DE102008050046B3 - Method for determining concentration, pressure and temperature profiles in exhaust gas of aircraft, involves implementing derivations of forward models based on equations for radiation transport by automatic differentiation process - Google Patents

Abstract

The method involves transferring profile functions by a discretization code in a state vector. Models of the spectra utilized for adjustment calculus are adapted to measuring spectra up to concurrence with respect to noise, in terms of a non-linear least-square-fit process by variation of the state vector. Different accuracies of independent variables such as observation angle or tangent altitude, are considered by an orthogonal distance regression process. Derivations of forward models based on equations for radiation transport are implemented by an automatic differentiation process.

Description

The Invention relates to a method for determining concentration, Pressure and temperature profiles in any, preferably gaseous media by contactless working spectrometric radiation measurement, wherein a sequence measured spectra in any spectral range and simultaneously in the sense of a "Global Fit "in is evaluated by an equalization method by individual spectra corresponding measurement vectors to a total measurement vector be concatenated.

The Measurement of gaseous state Media, characterized by pressure, temperature, composition and the like, including their spatial Distribution and temporal variability are among the common tasks in scientific, industrial and official Area.

The Temperature is a central variable of the Earth's atmosphere and the Knowledge of spatial Temperature distribution (height, Length and Width) on earth is an essential condition for the numerical weather forecast. The same applies to Pressure and water vapor. In addition, the detection of spatial Distribution of trace gases and their evolution over time essential task of atmospheric research (Climate change, stratospheric Ozone depletion, etc.) and meteorology.

The Determination of exhaust gases from combustion and other chemical processes Processes is important for the control and optimization of the respective plants or engines or aircraft engines (efficiency) and the monitoring of undesirable or prohibited components (emission regulations, immission control regulations, Etc.). Combustion products are often harmful to the environment and / or health and their registration is therefore for both (health, environment, ...) authorities as also for them Research of great importance. In many cases, the gas temperature is one decisive size, as they are both the efficiency of the combustion processes and the composition influenced by the exhaust gases.

The used measuring methods essentially in two big ones Classes are divided, namely in sampling and non-contact remote sensing measurements.

at sampling method will be performed at a defined location within taken a sample of the medium to be examined and then their Composition, e.g. By gas chromatography or mass spectrometry, analyzed. Pressure and / or temperature measurements are by appropriate sensor made directly at the considered location. With one possible dense scanning of the medium leaves thus a good spatial resolution achieve.

Trial participants Measuring methods, however, are associated with many disadvantages. sampling are time consuming and expensive. Sampling methods are Furthermore often gas-specific. A measurement with high spatial resolution Time-variable media can only be achieved through simultaneous use a variety of probenehmenden sensors can be achieved. falsifications of the gas medium by sampling should be avoided. Possibly are changes of the medium, z. B. by turbulence, to take into account. In case of unstable Gases must have the composition immediately after or in time with sampling. Sampling measuring methods are about that not suitable for measurement in hard or inaccessible areas, z. B. in very hot Gases or in the middle to high earth atmosphere.

Next The measuring methods used have been non - contact measuring methods of the Remote sensing established on the measurement of the gas emitted by the and / or scattered electromagnetic radiation in the microwave range, infrared, visible or ultraviolet spectral range. With a suitable choice of the spectral range and the spectral resolution as well the measuring geometry allows the spectrometric radiation measurement the spatially resolved determination of pressure p, temperature T and gas concentrations ρ (also inhomogeneous) Media of any composition. Especially for hot gases are infrared spectroscopic Measuring method suitable because the gases of interest here on the one hand have significant spectral absorption properties, on the other hand at higher Temperatures also show strong thermal emissions.

In atmospheric research, satellite-assisted remote sensing with spectroscopic methods is one of the most important methods for determining the spatial pressure, temperature, gas concentration or aerosol distribution. Compared to nadirblickenden sensors horizontally probing sensors, so-called limb sounder, have some significant advantages. 1 shows in a schematic view the geometry of the atmospheric remote sensing by a horizontally probing instrument (Limb Sounder), which is attached to a circulating around the earth satellites. As the radiation (in absorption versus the sun or in emission) is measured for a series of line of sight (s) passed tangentially to the earth, this allows a much higher vertical resolution, which is essentially due to the vertical distance of the tangent points and the finite size of the field of view of the spectrometer (Field of View, FoV) is determined. Due to the horizontal lines of sight through the earth's atmosphere, compared to a vertical sounding, significantly longer distances in the altitude range are found directly above the tangent point. Thus, even species with very low concentrations can be measured. In 1 the earth radius is denoted by R _e .

