the amount of energy available as input to either electric power production or thermal end use equipment are typically made using pyranometers artd pyrheliometers. Thermal detector-based instruments most often use clear windows which transmit all radiation between about 280 and 2800 nm. The other common solar energy survey instruments use silicon photodiode detectors. These respond to radiation in the 360-1100nm range. Less often, solar radiation is measured in spectral bands defined by glass or interference filters[l]. Ideally, the spectral distribution of solar radiation is measured with a carefully calibrated dispersive device such as a grating or prism spectrometer[2, 3]. Were cost no concern, a calibrated, low resolution grating or prism spectrometer at candidate photovoltaic sites would be the most desirable spectral survey option. However, the expense of building,

2. SPECTRAL ESTIMATION A problem common to the physical sciences, generally, is that of estimating a distribution from a finite, and often small, number of samples of that distribution. In solar energy, examples include determining a detector's sensitivity and measuring a mirror's reflectivity as a function of wavelength. In both cases the variation with wavelength is expected to be smooth enough that given many filters with narrow bandwidths, an acceptable wavelength dependence can be obtained by connecting measured points along the spectrum. In practice this method may require an

tMember of the ISES. 57

58

J. J. MICHALSKY and E. W. KLECKNER

impractically large number of filters. Further, it is not precise since points on the spectral curve are not exactly given by measurements through filters of finite width which average over a portion of the spectrum. The solar spectrum is not a smoothly varying function. A low resolution spectrum, e.g. sampled every 10 nm, is interrupted by abrupt emission and absorption features due either to the solar or terrestrial atmosphere. This is especially pronounced in the near-IR where water vapor and molecular oxygen have strong absorptions. However, if very low resolution spectral information is adequate for a given application, techniques exist for generating a useful spectrum from filter data.

3. WIENER ESTIMATION

Techniques are discussed in Pratt and Mancill[5] which have direct applicatioa to this problem. Applying their notation to the problem we wish to address, a measurement in one filter can be represented by xi = f c ( 2 ) s i ( 2 ) d2 + n,.

(1)

each yield oscillatory solutions for c. This is not surprising since the problem is highly underdetermined, i.e. there are many more resolution elements than filters. The estimation procedure found especially useful in eliminating the oscillations and most closely matching a simulated spectral distribution is a Wiener estimation procedure [5]. The Wiener estimate is a minimum-variance estimate[6] of c given by c = K ~ S r ( S K c S V + K , ) - ' x.

Here c is assumed to be an unknown random vector with known covariance matrix Kc and n is assumed to be an unknown random error vector with known covariance matrix K,. The observational noise vector n does not depend on the magnitude of the vector c, and ( S K S r + K,) is nonsingular. In practice, the difficult part of applying this procedure, obtaining the covariance matrices, is handled by modeling each. In[5] K c is modeled as a first-order Markov process covariance matrix given by

Kc=-~ ~c 2

1

p

p2 . . .

pQ-1 \

p p.2

l

p

pQ-2

...

/~Q-1 Here c(2) is the spectral irradiance of the solar radiation incident on our radiometer, si(2) is the spectral sensitivity of the ith measurement filter convolved with the transmission or reflection of each of the optical elements and the detector spectral sensitivity, xi is the detector output for filter i and ni is the noise in xi. To apply the inversion techniques which determine c(2), we use the discrete form of (1) X i = SiTc "3!- h i .

