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Comments on inverse atmospheric radiative transfer problems: A nonlinear minimization search method of solution
Physics of the Earth and Planetary Interiors, 21(1980)67—70 © Elsevier Scientific Publishing Company, Amsterdam — Printed in The Netherlands
67
REPLY TO COMMENTS ON INVERSE ATMOSPHERIC RADIATIVE TRANSFER PROBLEMS: A NONLINEAR MINIMIZATION SEARCH METHOD OF SOLUTION * A.L. FYMAT Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91103 (US.A.)
(Received January 7, 1979)
Fymat, A.L., 1980. Inverse atmospheric transfer problems: a nonlinear minimization search method of solution — a reply. Phys. Earth Planet. Inter., 21: 67—70.
Our Minimization Search Method (MSM) (Fymat, 1976) is a nonlinear retrieval technique which has been developed for “inverting” physical data in terms of characteristic parameters of the medium (or system) under investigation. While we have applied this method mainly to atmospheric and geophysical data (see also Mill and Drayson, 1978), it can equally be used in all other inversion problems which can be reduced to the determination of an optimized set of unknown parameters. The reformulation of the initial inversion problem as a minimization search problem, the termination of the search, and the interpretation of the corresponding results must, or course, be adapted to the particular problem of interest and its underlying physics. However, the mechanics of the search itself, that is, the strategy followed for iteratively improving earlier estimations of the unknown parameters, is essentially independent of the problem. It must be stressed that the accuracy of the inversion results is predominantly a function of the information content of the observables and of the accuracy of the corresponding measurements, rather than a function of the numerical search itself. Indeed, if the unknown parameters have more-or-less equal influence on the observables (in an order-of-magnitude sense), simulated experiments which we have carried out demonstrate for (ideal) noiseless measurements that a retrieval accuracy of at least three decimal places, or better, can *
S~eCerni and Pepin (1979).
routinely be obtained with the MSM (see, for example, Fymat and Kalaba, 1974; Fymat and Lenoble, 1979). On the other hand,if the parameters of interest have little effect on the measured observables, or if the measurements are too noisy, or both, there is little hope (if any) of carrying out a successful inversion, irrespective of the retrieval approach adopted. An increase in the number of unknowns mostly has the effect of increasing the computation time, in proportion to the first power of such variables. This feature is to be contrasted with most optimization methods, where times are proportional to the cube of the number of variables (Huelsman, 1968). In the examples we have chosen with more than two unknowns, the accuracy of the solution usually did not deteriorate noticeably unless these variables had widely different orders of magnitude (some being extremely small). We would also emphasize the nonlinear character of our inversion procedure, whose advantages have also been discussed by Mill and Drayson (1978). Having clarified some important features of our inversion method, we now turn to the problem of specifying those radiation observables (i.e. extinction measurements) that are sensitive to the complex refractive index of the aerosols. Extinction measurements contain, in principle, information on both the size distribution in the aerosol and on the refractive index. The extent of this information, however, depends stringently on the total spectral interval spanned by the measurements (in addi-
68 I 7.2
I
I
I
I
I
= 1.0 pm mrl.SO
-
-
=
L5.6
(
~4.O
16 ~
I
0.4
0.8 1.2 EXTINCTION
O(O.05)6.Oi.im
-
-
.
-
I
I
mr 1.50 m~O =
0(0.05) 6.Opm
) ) )
( (
2.4-
0.0
I
m~=
-
~~Ii::I;52:5I
1.6 2.0 2.4 2.8 3.2 EFFICIENCY FACTOR, ~ext
‘
~
________
3.6
0.0
OA
0.8 1.2 1.6 2.0 EXTINCTION EFFICIENCY
2.4 2.89ext3.2 FACTOR,
3.6
4.0
Fig. 1. Variations of the extinction efficiency curve Qextfr) (a) for several aerosol absorptions (b) for different light wavelengths (from Fymat and Mease, 1978).