Non-contact measuring methods of infrared spectroscopy have also proven themselves for the investigation of exhaust gases, in particular of jet engines or turbines. They allow the determination of exhaust gas components, ie gases such as CO ₂ , CO, NO ₂ , SO ₂ , H ₂ O and unburned hydrocarbons, particles or soot. For tomographic investigations of the spatial distribution of the exhaust gases and of pressure and temperature, several measurements are required in which spectra are observed for a number of lines of sight (preferably perpendicular to the exhaust gas jet). The measurements may be carried out for a sequence of parallel-displaced lines of sight, possibly also in two mutually perpendicular directions, or for a sequence of lines of sight which detect the exhaust jet at different angles, similar to the horizontal sounding of the earth's atmosphere.

2 shows a schematic representation of the tomographic exhaust jet measurement by a sequence of parallel lines of sight. The direction of the exhaust jet, which need not necessarily be exactly circular cross-section, is perpendicular to the plane of the drawing.

3 shows a schematic representation of the tomographic exhaust jet measurement through a sequence of differently directed lines of sight. The direction of the exhaust jet, which does not necessarily have to be exactly circular cross-section here, is also perpendicular to the plane of the drawing.

Either in the field of application of atmospheric research as well as in the field of investigation of exhaust gases based the so-called inversion, d. H. the evaluation of the sequence measured Spectra for the determination of pressure, temperature, gas concentration and aerosol profiles, usually on generally non-linear ones Least-Squares Fits (Gauss Compensation Method least squares). In the remote sensing of the atmosphere, the so-called onion-peeling process, ie the sequential evaluation of the spectra, starting with the outermost Spectrum (ie highest Tangent point) as far as possible by a simultaneous evaluation of Spectra replaced. Such a so-called "Global Fit "can also for them Analysis of exhaust gas measurements are taken as a basis.

The following prior art considerations are formulated primarily with respect to horizontally probing atmospheric remote sensing and may be transmitted in a technically known manner to tomographic exhaust gas examinations; the variable "tangent height" or observation angle is correspondingly by displacement or angle, cf. 2 respectively. 3 , to replace.

The starting point of the evaluation is the theory of radiation transport in gaseous media. Ignoring the scattering (aerosols and scattering are of no relevance to further discussion and are therefore neglected below), the radiance (intensity) I observed by an observer at location s is described by the integral form of the Schwarzschild equation:

Here, τ is the transmission as a function of the wavenumber ν, B (ν, T) the Planck function to temperature T, p the pressure, k _g and ρ _g the absorption cross section and the particle density of the gth gas, s' the path coordinate along the line of sight , I ₀ the source intensity and s ₀ the source value of the path coordinate s'. The sum in equation (2) extends over all the absorbing gases. In order to take account of instrumental effects, it is necessary to fold in a technically known manner with a suitable instrumental line profile function. In an analogous manner, the effect of the finite size of the instrumental visual field also has to be taken into account by folding.

In horizontally probing sensors, both instruments that measure in absorption (eg, against the sun) and instruments that measure the intrinsic emission of the atmosphere are common. Without limitation of generality, only emission spectroscopy will be dealt with below; in the case of absorption spectroscopy, only in equation (3) and the following equations is the vector of measured intensity I ^(obs) and intensity ^(modulated according to equation (1)) I ^{(mod) represented} by corresponding vectors (shown in bold) for the measured and replace modeled transmission (according to equation (2)).

Without restriction The general public should be assumed below that only a profile, eg. The temperature or concentration of a gas, is to be determined and all other profiles are known. The generalization in the case of several unknown profiles is done in technically known Wise.

The the profile to be determined by the measurement is in the simplest case a continuous function of the height z (related eg to the Earth's surface) in the general case also a function of the geographical length and Width. Without restriction The public will be neglected in the following length and width. from that also results that in the equations (1) and (2) as a function the path coordinate s occurring atmospheric profiles p (s), T (s), ρ (s) in unambiguous Way as a function of height z are representable.

