Information content of the radiative transfer theory

Rodolfo Guzzia, Simona Zoffolia,Stefano Corradinib, Marco Chiarinic

aAgenzia Spaziale Italiana, Viale Liegi 26, Rome, Italy,b Universita di Modena, c Carlo Gavazzi Space

ABSTRACT

The new generation of instruments to measure the atmosphere gives a large amount of information whose contentis not immediately obvious. For this reason the information content (in the Shannon sense) and the number ofdegrees of freedom for signal related to parameters to retrieve, are crucial to carry out, for instance, the selectionof microwindows or subset of channels for retrieval. This seems to be even true for the miniaturized devices likethose obtained by MEOMS technologies that have high flexibility because are highly integrated. In this paperwe present the methodologies to obtain the information content related to wavelengths selection from satellitewith limb and nadir view with in mind the MEOMS technology.

Keywords: Information content, remote sounding, spectroscopy, atmosphere, aerosol retrieval

1. INTRODUCTION

Remote sensing measurements have intrinsic limitations due to their complexity (Guzzi and Smokty1). The mat-ter has also been introduced by different authors (among those we cite Twomey,2 Rodgers3) by a regularisationand bayesian approach, as a function of the complexity of the problem to solve.

A new generation of spectral instruments have introduced new analysis methods to retrieve the atmosphericparameters. In the near future a dramatic improvement of spectroscopic instruments will be obtained by theintroduction of MEOMS (Micro-Electro-Optical-Machining System) technologies. Such technologies, not onlywill reduce the size of the instruments integrating the optics with the sensors, but will offer high flexibility andability to build up intelligent devices.

In our laboratories are under developing small Mach-Zehnder interferometers and a small imager interferom-eter is under assembling as it will briefly described in the next paragraph. These instruments offer a very largespectrum of applications: from medical to space.

In order to evaluate the performance of such device, due to the wide spectrum of applications, the firstrequirement is to study the information content of the signal measured by the instrument, like, for instance, incase of an interferometer where a large amount of information is obtained. Then the question arising is how thedata should be treated or filtered in operational retrieval of a predefined parameter.

We present here the information content of some special applications related to Earth Observing System inorder to retrieve the atmospheric aerosol by spectral measurements.

We are aware that an interferometer is not the best instruments to retrieve the aerosol, because it is a currentinstrument to study the gas absorption. But, as it is described in the next paragraph, a new device like MEOMSis able to be used in different modes useful to retrieve gases and aerosol, reaching the accuracy required. Analysison aerosol retrieval follows that done by Rodgers3 has treated the argument of how is the information contentof an interferometer measuring the atmospheric gases.

In this paper we discuss the information content related to aerosol retrieval from limb and nadir spectralmeasurements from satellite.

Further author information: (Send correspondence to Rodolfo Guzzi)R.G.: E-mail: rodolfo.guzzi@asi.it, Telephone: +39 06 8567 458

2. BRIEF DESCRIPTION OF MEOMS DEVICE

MEOMS is an advance technique able to make small optical device highly flexible in operational mode. Thedifferent parts of an instrument, like those of an interferometer, are integrated over the same substrate applyingsemiconductor technologies. An optical device to make the interferometry has been built up on Lithium Niobatemaking a Mach-Zehnder interferometer (see Fig. 1). The two arms have been obtained by creating two opticalguides by ion implantation technique. We do not enter into the technology because is widely described in twopapers (Bentini et al.,4 Bentini et al.5). However we may say the interferometer has a branch active biased bya voltage ramp while the second branch is passive. No parts are in movement and the device is protected byradiation. Starting from single device we have under developing an imager using a bundle of fiber optics linkedto each interferometer. One of the feature of such device is to have different mode of operation. It may be usedas interferometer (dynamics mode), but also as interference filter (static mode) with step by step selection ofwavelengths. Infact the selection may be carried out applying the correct bias to the active branch such that theoptical path between the two branches is: ∆φ = 2πL∆n/λ with ∆n = 1/2r33n

3V/d where λ is the wavelengthselected, n is the refractive index of the Lithium Niobate, r33 is the electro-optics coefficient, L is the branchlength and d is the distance between the electrodes with ramp voltage V.