tion to the other factors mentioned earlier). A demonstration of this dependence is provided by the following illustrations. Fig. la displays the variations of the Mie extinction efficiency factor Qext (computed, for simplicity, and without loss of generality, from van de Hulst’s anomalous approximation) as a function of aerosol radius r at wavelength X = 1 pm, and for aerosol refractive index components mr = 1.33, m 1 = 0,0.135, 0.290 and 0.50. It can be seen that if the aerosol is transparent (m~= 0), the efficiency factOr follows a train of exponentially damped oscillations about the value Qext = 2, which is the asymptotic value (even iii the Mie case) when r becomes very large (theoretically infinite). Maximal contribution to this effect from aerosol’ radii comes near the first curve maximum, in this case r0 the 0.67 pm. As absorption progressively introduced, oscillations are furtherisdamped, and this contribution is correspondingly reduced. With sufficient absorption, the curve becomes nearly featureless for r > r 0 with little (if any) dependence on r. The same general characteristics prevail for different mr values, except that the oscillatory pattern compresses with a slight general shift toward somewhat lower r-values as mr increases. Fig. lb showsthe wavelength —
variations of Qext as a function of r. The same general features described above persist, with compression and shift toward lower r-values as X decreases. In the wavelength interval 0.3—0.4 pm, the compression length is z~.r 0.27 pm; this means, in the example illustrated, that observations in that spectral interval are mostly contributed by particles in the approximate size range 0.27—0.54 pm. The latter range would be even narrower for a smaller spectral interval. In summary, if there were equal numbers of aerosols at all sizes (or, equivalently, a rectangular size-distribution), the measurements would reflect mostly effects from particles in the above narrow size range. Next, the influence of a model (gamma-type) size distribution n(r) isofshowii in Fig. 2, where 2Qextn(r) the extinction integralthe hasinte. been grand irr plotted as a function of r for several X-values. It is seen that n(r) acts as a filter, in “clipping” and modulating only a portion of the Qext curve. The integrands for this case are seen to be sharply peaked, with maximal contributions near the peaks. The locations of these peaks are, approximately, r(X = 0.8 pm) 0.58 pm, r(X = 0.6 pm) 0.5 pm and r(X = 0.4 pm) 0.72 pm, spanning the very narrow size range of 0.22 pm. In “-
~
69 3.2
,
I
I
I
T
I
2.4 -
a =2/3pm, rm =0.333p,
DISTRIBUTION b=
!/6 h =oapm
,,rrrj
zo.0
;
author concluded that there is no size resolution beyond approximately r = 2 pm at the wavelengths he studied. This is why Grass1 was forced to extend the spectral interval of his measurements into the IR range (10.4 pm) in order to sample particles of 6-pm radius. In summary, the effective size range of “active” aerosols is directly related to the width of the spectral interval investigated. We have advocated the use of a narrow spectral interval so that the measurements would be little affected by the size distribution. As just shown, they would be affected only by a very narrow range of sizes, say (rl, r2). We can reasonably assume that the refractive index is nearly constant within such a range. Using the first mean value theorem, we have furthermore:
I
GAMMA
-,-----
_____---A
0---
0.0
----7 0.4
0.8
:
1.2
1.6
2.0
2.4
2.8
3.2
3.6
E(X) = rr 7 \.---’ 0.8.
--4
-----__
0.0
a.4
0.6
-=--
_-
ooX--I~_-__------
_..
r2QeXt(h , m, r)n(rW
- JQ@;
mk2
Yl
1.2
1.6 2.0 INTEGRAND,
2.4 2.8 n2C&“Vl
3.2
3.6
where r2
4.0
Fig. 2. Illustrating the filtering effect on the extinction curve performed by a model size distribution (from Fymat and Mease, 1978).