Since only a finite number of parameters can be determined from a measurement, the unknown profile f (z) is discretized in a technically known manner, for. By applying a quadrature method to calculate the path integrals in equations (1) and (2), ie approximately ∫k (ν, p (z), T (z),) ρ (z) dz ≈ Σ n j = 1 w j k (ν, p (z j ), T (z j )) Ρ (z j ) or by developing according to a suitable functional system f (z) ≈ Σ n j = 1 x j φ j (Z) , The n-dimensional parameter vector x representing the unknown profile f (z) is thus formed from the function values x _j = f (z _j ) at the quadrature nodes z _j or from the development coefficients x _{j of} the function development.

At a spectrum measured for m discrete wavenumbers (or frequencies) ν ₁ , ..., ν _m I (Obs) = (I (Obs) 1 , ..., I (Obs) m ) Now, by appropriate variation of the parameter vector x to be determined, a spectrum I ^(mod) modeled according to equation (1) is changed until the norm of the m-dimensional residual vector is minimal:

There the spectrum generally in a non-linear manner from the parameter vector x (or the x represented by Profile), This nonlinear least squares problem is known in the art Way z. By Gauss-Newton or Levenberg-Marquardt method.

durch einen im Allgemeinen nichtlinearen Least-Squares Fit bestimmt.In the case of horizontal sounding, a complete measurement consists of a sequence of several spectra measured for different tangent heights (I (Obs) 1 , ..., I (Obs) L ) , Due to the unavoidable error propagation in the sequential evaluation of such a series of measurements (Onion Peeling), the simultaneous evaluation of the spectra (Global Fit) results in the measurement vectors I (obs) / l of length m _l corresponding to the individual spectra becoming an overall measurement vector I (Obs) = (I (Obs) 1 , ..., I (Obs) L ) (4) of length M = m ₁ + ... + m _L concatenated. Without limiting the generality, all spectra should have the same number of measuring points, ie m ₁ = m ₂ =... = M _L = m and thus M = L * m. Analogously, an overall model vector is formed and the unknown parameter vector x is obtained by generalization of equation (3).

determined by a generally non-linear least-squares fit.

• a) Ungenaue Mess-Geometrie (Winkel, Höhen, ...) Wie vorstehend in Verbindungen mit den Überlegungen zum Stand der Technik erläutert wurde, ist ein einzelner Messwert bei der Horizontalsondierung eine Funktion von zwei Variablen, nämlich von der Wellenzahl ν und der Tangentenhöhe h_t (bzw. Elevationswinkel des Sensors). An dieser Stelle wird darauf hingewiesen, dass im Folgenden teilweise die Tangentenhöhe mit ”t” als Superskript geschrieben wird, wenn zusätzlich ein Laufindex l erforderlich ist. Im Gegensatz zu den Wellenzahlen ist die Tangentenhöhe jedoch nur mit eingeschränkter Genauigkeit zu bestimmen. Die aus den Messungen abgeleiteten Profile sind daher mitunter in der Höhe verschoben oder verzerrt. Daher ist es bei der Auswertung einer Horizontalsondierungsmessung vielfach üblich, auch die Tangentenhöhen aus den gemessenen Spektren durch Least-Squares zu bestimmen. Der unbekannte Parametervektor des Profils wird somit ergänzt zu x → X = (x1, ..., xn, ht1 , ..., htL ) und das in der Gleichung (5) angesprochene Minimalisierungsproblem entsprechend verallgemeinert. Dies hat jedoch eine schlechtere Konditionszahl der Jacobi-Matrix zur Folge (lineare Abhängigkeiten der Spaltenvektoren). Dieser Ansatz ignoriert zudem jedoch die unterschiedliche Natur der unabhängigen Größen Wellenzahl und Tangentenhöhe.
• b) Zusammenhänge zwischen den Einzelspektren Beim ”Global Fit”-Verfahren werden die einzelnen Elemente des Gesamt-Messvektors als unabhängig voneinander betrachtet, d. h. die m × L gemessenen Intensitätswerte werden als Funktion unabhängiger Messpunkte ((ν1, ht1 ), (ν2ht1 ), ..., (νm, ht1 ), (ν1, ht2 ), ..., (νm, htL )) betrachtet. Beziehungen zwischen den Messpunkten werden dabei vernachlässigt, also z. B. funktionale Zusammenhänge zwischen den Intensitäten einer bestimmten Wellenzahl, I(νi, ht1 ) ↔ I(νi, ht2 ) ↔ ... ↔ I(ν1, htL ). Tatsächlich jedoch ist dies als eine Matrix mit einer ”Matrix” unabhängiger Variablen und einer entsprechenden Matrix abhängiger Variablen zu betrachten, vgl. die später angegebenen Gleichungen (17) und (18).
• c) Schlechtgestelltes inverses Problem Inverse Probleme, wie die Bestimmung von Profilen aus spektroskopischen Messungen, sind vielfach ”schlechtgestellt”, d. h. insbesondere, dass kleine Änderungen des Mess-Vektors zu starken Änderungen des Lösungsvektors führen. Standard-Lösungsverfahren des Least-Squares-Problems ergeben meist physikalisch ”unsinnige” Lösungen, z. B. negative Temperaturen und Gaskonzentrationen oder stark oszillierende Profile (bzw. deren diskretisierte Darstellung).
• d) Ableitung nach Tangentenhöhen nicht analytisch Bei der Lösung eines nichtlinearen Least-Squares-Problems wie in den Gleichungen (3) oder (5) werden die partiellen Ableitungen der Modellfunktion nach den Komponenten des zu bestimmenden Parametervektors x benötigt (Jacobi-Matrix, im Falle des einfachen Least- Squares Fits der Gleichung (3) also eine m×n-Matrix). Während sich die Ableitungen nach Druck, Temperatur oder Gaskonzentrationen (genauer die Ableitungen nach den Komponenten ihres diskreten Darstellungsvektors x) einfach aus der analytischen Darstellung des Modellspektrums gemäß den Gleichungen (1) und (2) berechnen lassen, sind die Ableitungen nach den Tangentenhöhen nur extrem aufwändig analytisch bestimmbar und werden daher in der Regel durch finite Differenzen
  