Now sir and e are vectors (T superscript denotes transpose) composed of Q components. A rule of thumb is to use a Q which will yield a resolution between five and ten times the filter spacing, e.g. for filters 100 nm apart try 10-20 nm resolution. This is appropriate to indicate trend, and higher resolution cannot produce any further information. A set of i = 1,2. . . . . P observations equal to the number of filters which sample across the spectrum can be arranged into a vector equation of the form

1

)

where 0 ~

where a, 2 is the noise energy and I is the identity matrix. If we define a new matrix K'~ = K~(a<21Q ) - l ,

then we can rewrite (3) as c = K'~Sr(SK'cSr+ a,2/aczI) -1 x.

x = Sc + n,

(4)

(2)

where sir is the ith row of the matrix S. As shown in Pratt and Mancill[5], the generalized inverse estimate e = S r ( S S r ) -~ x,

and an inverse estimate with a generalized smoothing matrix M applied c = M-ISr(SM-ISr)

(3)

-I x,

In this form all the multiplicative scalars are eliminated with the exception of ¢rc2 and a, 2 which appear as the noise-to-signal ratio. To solve this problem we measure the transmission, reflectance, or spectral response of every element in the optical train to derive the spectral sensitivity matrix components. The signal-to-noise ratio is taken from observations. The correlation coefficient is assumed and adjusted to just eliminate oscillations. The procedure will be discussed further in the example that follows.

Continuous solar spectral distributions

1.60

Wiener estimation example To demonstrate the technique we estimate the direct solar spectrum between 360 and 1030 nm given a set of measurements through seven broad-band filters which span this wavelength range. Applying Wiener estimation to this problem, we undertook a measurement which put minimal demand on our ability to specify the elements of the sensitivity matrix S in (4). A typical filter radiometer has a variety of optical components and a detector. The elements of the sensitivity matrix are determined by the transmission, reflection, or response characteristics of each of these components as a function of wavelength. To simplify the problem, we used a self-calibrating cavity radiometer, the output of which is the irradiance within a 2.5 ° field of view. At full sun the device measures direct plus some circumsolar insolation to an accuracy of 0.5 per cent. At lower values of the insolation, as occur when filters are placed before the cavity instrument, a smaller absolute irradiance error, but higher fractional error, results. To improve the accuracy, repeated measurements are made near solar noon with all the filters in the set. Making near-noon measurements minimizes the change in airmass and, therefore, the source (atmosphere-filtered sun) output. The total solar output is tracked with another pyrheliometer during the course of these measurements to provide guidance in grouping filter measurement sets. To estimate the solar spectral distribution according to eqn (4), we need seven numbers specifying the solar irradiance through each of the seven filters. We also require a trial correlation coefficient and the measured signal-to-noise ratio. For the sensitivity matrix we use the transmission as a function of wavelength in 10 nm increments (giving a Q equal to 68 from 360 to 1030 nm) for each of the seven filters in order to specify the rows of the sensitivity matrix. It is assumed that there is no transmission of radiation to which our detector is sensitive outside the spectral region we are trying to approximate; otherwise, the estimated curve will be too high in the filters which leak out-of-band radiation. On 1 July 1981, we collected eight sets of filter measurements in a period of one and one-half hours centered on 12:49 local time. The total irradiance as monitored by the companion pyrheliometer varied between 954 and 966 W m - : during that period. The eight seven-filter sets were averaged to produce seven values which yielded the estimated spectrum of Fig. 1. The relative transmission of the filters at 10-nm resolution is given at the bottom of the figure. Our initial reaction was that the spectrum looked reasonable overall. However, without an absolute spectrometer making simultaneous measurements, there was no immediate way to validate this conclusion. As an alternative to direct comparison with an absolute spectrometer, we compared our results to a theoretical model. The Air Force Geophysical Laboratory has modeled both the effects that the atmosphere has on transmitted radiation and the

59

1.40

:.;.:" ....................... "'" "..

7EE 1.20 ?E

".,"... "',..

/

1.00 ...,..:':

t~j

z

"... ""..

""

_< a 080

".-.. "..

.....