P2
=
s
r2n(r)dr,
and 0 =
Q,,dk w r>,
Yl
whererr
We can therefore write other words, the observations corresponding to this case will be governed essentially by aerosols in this size range. Thus, while the distribution extended from Y= 0 to r = 2.0 pm, only particles in the range r = OS-O.72 /.mr substantially affected the transmission. The conclusion, consistent with our earlier claim, is that within a narrow spectral interval, transmission observations contain information on only a narrow range of sizes. In order to observe effects outside of this range, it is necessary to broaden the spectral interval. This observation finds confirmation in the works of Yamamoto and Tanaka (1969) Quenzel (1970) and Grass1 (1971). The first authors considered size distributions extending up to r = 12 pm, and concluded that a spectral interval of 0.35-2.27 pm was required to reproduce the portion of such distributions within the range 0.1 < r < 5 pm. Quenzel needed a spectral interval from 0.4 to 1.6 pm in order to sample particles in the size range 0.3-l .3 /.m. While these size limits are not sharply defined, because of the width of the integrand-curves, this
which shows that the spectral transmission ratios within a narrow spectral interval are fairly insensitive to the size distribution, and are mostly functions of the refractive index. Such would not be the case, however, in the cases of wide spectral intervals, referred to by Cerni and Pepin. While the computations we have reported would appear to be erroneous, the physical basis of our method is certainly correct. We must note in passing that the refractive index thus determined would strictly characterize only those active aerosols at the wavelengths used. This opens the possibility of determining the spectral variations of the index, by experimenting with different sets of narrow spectral intervals. Equation 1 of the article in question was reproduced from eq. 28 of our earlier article (Fymat, 1975): reference to this latter would have immediately shown that the leftmost equality was a typographical error. This can further be verified from eq. 16c of
70
our subsequent publication (Fymat and Mease, 1978). Lastly, Cerni and Pepin remarked that in the case of the power-law size distribution n(r) = Cr~~1’, “extinction ratios are sensitive only to size distribution and not to refractive index”. We shall demonstrate below that proper derivation shows the claimed sensitivity to be absent. From the definition
This surprising result may only be an artifact of the particular distribution model adopted. If this were the case, it would render such a model inappropriate for the present inversion problem.
References Cerni, T.A. and Pepin, T.J., 1979. Inverse atmospheric radia-
r
N
—
f
2
n(r)dr
tive transfer problems: a nonlinear minimization search method of solution — comments. Phys. Earth Planet. Inter., 21: 61—66. Fymat, A.L., 1975. Appl. Mat. Comput., 1: 131.
rj
the explicit expression of the “constant” C is obtained as C Nu*/(r~* ri~~*). This yields for the extinction integral —
ivirp
*
E(X) = 2(xj~’ x~ ) F(m; p where k * —
F(m; p*; x 1, x2) =
*
I
x1, x2)
Qext(mX)XI_j~*dX.
Thus, = 2(X~IX~)2 ~This expression E(Xu)/E(Xu) = (k~/k~) shows that for a power-law size distri. bution, ratios of spectral extinction are independent both of the size distribution and of the refractive index.
Fymat, A.L., 1976. Inverse atmospheric radiative transfer problems: a nonlinear minimization search method of solution. Phys. Earth Planet. Inter., 12: 273. Fymat, A.L. and Kalaba, R.E., 1974. J. Quant. Spectrosc. Fymat, Radiat. A.L.Transfer, and Lenoble, 14: 919. J., 1979. J. Quant. Spectrosc. Radiat. Transfer, in press. Fymat, A.L. and Mease, K.D., 1978. In: A.L. Fymat and V.E. Zuev (Editors), Remote Sensing of the Atmosphere: Inversion Methods and Applications. Elsevier, New York, p. 233. Grassi, H., 1971. Appl. Opt., 10: 2534. Huelsman, L.P., 1968. University of Arizona, Tucson, Ariz., Ref., pp. 26—29. Mill,Zuev J.D.(Editors), and Drayson, S.R.,Sensing 1978. of In: the A.L.Atmosphere: Fymat and V.E. Remote Inversion 123. Methods and Applications. Elsevier, New York, p. Quenzel, H., 1970. J. Geophys. Res., 75: 2915. Yamamoto, G. and Tanaka, M., 1969. AppI. Opt., 8: 447.