  bestimmt. Die Berechnung von Ableitungen durch Störung ist jedoch sehr zeitaufwändig und fehlerbehaftet. Da eine fehlerhafte bzw. ungenaue Jacobi-Matrix zudem das Konvergenzverhalten des Optimierungsproblems negativ beeinflusst, führt dies zu einer weiteren Verzögerung und unter Umständen konvergiert der Fit nicht.

There are a number of disadvantages with the above-mentioned gas profile analysis methods currently used:

• a) Inaccurate Measurement Geometry (Angles, Heights, ...) As discussed above in connection with the prior art considerations, a single horizontal probing reading is a function of two variables, viz., wavenumber ν and tangent height h _t (or elevation angle of the sensor). At this point it depends showed that in the following part of the tangent height with "t" is written as a superscript, if in addition a running index l is required. In contrast to the wave numbers, however, the tangent height can only be determined with limited accuracy. The profiles derived from the measurements are therefore sometimes shifted in height or distorted. Therefore, in the evaluation of a horizontal probe measurement, it is often customary to also determine the tangent heights from the measured spectra by least squares. The unknown parameter vector of the profile is thus added x → X = (x 1 , ..., x n , H t 1 , ..., H t L ) and generalize the minimization problem addressed in equation (5) accordingly. However, this results in a worse condition number of the Jacobi matrix (linear dependencies of the column vectors). However, this approach also ignores the different nature of the independent quantities wavenumber and tangent height.
• b) Relationships between the individual spectra In the "Global Fit" method, the individual elements of the overall measurement vector are considered to be independent of each other, ie the m × L measured intensity values become as a function of independent measurement points ((Ν 1 , H t 1 ), (ν 2 H t 1 ), ..., (ν m , H t 1 ), (ν 1 , H t 2 ), ..., (ν m , H t L )) considered. Relations between the measuring points are neglected, so z. B. functional relationships between the intensities of a given wavenumber, I (ν i , H t 1 ) ↔ I (ν i , H t 2 ) ↔ ... ↔ I (ν 1 , H t L ). In fact, however, this is considered to be a matrix with a "matrix" of independent variables and a corresponding matrix of dependent variables, cf. equations (17) and (18) given later.
• c) Deficient inverse problem Inverse problems, such as the determination of profiles from spectroscopic measurements, are in many cases "bad", ie in particular that small changes in the measuring vector lead to strong changes in the solution vector. Standard solution methods of the least squares problem usually result in physically "nonsensical" solutions, eg. As negative temperatures and gas concentrations or highly oscillating profiles (or their discretized representation).
• d) Derivation by tangent heights not analytically In solving a non-linear least-squares problem as in equations (3) or (5), the partial derivatives of the model function are required for the components of the parameter vector x to be determined (Jacobi matrix, in the case of the simple least-squares fit of equation (3), ie an m × n matrix). While the derivatives of pressure, temperature, or gas concentrations (or, more precisely, the derivatives of their discrete vector x) can be easily calculated from the analytic representation of the model spectrum according to Equations (1) and (2), the derivatives are only extreme after the tangent heights Complex analytically determinable and therefore usually by finite differences
  