~__ j 060

'".,.

i 0.40 (,9 0.20 /'"1 /

I! il 355

~ /

~/ 455

\,/

~,

\

/~

/''",~

\11 /

555 655 755 855 WAVELENGTH (nm)

/I~

\/~1 955

Fig. 1. Wiener estimated spectrum of near-noon direct solar spectral irradiance based on measurements through indicated filters.

radiation that is emitted by the atmosphere at infrared wavelengths. Their present model is called LOWTRAN5[7], the fourth published update of the original code. Aside from seasonal and geometrical factors, the principal inputs to the LOWTRAN5 code for the visible and near-infrared spectrum are the visibility (to specify the aerosol component) and the total water vapor content. To calculate the spectral distribution of direct solar radiation, we used SOLTRAN5 which is the transmission calculated from LOWTRAN5 multiplied by the extraterrestrial spectrum according to the prescription of Bird and Hulstrom[8]. We used a measured turbidity (equivalent to 99 km visual range with a moderate volcanic profile and aged volcanic extinction) and an assumed water vapor content appropriate to the 1 July 1981 date. The solar zenith angle was that calculated for 20:49 UT for our latitude of 46.40 ° N'and 119.60° W. The resultant spectrum sampled every 60 cm -l with a 20 cm -~ bandpass appears as the solid curve in Fig. 2. The dotted curve is that of Fig. 1 representing the spectrum estimated from the cavity measurements. It generally follows the shape of the theoretical spectrum, but does not agree in detail. Integrating to find the total irradiance between 360 and 1030nm, we calculate 743 Wm -2 for the estimated spectrum and 718 Wm -2 for the theoretical one. This 3.5 per cent excess in the estimated value is principally due to higher estimated values in the ultraviolet and near-IR which are only partially offset by lower values in the mid-visible. There are several factors which may serve to explain the discrepancies between the measured and theoretical spectra. Subsequent to this comparison, we were able to obtain extended filter transmission

60

J.J. MICHALSKYand E. W. KLECKNER 1.60

1.60

1.40

f

~ . .

1,40

..

.oof,

E 1,20 E E 1.00 z

0

£3

0.80

=<

"% 0.60

=<

o,0 [

0.20t'-,".-.

,r-~

,F\

I/ )t, 355

455

]

555

"-~", r! /

kl I~W I 655 755 855 WAVELENGTH (nm)

A I7 955

co

0.20 ~ . I / ~

/

\1

Fig. 2. Estimated spectrum from Fig. 1 and SOLTRAN5 direct solar spectrum calculated for time of measurements.

measurements out to 2400nm. The bluest filter leaked radiation beyond the region of interest and, consequently, gave a spuriously high irradiance since the cavity instrument responds to the leaked infrared radiation. This could explain the higher estimated ultraviolet values. In assuming a water vapor profile, we chose a rather low total precipitable water vapor, and, therefore, it is rather surprising that the estimated near-IR spectrum is higher than the theoretical one. Some of this discrepancy could be due to the errors associated with the cavity measured irradiance values for the two narrow infrared filters; for these, the standard deviations of the eight runs were on the order of 3 per cent. We have not found any simple explanation for the apparent underestimation in the mid-visible range. Of course, errors in the theoretical spectrum could also be responsible for the differences between that curve and our results. Efforts to validate LOWTRAN5 in the spectral region studied are under way at the Solar Energy Research Institute[2]. It may be that the estimated spectrum is only in error in the UV portion. Simultaneous absolute spectroradiometry would be the best calibration/comparison procedure. In view of the various experimental and theoretical factors discussed above, the comparison between our estimated spectrum and the theoretical curve may not serve as a definitive test of the Wiener estimation procedure. Consequently, we applied the estimation procedure to a set of "measurements" generated by convolving the SOLTRAN5 spectrum with the actual filter transmission functions. The estimated spectrum which results from this procedure is given in Fig. 3 along with the SOLTRAN5 spectrum on which it is based. The integrated irradiance for this spectral

355

455

555

655 755 855 WAVELENGTH (nm)

955

Fig. 3. SOLTRAN5 spectrum from Fig. 2 and Wiener estimated spectrum resulting from measurement simulation based on convolution of SOLTRAN5 spectrum and filters.