67
REPLY TO COMMENTS ON INVERSE ATMOSPHERIC RADIATIVE TRANSFER PROBLEMS: A NONLINEAR MINIMIZATION SEARCH METHOD OF SOLUTION * A.L. FYMAT Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91103 (US.A.)
(Received January 7, 1979)
Fymat, A.L., 1980. Inverse atmospheric transfer problems: a nonlinear minimization search method of solution — a reply. Phys. Earth Planet. Inter., 21: 67—70.
Our Minimization Search Method (MSM) (Fymat, 1976) is a nonlinear retrieval technique which has been developed for “inverting” physical data in terms of characteristic parameters of the medium (or system) under investigation. While we have applied this method mainly to atmospheric and geophysical data (see also Mill and Drayson, 1978), it can equally be used in all other inversion problems which can be reduced to the determination of an optimized set of unknown parameters. The reformulation of the initial inversion problem as a minimization search problem, the termination of the search, and the interpretation of the corresponding results must, or course, be adapted to the particular problem of interest and its underlying physics. However, the mechanics of the search itself, that is, the strategy followed for iteratively improving earlier estimations of the unknown parameters, is essentially independent of the problem. It must be stressed that the accuracy of the inversion results is predominantly a function of the information content of the observables and of the accuracy of the corresponding measurements, rather than a function of the numerical search itself. Indeed, if the unknown parameters have more-or-less equal influence on the observables (in an order-of-magnitude sense), simulated experiments which we have carried out demonstrate for (ideal) noiseless measurements that a retrieval accuracy of at least three decimal places, or better, can *
S~eCerni and Pepin (1979).
routinely be obtained with the MSM (see, for example, Fymat and Kalaba, 1974; Fymat and Lenoble, 1979). On the other hand,if the parameters of interest have little effect on the measured observables, or if the measurements are too noisy, or both, there is little hope (if any) of carrying out a successful inversion, irrespective of the retrieval approach adopted. An increase in the number of unknowns mostly has the effect of increasing the computation time, in proportion to the first power of such variables. This feature is to be contrasted with most optimization methods, where times are proportional to the cube of the number of variables (Huelsman, 1968). In the examples we have chosen with more than two unknowns, the accuracy of the solution usually did not deteriorate noticeably unless these variables had widely different orders of magnitude (some being extremely small). We would also emphasize the nonlinear character of our inversion procedure, whose advantages have also been discussed by Mill and Drayson (1978). Having clarified some important features of our inversion method, we now turn to the problem of specifying those radiation observables (i.e. extinction measurements) that are sensitive to the complex refractive index of the aerosols. Extinction measurements contain, in principle, information on both the size distribution in the aerosol and on the refractive index. The extent of this information, however, depends stringently on the total spectral interval spanned by the measurements (in addi-
68 I 7.2
I
I
I
I
I
= 1.0 pm mrl.SO
-
-
=
L5.6
(
~4.O
16 ~
I
0.4
0.8 1.2 EXTINCTION
O(O.05)6.Oi.im
-
-
.
-
I
I
mr 1.50 m~O =
0(0.05) 6.Opm
) ) )
( (
2.4-
0.0
I
m~=
-
~~Ii::I;52:5I
1.6 2.0 2.4 2.8 3.2 EFFICIENCY FACTOR, ~ext
‘
~
________
3.6
0.0
OA
0.8 1.2 1.6 2.0 EXTINCTION EFFICIENCY
2.4 2.89ext3.2 FACTOR,
3.6
4.0
Fig. 1. Variations of the extinction efficiency curve Qextfr) (a) for several aerosol absorptions (b) for different light wavelengths (from Fymat and Mease, 1978).