3. INFORMATION CONTENT

Aerosol retrieval is currently carried out by various wavelengths combination by multispectral radiometer. Look-ing at limb of the atmosphere the information content is related, if we do not take into account the curvature of theatmosphere, to the attenuation to the different wavelengths of the sunlight I0λ passing through the atmosphereby the relation:

Iλ = I0λexp(−τ) (1)

where τ is the aerosol optical thickness defined as:

τ(λ) =∫

πr2n(r)Q(m, r, λ)dr (2)

where Q(m, r, λ) is the Mie extinction efficiency factor; m is the refractive index of particles and n(r) is the sizedistribution of the aerosol.

Twomey2 has shown that by looking at the eigenvalues of the kernel covariance matrix it is possible todetermine how many independent pieces of information can be obtained from a given measurement. The approachis based on the retrieving the spectral extinction written as:

gi =∫

Ki(x)f(x)dx + εi (3)

where Ki is the kernel function and εi is the measurement error. Ki = λiQi(m, r, λi)/2πr and f(r) = n(r).

The eigenvalues analysis is carried out computing the covariance matrix that is given by:

Cij =∫

Ki(x)Kj(x)dx (4)

The pieces of information independent are given by the number of eigenvalues for which the eigenvaluesλj > e2 where e is the relative error. This analysis has been performed by Box et al.6 and will not be repeatedhere. They showed the information content to retrieve the size distribution of the aerosol particle can be higherwhen the number of wavelengths increases, but it is greatly dependent on the measurement errors in the visibleand near infrared spectral range.

Let now deal with a more complex case in which not only the spectral transmissivity of the atmosphereis involved, but the radiance by which the aerosol profile may be retrieved. In this case a key component isthe forward model. Infact, given proper observing conditions and the response function of the instrument, theforward model (radiative transfer model) simulates the radiance measured by satellite instrument that must becompared to the measured one.

Figure 1. Mach-Zendher interferometer. A simulation of the interferometer mode is shown for NO2 absorption as afunction of voltage applied, bottom figure.

Since the information content of a measurement may be better analyzed by considering a linear problem, alinearisation of the forward model about a reference status x0 may be written as:

y − F (x0) =δF (x)

δx(x − x0) + ε = K(x − x0) + ε (5)

where K represent the forward mapping from the space of state vector to the measurement space vector. Thevertical distribution requires to define the Jacobian matrix of the derivates of the simulated measurements withrespect to the retrieval parameters. The set of these derivatives are referred as weighting function matrix.

Optimal estimation provides the solution of ill-conditioning problems found in remote sensing; it combinesinformation of the state vector contained in the measurements with a priori information of the state vector itself.

The linear inversion is based on Bayes’ theorem for the determination of the posterior probability densityfunction p.d.f. P (x|y) for state x given measurement y in term of a priori knowledge characterized by P (x) andthe conditional p.d.f. P (y|x) for y given x. The posterior p.d.f. is related to the prior p.d.f. of measurementsP (y) by:

P (x|y) = P (y|x)/P (y) (6)

.

In case of Gaussian p.d.f.

P (x) =1

(2π)n/2|Sa|1/2exp(−1

2(x − xa)T S−1

a (x − xa) (7)

where xa is the prior estimate of x; Sa is a prior covariance matrix. Then the posterior estimate of x has agaussian p.d.f. with covariance given by:

S−1 = KT S−1ε K + S−1

a (8)

where Sε is the covariance of measurements error.

The information gained by making a measurement can be described by the entropy of a p.d.f.:

H(P ) = −∫

P (x)log[P (x)/M(x)]dx (9)

where M(x) is a measure function that can be constant. The information content is the change in entropy:

S = H(P (x)) − H(P (x|y) =∑

i

12log(1 + λ2

i ) (10)

while the degrees of freedom for signal is given from :

ds =∑

i

λ2i

λ2i + 1

(11)

where λi are the eigenvalues of the transformed Jacobian of the forward model linearisation y = Kx:

K = S−1/2ε KS1/2

a (12)

Such analysis can be found in the Rodgers’ book.3

An explicit solution for optimal estimation is given by:

∆xoe = ∆xa + (S−1a + KT S−1

ε K)−1KT S−1ε [∆y − K∆xa] (13)

with ∆xa = xa − x0 and ∆ya = y −F (x0), where xa is the a-priori profile and y is the vector of measurements.

Figure 2. Aerosol retrieval profile related to the first set of data of wavelengths. The “true” profile is related to maritimeaerosol obtained by Modtran 4 atmospheric data base with visibility of 8 Km. The a-priori profile is also drawn.