interval for the estimated spectrum is 0.3 per cent higher than the SOLTRAN5 spectrum. The agreement in broad spectral regions is also quite good, with the estimated spectrum displaying a reasonable averaging through regions of absorption. The largest departure between the original and estimated curves is in the near-IR spectrum in the vicinity of the 820 and 940 nm water vapor bands. Our original selection of the very broad filter indicated in each of Figs. 1-3 for this spectral interval was based on an apparently erroneous terrestrial spectrum [9]. The differences in this spectrum and the LOWTRAN5 results are addressed in [10]. The original goal was to select continuum filters on either side of the broad water vapor absorption region formed by the bands at 820 and 940 nm and a broad filter centered on this region to estimate total water vapor. In light of the LOWTRAN5 model, a much better selection of continuum and absorption feature filters could now be made for this region. Another discrepancy is the high value of the spectrum at the short and at the long wavelength cutoffs. These can be explained by the fact that no information is available beyond these cutoffs. The higher values are influenced by the high correlation coefficient assigned in the covariance matrix. A lower correlation coefficient reduces the influence of the adjacent spectrum thereby lowering these values to produce better agreement at the cutoffs. However, reducing the correlation coefficient to a value much lower than 0.95 results in some low-amplitude oscillations in the spectra.

4. PARK-HUCKESTIMATION Another technique, based on the work of Park and Huck[11], was employed to estimate the solar spec-

61

Continuous solar spectral distributions trum. The relationship represented by eqn (1) may be cast as a set of linear equations whose solution vector is the set of coefficients which multiply the cubic splines used as basis functions in an estimate of c(2). With this approach, there is no need to make d priori assumptions about the degree of correlation, as done in the Weiner method. In a broader sense, however, the methods are equivalent and there are assumptions involved in the Park-Huck method which we identify below. From the computational point of view, the stability of the algorithm and the speed when applied to large amounts of data may favor the Park-Huck method over the Weiner technique. Recall that in eqn (1) we defined the overall instrument response as s~(2). It is convenient to decompose s~(2) as the product of a system calibration constant, r~, and a system transfer function Tffh). Each calibration constant, r~, can be chosen such that f0°~ Ti(2) d2 = 1, 7,.(2) > 0 for all 2.

b i = ~ a o a j , l~
(10)

j=l

where ei = 0 at each measurement point. In other words, we have chosen to have the sample and its estimate identical at each measurement point. The choice of the basis set hi(2) corresponds to the priori assumptions concerning correlation used in the Weiner method. A natural choice is a set of cubic spline basis functions which have certain desired properties of smoothness. This choice is described in the following paragraph following Park and Huck[11]. Cubic splines are curves defined on a sequence of adjacent sub-intervals; these sub-intervals are connected at points known as knots. On each subinterval the function is a cubic polynomial, and at each knot the curves are joined such that the 0th through 2nd derivatives are continuous. Outside of the wavelength range a ~< 2 ~< b, the system transfer function is effectively zero. A set of equally spaced knots 2-1, 2-2. . . . . 2-m can be defined as ~ = Zt + ( j - - 1)A, j = 2 , 3 . . . . . m,

We then have

b~ - xJr i = f c(2)Ti(2 ) d2,

(5)

where bi are the spectral samples. This is the basic form we shall work with. Equation (5) relates the solar spectrum c(2) and the set of multi-spectral samples {b~. . . . . bin}. We may form an estimate of the solar spectrum c(2) by expanding in a set of linearly independent basis functions, hi(2), i.e.

where hi/> a and hm ~< b, and A = (h,, - hl)/(m - 1). The basis functions are centered on the knots and given by h](2)=c(2-~),

l~
where

I1/6A3[A3 +

3A2(A --I,ll)

+3A(A -121) 2

c(2)=,~ -3(A-121) 31 a(,~) = ~, ~,jhj(,~).