  certainly. However, the calculation of derivatives by interference is very time consuming and error prone. Moreover, since a faulty or inaccurate Jacobi matrix adversely affects the convergence behavior of the optimization problem, this leads to a further delay and under certain circumstances the fit does not converge.

On Reason of the above-mentioned disadvantages lead "standard method" for measurement of temperature, pressure and / or concentration distributions of gaseous media at elevated Computing effort to worse results, d. H. to inaccuracies, associated with adulterated spatial Assignment.

A standard method in this sense for the quantitative analysis of gas volumes, in particular exhaust gases from incinerators, by means of emission or absorption spectrometry in any spectral range is also out EP 0 959 341 B1 known. Here, a geometrically defined and reproducibly adjustable observation plane is defined, which is aligned in each case perpendicular to the exhaust gas jet longitudinal axis and displaceable along this axis. In a first series of measurements, a number a of spectral measurements is carried out, with the optical axis of a spectrometer always lying in the respective observation plane, but offset from one measurement to the next in each case by a first distance in parallel. In a second series of measurements, b measurements are then carried out, the optical axis aligned this time perpendicularly to the previously mentioned optical axis again being in the observation plane and displaced parallel from measurement to measurement by a second distance. As a result of the arrangement of the optical axis in the observation plane, two (orthogonal) measurements result in two sets of orthogonal radiation beams that form a lattice with (a × b) intersection volumes whose size is given by the geometry. With the aid of the set of (a + b) measurements, the spectral transmission or the spectral emission results, which is integrated over the entire gas volume in the beam of the field of view of the spectrometer. In this known method, a plurality of spectra can be measured and simultaneously evaluated in the sense of a "global fit" in a least-squares compensation method by concatenating the measurement vectors corresponding to the individual spectra to form an overall vector and applying the minimization condition to this vector.

Out DE 198 40 794 C1 discloses a method for detecting the infrared radiation properties of exhaust gases from jet engines or turbines, wherein both emission and transmission measurements are carried out at short time intervals and the measurements in two mutually perpendicular directions transverse to determine the spatial distribution of pressure, temperature and gas concentration be carried out to the exhaust jet with an adjustable step along the axis of the exhaust gas jet.

In DE 10 2004 001 748 A1 For example, a method of analyzing gaseous media is described. The gas analysis is carried out here by an infrared spectrometer and by information about the temperatures of the surface portions of the viewed by the spectrometer background seen thermal imaging camera.

In DE 10 2005 060 245 B3 is a method for determining concentration, temperature and pressure profiles of gases in combustion processes and their exhaust gas streams and clouds treated. For this purpose, the absorption coefficients necessary for a calculation of model spectra for each wave number as a function of pressure and temperature are considered in a compensation calculation. These functions are approximated by means of interpolating, two-dimensional, cubic spline functions, and in a pressure-temperature plane, a desired pressure and temperature range is subdivided into rectangular, axis-parallel, but non-overlapping subregions. For each partial rectangle, a polynomial is considered to be an interpolating function whose first derivatives are equal in terms of pressure or temperature and the mixed derivatives according to pressure and temperature at the corners.

task The present invention is in a method of measurement of temperature, pressure and / or concentration distributions in gaseous or similar Media while avoiding the disadvantages mentioned above an evaluation several for different elevation angles or tangent heights of measured spectra create, taking into account will that the observation geometry (angle and the like) only is known with finite precision and that the spectra of the Measurement sequence are measured on the same wavenumber grid, additional information or ancillary conditions become a stable, physically meaningful solution of the allow inverse problem, and derivations that are not analytically are easy to calculate by automatic differentiation methods instead of being calculated by fault become.