tion to the other factors mentioned earlier). A demonstration of this dependence is provided by the following illustrations. Fig. la displays the variations of the Mie extinction efficiency factor Qext (computed, for simplicity, and without loss of generality, from van de Hulst’s anomalous approximation) as a function of aerosol radius r at wavelength X = 1 pm, and for aerosol refractive index components mr = 1.33, m 1 = 0,0.135, 0.290 and 0.50. It can be seen that if the aerosol is transparent (m~= 0), the efficiency factOr follows a train of exponentially damped oscillations about the value Qext = 2, which is the asymptotic value (even iii the Mie case) when r becomes very large (theoretically infinite). Maximal contribution to this effect from aerosol’ radii comes near the first curve maximum, in this case r0 the 0.67 pm. As absorption progressively introduced, oscillations are furtherisdamped, and this contribution is correspondingly reduced. With sufficient absorption, the curve becomes nearly featureless for r > r 0 with little (if any) dependence on r. The same general characteristics prevail for different mr values, except that the oscillatory pattern compresses with a slight general shift toward somewhat lower r-values as mr increases. Fig. lb showsthe wavelength —
variations of Qext as a function of r. The same general features described above persist, with compression and shift toward lower r-values as X decreases. In the wavelength interval 0.3—0.4 pm, the compression length is z~.r 0.27 pm; this means, in the example illustrated, that observations in that spectral interval are mostly contributed by particles in the approximate size range 0.27—0.54 pm. The latter range would be even narrower for a smaller spectral interval. In summary, if there were equal numbers of aerosols at all sizes (or, equivalently, a rectangular size-distribution), the measurements would reflect mostly effects from particles in the above narrow size range. Next, the influence of a model (gamma-type) size distribution n(r) isofshowii in Fig. 2, where 2Qextn(r) the extinction integralthe hasinte. been grand irr plotted as a function of r for several X-values. It is seen that n(r) acts as a filter, in “clipping” and modulating only a portion of the Qext curve. The integrands for this case are seen to be sharply peaked, with maximal contributions near the peaks. The locations of these peaks are, approximately, r(X = 0.8 pm) 0.58 pm, r(X = 0.6 pm) 0.5 pm and r(X = 0.4 pm) 0.72 pm, spanning the very narrow size range of 0.22 pm. In “-
~
69 3.2
,
I
I
I
T
I
2.4 -
a =2/3pm, rm =0.333p,
DISTRIBUTION b=
!/6 h =oapm
,,rrrj
zo.0
;
author concluded that there is no size resolution beyond approximately r = 2 pm at the wavelengths he studied. This is why Grass1 was forced to extend the spectral interval of his measurements into the IR range (10.4 pm) in order to sample particles of 6-pm radius. In summary, the effective size range of “active” aerosols is directly related to the width of the spectral interval investigated. We have advocated the use of a narrow spectral interval so that the measurements would be little affected by the size distribution. As just shown, they would be affected only by a very narrow range of sizes, say (rl, r2). We can reasonably assume that the refractive index is nearly constant within such a range. Using the first mean value theorem, we have furthermore:
I
GAMMA
-,-----
_____---A
0---
0.0
----7 0.4
0.8
:
1.2
1.6
2.0
2.4
2.8
3.2
3.6
E(X) = rr 7 \.---’ 0.8.
--4
-----__
0.0
a.4
0.6
-=--
_-
ooX--I~_-__------
_..
r2QeXt(h , m, r)n(rW
- JQ@;
mk2
Yl
1.2
1.6 2.0 INTEGRAND,
2.4 2.8 n2C&“Vl
3.2
3.6
where r2
4.0
Fig. 2. Illustrating the filtering effect on the extinction curve performed by a model size distribution (from Fymat and Mease, 1978).