I2[~
g/6A3(2A-Izl)

(6)

A<~

~ 2(Al l ) 2 A .

j=l

This estimate is valid provided the coefficients c9 are chosen to minimize some measure of the error e(h) ~ c(X) - d(h). We can determine the expansion coefficients by multiplying both sides of (6) by T~(2) and integrating over wavelength. This goes .L e ~ = b ~ - ~ a,jc% l<~i~
(7)

One can write therefore, t(h) = ~

ajc(X - ks).

(12)

j=l

The coefficients aj are found by solving eqn (8)

j=l

a0 =

where

T,(2)c (2 -- ~) d2. 0

j

r oo

%=

T,(2)hj(2) d2,

(8)

7",.(2)e(2) d2.

(9)

0

and

This estimate may be improved by the addition of two basis functions, one to each end of the interval, i.e. at 2-0 = ~ ' j - A and 2-m+I=2-m+A" Then it is necessary to write

l cx~

e, =

t

m+l

~(2)= ~ c9c(2-~),

0

Since (7) expresses the errors ei in terms of the coefficients ct, these may be chosen to minimize the selected measure of error. In particular, we choose n = m and solve the square system:

(13)

j=O

and m+l

bi= ~. aoaj, j=0

l<~i<~m.

(14)

62

J. J. MICHALSKY and E. W. KLECKNER

There are two degrees of freedom undetermined by this system, and a conventional, and in our application, reasonable choice is to require the estimate given by (13) to have zero second derivatives at ,['l and Am. This merely says that we have no knowledge of the curvature outside the range of our measurements. The solution for (z is determined by solving A~t = b, i.e. eqn (14). An alternative form provides a faster algorithm when computing a large data set. In this new representation we write

16o ..-.....

1.40 _ TE 120 c E

~ l o0 z

-< 0 8 0

.J 0.60

m

~(~) = Y b£(,~),

05)

=< 0.40

where f ( 2 ) are what are called the system characteristic functions in Park and Huck[11], i.e. the f ( 2 ) are the unique cubic spline estimates corresponding to the ideal multispectral samples with b i = 1 and bj(j ¢ i) = 0. Equation (12) can be written as

co

0.20

355

^ , , , 455

555

r-.._

655 755 855 WAVELENGTH (nm)

955

Fig, 5. Same as Fig. 3 except P a r k - H u c k estimated spectrum.

where h0, ) is the column vector of components c ( Z - 2 j ) . Inverting this yields ?(2) =

br(A -')rh(2),

or

identifying f(h) with (A-l)rh(h), we have eqn (15). Once the characteristic functions are known, the estimates are the simple vector products of the samples and the characteristic functions. We have applied the Park-Huck analysis to the same data used in the evaluation of the Weiner method. Figure 4 represents the estimate formed using the actual active cavity radiometer measurements (smooth curve) and the theoretical spectrum. If we integrate to find the total irradiance for the chosen spectral range, we obtain a value of

742 Wm 2. When this is compared to the Weiner result (Fig. l), it is seen to provide the same estimate within the limits of experimental error. As in the Weiner method, the estimate overcompensates in the ultraviolet and infrared and underestimates in a broad band in the visible. Fig. 5 presents the results of convolving the SOLTRAN5 spectrum with the system transfer functions. In this case, the agreement is very good: an estimated value of 716 Wm -2 vs an actual value of 718 Wm -2. The agreement at the UV end of the spectral range has improved, quite possibly due to the reasons discussed in Section 3 on the problem of the ultraviolet filter leaking outside of its passband.

1.60

5. CONCLUSIONS 1.40

C

E

c 1.20

E

~ 1.00 Z <

<

0.80

0.60

o.2o~-/-,~/'\ /~'\//~'\ I~ ~ . h II ,I 3 \I~ I 355

455

885 655 756 866 WAVELENGTH (nm)

958

Fig. 4. Same as Fig. 2 except P a r k - H u c k estimated spectrum.

