integral equations, 27.09.2007 inverse methods for the remote sensing of atmospheric parameters...

Integral Equations, 27.09.2007

Inverse methods for the remote sensingof atmospheric parameters

Rolando Rizzi

Atmospheric Dynamics Group, Department of Physics

University of Bologna


Purpose

Show an example of the intimate relationship between important physical processes and the mathematical tools required for the their quantitative description.

It is a presentation of the physical interpretation behind an integral equation that describes modern remote sensing techniques to retrieve properties of the atmospheres, and of current methodologies applied to solve this problem.

Therefore it is not intended to be a presentation of up-to-date personal research work.


Radiances from Earth

Over the Antarctic

Over the Sahara Desert

Measured by IRIS-D on Nimbus-4


Radiances from Mars

Over lat. 21 ° South

Measured by IRIS-D on Mariner 9

Near lat. 66 ° North


How can we interpret these spectra?

Qualitative: knowledge about spectroscopy of gaseous substances, that is how they interact with light;

Quantitative: an equation that predicts the absorption and emission of light by atmospheric (Earth, Mars,…) gases


Relationship bewteen radiance and matter

There is a relationship between the energy flowing in a certain direction per unit time in a specific wavelength interval and certain properties of the surface and of the atmosphere:

Simplifications:

•plane parallel atmosphere

•horizontal homogeneity

•dependence on time implicit

•No particulate matter/clouds


Relationship bewteen radiance and matter

Simplification; we consider energy flowing at zenith at TOA;

It is also understood that radiance L, transmittance τ and absorption

coefficient k all depend on Ts,T,qi,…


The meaning of the Schwarzchild equation

d

s

s

dzqvkdzqvk

TOA

T

T

p

p

vv

Svk

eevzz

TOAz

z

iii

TOAz

z

;)(

)(

),,(

220

)()(

I II


Limiting cases

Case 1: k very small: τ(0,zTOA)~1, term I >> term II

Radiance represents surface properties

Case 2: k very large: τ(0,zTOA)~0, term I << term II

Radiance represents some atmospheric properties


Some atmospheric properties?


Transmittance and Weighting function:example


Transmittance and Weighting function


Application to remote sensing?

We have seen that by increasing the absorption coefficient k of a particular gas we can measure radiance that comes from broad atmospheric layers closer to the measuring device.

How can we change the absorption coefficient?

By changing the wavelength!

Spectrometers are designed to measure spectral radiance. A sequence of spectral radiances contain information on how some atmospheric

parameters (T,qi) change with height.


Some preliminary steps

From radiance measured at wavelengths where k is very small we can derive the magnitude of the first term:

We can extrapolate from these measurements the properties of the

surface (the product of emissivity ε and Planck radiance B) also at

wavelengths where k is very large (emissivity ε varies slowly with

wavelength and we know the wavelength dependence of B)


The integral equation

Finally we rewrite the integral equation with the dependence on T(z) and q(z) and…taking into consideration the ever present measurement errors!

The temperature T can be derived from analytical inversion once the

Planck function B(z,v) is known.

The other unknown (q) is embedded into the kernel. It looks as the

problem of retrieving q is not the simplest problem.


The integral equation

To retrieve the Temperature(z) we would ideally exploit the properties of a gas whose concentration is precisely known. Until ten years ago CO2 concentration was considered well known…

Assume that M channels within the CO2 15 m band have been chosen to

recover the temperature profile. Thus, a set of M observations give M integral equations as:

Index i identifies the selected wavelengths.

Note that the spectral dependence of B has been factorised into the kernel function.


This Freedholm integral equation of first kind is clearly an ill-posed problem, since the unknown is a continuous function of height but there are only a finite number of observations.

It is clearly both mathematically and physically incorrect to solve an under-constrained set of equations. There is an infinite manifold of solutions which satisfy them exactly, and an even larger number of solutions which satisfy them within experimental error.


An example of an initial error

Most linearisation schemes (such as expansion on a complete basis) can be expressed as:

MifAg i

N

jjiji ,,2,1;

1

εAfg

gT1T AA)(Af

The straightforward least squares solution (that neglects experimental errors) provides an unstable

solution. matrix ATA has a small determinant, due to the structure of the weighting functions for the temperature (and humidity) and is very sensitive to very small measurement errors. Therefore the matrix inverse operation amplifies these small differences.