Figure 3. Aerosol retrieval profile related to the second set of data of wavelenghts. The “true” profile is related tomaritime aerosol obtained by Modtran 4 atmospheric data base with visibility of 8 Km. The a-priori profile is alsodrawn.

4. RESULT AND DISCUSSION

Let’s now apply the optimal estimation to aerosol profile retrieval. In such a case the complexity of the systemdeserves the use of a linearized Radiative Transfer Model as that one developed by Spurr.7 Results, for satellitelooking at nadir view, related to a maritime aerosol profile, obtained by Modtran 4 aerosol data set, at differentvisibilities, are presented in Table 2, for two different sets of wavelengths reported in Table 1. Measurementswere simulated by Modtran 4.

Table 1. set of wavelengths selected

λ(µ) λ(µ)

0.446 0.446

0.558 0.558

0.672 0.672

0.866 0.866

- 0.870

- 0.885

- 1.025

- 1.045

- 1.067

- 1.590

- 2.313

As it is evident from results in Table 2, comparing results obtained by the two wavelength sets of Table 1,no more information on aerosol profile comes adding new wavelengths.

In Fig. 2 and Fig. 3 are shown the retrieval using Equation (13). Comparing the two last figures it is clear whatthe analysis of information content has shown: the aerosol profile is not dependent on the spectral range used.This means the sensor should be addressed to measure the aerosol load, minimising the number of wavelengthsable to retrieve the aerosol as it is also shown in the analysis coming out from Equation (4). In such a way theMEOMS device may be properly activated, on selected wavelengths, in interference filter mode.

Further analysis will be carried out in future using the Oxygen band, where the effect of the aerosol on gasesmay be enhanced in such a way to be better analyzed. Off-nadir measurements will be also included to cope thestructure of MISR instrument mounted on TERRA satellite. All this modalities that are currently obtained bycomplex instruments, will be more easily obtained by MEOMS, as the device we have presented, because it hasno parts in movement and all operational modes can be obtained by a really integrated system.

In conclusion the analysis on information content obtained applying optimal estimation gives useful infor-mation to characterize the measurements modes. Such approach will be in near future more and more evidentwhen MEOMS devices will be operationally used in many fields of remote sensing.

REFERENCES1. R.Guzzi and O. Smokty “ Information content problems in radiative transfer theory”Exploring the Atmosphere

by Remote Sensing Techniques, R. Guzzi ed., Springer Verlag, Heidelberg 2003.2. S. Twomey Introduction to Mathematics of Inversion in Remote Sensing and Indirect Measurements, Elsevier,

New York, 19773. C. D. Rodgers Inverse Methods for Atmospheric Sounding. Theory and Practice, World Scientific, 2000

Table 2. Computation of DFS and H per the two wavelength sets of Table 1

Vis (Km) DFS(1) DFS(2) H(1) H(2)

50 0.226 0.285 0.166 0.322

23 0.896 0.834 1.124 1.245

12 0.972 0.965 2.432 2.233

8 0.977 0.989 3.103 3.123

6 0.985 0.994 3.608 3.765

4 0.988 0.998 4.249 4.433

2 1.123 1.107 5.102 5.290

4. G.G. Bentini, M. Bianconi, M. Chiarini, G. Correra, P. Maccagnani, F. Tamarri, R. Guzzi ”Theory and designof an integrated optical sensor based on a MACH-ZEHNDER interferometer”.”Sensors and Microsystems”World Scientific Publishing Co. Pte. Ltd. Singapore, Vol. VII, 2002

5. G. G. Bentini, M. Bianconi, M. Chiarini, G. Correra, C. Sada, M. Mazzoldi, N.Argiolas, B.Bazzan,R.Guzzi“Effect of low dose high energy O3+ implantation on refractive index and linear electro-optic prop-erties in X-cut LiNbO3: Planar optical waveguide formation and characterization.” J.App.Phys. vol 92 no11 pp 6477-6483, 2002

6. G. P. Box, M. A.Box and J.Krucker “ Information content and wavelength selection for multispectral ra-diometers”, J. Geoph. Res., vol. 101, no D14, pp 19211-19214, 1996

7. R. D.Spurr Linearized Radiative Transfer Theory Ph. Thesis, Technische Universiteit Eindoven, 2001