Two methods are described for estimating a continuous direct solar spectrum based on seven filter measurements between 360 and 1030 nm. Both produce integrated values that are virtually identical to the test spectrum. Reasonable averaging of the absorption features and the continuum obtains over most of the spectrum. In Fig. 6, we have plotted the differences in the Wiener and Park-Huck estimates to the L O W T R A N 5 spectrum as given in Figs. 3 and 5, respectively. Except for the end points, which can be adjusted according to the assumptions at these boundaries, there is always less than a two per cent difference, and most often less than a one per cent difference, throughout the spectrum. In particular, note that the difference in the estimates is near a minimum at the central wavelengths of the middle five filters. We conclude that either the Wiener or the Park-Huck estimation procedure provides a satisfactory method of estimating spectra from filter data.

Continuous solar spectral distributions

63

10

8

do

[

400

1

500

1

600

I

700

I

800

1

900

I

100(3

WAVE LENGTH (NM)

Fig. 6. A plot of differences in estimates produced by Wiener method in Fig. 3 and Park-Huck method in Fig. 5.

Improvements are possible. Better agreement may be made in the near-IR water band region by a careful selection of filters which sample the center of the bands and the continua to either side. In fact, the filters for the DOE spectral survey program were chosen to optimize the fit to what was later found[10] to be an apparently erroneous spectrum[9]. Presently, we are proceeding with attempts to validate the cavity radiometer-produced spectra. Besides the SOLTRAN5 comparisons, we hope to make a number of simultaneous measurements with absolute spectroradiometers. As confidence in the cavity spectra is realized, it is intended that these serve as input in eqn (1). Then the procedures will be used to derive the sensitivity function s,(2). In practice it can be very difficult, if not impossible, to measure with high accuracy the components of the sensitivity function which include transmission, reflection or response of each component of an optical system. Once this sensitivity matrix is determined by this alternate approach, the system is absolutely calibrated. On completion of these improvements and validation, the proposed approach should be an inexpensive and effective method of obtaining direct solar spectra routinely. The technique can be applied to diffuse spectral estimation as well. Besides being useful to nonconcentrating solar collectors, this should be useful to daylighting studies since the C.I.E. color matching functions x, y and z can be convolved with the spectral distribution to yield tristimulus values X, Y and Z which determine a typical observer's response. In summary, we feel that reasonable, continuous estimates of spectral radiation are more acceptable to potential users than discrete samples of the spectrum. Low resolution data should be useful in devices and processes whose spectral response varies smoothly with wavelength such as photovoltaic cells. Many

more applications exist both within and outside solar energy. In conclusion, we believe that we have demonstrated that stable numerical techniques exist for producing useful low resolution spectra from very few well-considered filter measurements. If further detail in applying these techniques is desired, the authors would be pleased to be of assistance.

Acknowledgements--The authors would like to thank several summer students who undertook various aspects of this work and were funded by the DOE's summer program at PNL: Kevin Kennedy, Christopher Curzon and Wayde Hudlow. We are grateful to Douglas Johnson and Erik Pearson for suggestions which significantly improve the manuscript. This research was funded by the Office of Basic Research and the Office of Solar and Conservation within the U.S. Department of Energy under contract DEACO6-76RLO-1830. REFERENCES

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J. J. MICHAUKYand E. W. KLECKNER

8. R. E. Bird and R. L. Hulstrom, Availability of SOLTRANS solar spectral model. Solar Energy 30, 379 (1983). 9. M. Thekaekara, The solar constant and spectral distribution of solar radiant flux. Solar Energy 9, 7 (1965). 10. R. Hulstrom, Insolation models, data and algorithms:

annual report FY-78. N.T.I.S. Rep. No. SERI/ TR-36110 (1978). Il. S. K. Park and F. 0. Huck, Estimation of spectral reflectance curves from multispectral image data. Appl. Opt. 16, 3107 (1977).