Given the measured radiance vector g, the statistics of the experimental error

ε, the weighting functions K, and any other relevant information, what can

be said, in a physically meaningful way, about the unknown f ?


Other relevant information?

Other relevant information, beyond those provided by the measurements, is needed, in the form of constraints on the solution profile, in order that the error variance of the solution be finite.

If no constraints are available, or the constraints are insufficient, the only valid approach is to accept to find a smoothed version of the true profile, with a known smoothing function, by some linear combination of the measurements.

The most enlightening approach to understanding the role of the added information is to regard them as virtual measurements, that is, as being of the same nature as real measurements.

They say something about the unknown profile just as the measurements do, and together with the measurements they determine whether the problem is well-posed.


Virtual measurements

Many of the constraints that have been used in the literature fall into the class of linear constraints. Some of them are described below:

– Statistics. The climatological mean profile and its covariance may be regarded as a measurement and its uncertainty.

– Forecast profile. The forecast profile and its error covariance may similarly be regarded as a measurement and its uncertainty. However, the error covariance is often not known very well, so must be overestimated.

– Twomey-Tikhonov. The virtual measurement is an a priori guessed profile. The error covariance is proportional to Twomey's constraint matrix H, often taken to be a unit matrix.

– Discretisation. It is usual to express continuous functions such as the unknown profile and the weighting functions in a discrete form n order to simplify the numerical calculations, and so that the algebra of vectors and matrices may be used instead of Hilbert space. This is of course a linear representation and may be regarded as a linear constraint.


A few examples of inversion methods

Constrained linear inversion (Phillips (1962), Twomey(1963): minimization of the deviations from the mean

gAγH)A(Af T1T ˆ111

111

111

1

1

1

NNN

NNN

NNN

H

Many versions of constrained linear inversion differing a) in the quantity to

be minimised, b) for methods to compute γ and c) for the structure of

matrix H. For example we can minimise difference from a climatology

vector fc:

)ˆ( C1T fgAI)A(Af Tγ


Maximum probability solution

The maximum probability solution: the inversion problem is formulated as a variational problem: a cost function which has to be minimized :

J(x) = (x-xb)T B-1 ( x-xb) + (ym - y{x})T Y-1 ( ym - y{x})= minimum

x represents the atmospheric state (a vector of the vertical temperature profile and other

atmospheric variables such as water vapour concentration and surface temperature.

xb is a background profile

ym is the measurement vector

y{x} is the forward model

B-1 is the covariance of the background information

Y-1 is the covariance of the measurements and of the forward model



The most probable (and therefore optimal) profile is found by minimizing

J(x) or by solving its gradient equation: if J’(x) represents the gradient

of J(x) with respect to x, then

J'(x) = B-1 (x-xb) - K(x)T Y-1 (ym -y{x}) = 0

Where K(x) is a matrix containing the Jacobian x

x

y



In general, this problem is not trivial and there is no general analytic solution to this equation; but many techniques exist for solving it numerically for problems of interest. Here we shall restrict only the linear case when

K(x) = K(xb) = K

where K is taken to be a constant matrix.

x = xb + (B-1 + KT Y-1 K)-1 KT Y-1 (ym -y{ xb })


Conclusions

This talk ends where your interests start!

I could not possibly say anything that is new, from the mathematical point of view, of the hundreds of solutions that have been attempted to solve this difficult problem tend to be iterative versions of the constrained and maximum probability solutions to partially deal with non-linearities,

especially for the retrieval of q.

Atmospheric physicists with expertise in radiative transfer have succeded in convincing WMO and international meteorological centres that temperature and humidity profiles derived from satellite “sounders” are data that should be used.


Climatology will obtain a great advantage from this kind of data. The new, higher resolution, sounders (NASA-AIRS and European IASI) represent important positive steps forward.

In particular IASI is demonstrating a clear positive impact on operational numerical weather prediction (application) and is certainly a great instrument to work with.


Molte grazie per l’attenzione.

integral equations, 27.09.2007 inverse methods for the remote sensing of atmospheric parameters...

Documents

integral equations

gases slide

north slide

particulate matterclouds

relationship bewteen

atmospheric earth

intimate relationship

certain properties