P2
=
s
r2n(r)dr,
and 0 =
Q,,dk w r>,
Yl
whererr
We can therefore write other words, the observations corresponding to this case will be governed essentially by aerosols in this size range. Thus, while the distribution extended from Y= 0 to r = 2.0 pm, only particles in the range r = OS-O.72 /.mr substantially affected the transmission. The conclusion, consistent with our earlier claim, is that within a narrow spectral interval, transmission observations contain information on only a narrow range of sizes. In order to observe effects outside of this range, it is necessary to broaden the spectral interval. This observation finds confirmation in the works of Yamamoto and Tanaka (1969) Quenzel (1970) and Grass1 (1971). The first authors considered size distributions extending up to r = 12 pm, and concluded that a spectral interval of 0.35-2.27 pm was required to reproduce the portion of such distributions within the range 0.1 < r < 5 pm. Quenzel needed a spectral interval from 0.4 to 1.6 pm in order to sample particles in the size range 0.3-l .3 /.m. While these size limits are not sharply defined, because of the width of the integrand-curves, this
which shows that the spectral transmission ratios within a narrow spectral interval are fairly insensitive to the size distribution, and are mostly functions of the refractive index. Such would not be the case, however, in the cases of wide spectral intervals, referred to by Cerni and Pepin. While the computations we have reported would appear to be erroneous, the physical basis of our method is certainly correct. We must note in passing that the refractive index thus determined would strictly characterize only those active aerosols at the wavelengths used. This opens the possibility of determining the spectral variations of the index, by experimenting with different sets of narrow spectral intervals. Equation 1 of the article in question was reproduced from eq. 28 of our earlier article (Fymat, 1975): reference to this latter would have immediately shown that the leftmost equality was a typographical error. This can further be verified from eq. 16c of
70
our subsequent publication (Fymat and Mease, 1978). Lastly, Cerni and Pepin remarked that in the case of the power-law size distribution n(r) = Cr~~1’, “extinction ratios are sensitive only to size distribution and not to refractive index”. We shall demonstrate below that proper derivation shows the claimed sensitivity to be absent. From the definition
This surprising result may only be an artifact of the particular distribution model adopted. If this were the case, it would render such a model inappropriate for the present inversion problem.
References Cerni, T.A. and Pepin, T.J., 1979. Inverse atmospheric radia-
r
N
—
f
2
n(r)dr
tive transfer problems: a nonlinear minimization search method of solution — comments. Phys. Earth Planet. Inter., 21: 61—66. Fymat, A.L., 1975. Appl. Mat. Comput., 1: 131.
rj
the explicit expression of the “constant” C is obtained as C Nu*/(r~* ri~~*). This yields for the extinction integral —
ivirp
*
E(X) = 2(xj~’ x~ ) F(m; p where k * —
F(m; p*; x 1, x2) =
*
I
x1, x2)
Qext(mX)XI_j~*dX.
Thus, = 2(X~IX~)2 ~This expression E(Xu)/E(Xu) = (k~/k~) shows that for a power-law size distri. bution, ratios of spectral extinction are independent both of the size distribution and of the refractive index.
Fymat, A.L., 1976. Inverse atmospheric radiative transfer problems: a nonlinear minimization search method of solution. Phys. Earth Planet. Inter., 12: 273. Fymat, A.L. and Kalaba, R.E., 1974. J. Quant. Spectrosc. Fymat, Radiat. A.L.Transfer, and Lenoble, 14: 919. J., 1979. J. Quant. Spectrosc. Radiat. Transfer, in press. Fymat, A.L. and Mease, K.D., 1978. In: A.L. Fymat and V.E. Zuev (Editors), Remote Sensing of the Atmosphere: Inversion Methods and Applications. Elsevier, New York, p. 233. Grassi, H., 1971. Appl. Opt., 10: 2534. Huelsman, L.P., 1968. University of Arizona, Tucson, Ariz., Ref., pp. 26—29. Mill,Zuev J.D.(Editors), and Drayson, S.R.,Sensing 1978. of In: the A.L.Atmosphere: Fymat and V.E. Remote Inversion 123. Methods and Applications. Elsevier, New York, p. Quenzel, H., 1970. J. Geophys. Res., 75: 2915. Yamamoto, G. and Tanaka, M., 1969. AppI. Opt., 8: 447.