basierenden Vorwärtsmodells durch Automatische Differenzierungsverfahren implementiert werden, wobei I die von einem Beobachter am Ort s beobachtete Strahldichte (Intensität), τ die Transmission als Funktion der Wellenzahl ν, B(ν, T) die Planckfunktion zur Temperatur T, p der Druck, k_g und ρ_g der Absorptionsquerschnitt und die Teilchendichte des g-ten Gases, s' die Wegkoordinate entlang des Sichtlinienstrahles, I₀ die Quellintensität und s₀ der Quellwert der Wegkoordinate s' sind.According to the invention, which relates to a method of the type mentioned above, this object is achieved in an advantageous manner that the sought profile function or the desired profile functions is converted by a Diskretisierungsvorschrift in a state vector or are that used for the equalization Adapt the models of the spectra in terms of a generally non-linear least-squares fit method by varying the state vector (model parameter) to the corresponding measurement spectra to match the noise that the different accuracies of the independent variables, namely the observation angle or the tangent height on the one hand and the wavenumber on the other hand, by orthogonal distance regression instead of standard least squares are taken into account and that derivatives of the known equation for radiative transfer

For example, where I is the radiance (intensity) observed by an observer at location s, τ is the transmission as a function of wavenumber ν, B (v, T) is the Planck function to temperature T, p is the pressure, k _g and ρ _{g is} the absorption cross section and the particle density of the gth gas, s 'is the path coordinate along the line of sight, I _{0 is} the source intensity, and s _{0 is} the source value of the path coordinate s'.

The Method according to the present invention The invention is particularly in the context of atmospheric / meteorological remote sensing or in the measurement of gases in combustion processes, exhaust gas streams or clouds of high importance.

A Sequence of spectra can z. B. from a horizontally probing Instrument on a satellite or similar carrier for remote sensing of the atmosphere in any spectral range, preferably ultraviolet, infrared, or millimeter range, and then becomes one in the sense of a "global fit" single "measuring vector" merged (concatenated).

at the formulation of the evaluation representing an inversion problem the sequence of measured spectra for determining the profile function or the profile functions in the sense of an orthogonal distance regression The measurement uncertainties are advantageously pre-whitened considered.

at the iterative solution the weighted orthogonal distance regression by Gauss-Newtonian or Levenberg-Marquardt method becomes the conditioning of the Jacobi matrix forming a derivative matrix advantageously improved by column normalization.

at the so - called inverse problem representing determination of the Profiles or profiles from the spectroscopic measurements a numerically stable and physically meaningful solution in expedient manner by additional Regularization by means of barriers, z. B. positivity, and / or by means of quadratic secondary conditions, for. B. smoothness determined.

to Evaluation of a sequence of spectra for a measurement of engine exhaust gases is the inventive method Advantageously characterized in that the for the evaluation of a sequence of Limb spectra as part of atmospheric remote sensing Methods on similar Evaluation problems transmitted analogously become.

advantageous and appropriate training and embodiments of the method according to the present invention are specified in the reclaiming claims on claim 1.

The Invention will be explained below in detail with reference to drawings. It demonstrate:

1 an already described geometry of atmospheric remote sensing by a horizontally probing instrument (Limb Sounder),

2 a likewise previously explained schematic representation of a tomographic Ahgasstrahlmessung by a sequence of parallel lines of sight, wherein the direction of the not necessarily exactly circular cross-section-shaped exhaust gas jet is perpendicular to the plane,

3 a likewise already explained above schematic representation of the tomographic exhaust jet measurement by a sequence of differently directed lines of sight, wherein the direction of the not necessarily exactly circular cross-section-shaped exhaust gas jet is perpendicular to the plane,

4 a schematic representation of the ordinary least-squares fit, and

5 a schematic representation of the orthogonal distance regression ODR used according to the invention.

The working according to the invention Method Uses Fits (Gauss Compensation Method) Instead of a Simple Least Squares the so-called Orthogonal Distance Regression (Orthogonal Distance regression; ODR). In the case of purely linear relationships the so-called "Total Least Squares "instead the normal "Least Squares "used.

The basic idea of these equalization methods is, besides errors in the dependent variable y (in the present case this is the intensity), also to consider errors in one or more of the independent variables. If f (x, t) is a model of the quantity y measured as a function of the independent variable x for different parameters t with errors ε, that is f (x, t _i ) ≈y _i + ε _i for i = 1,. m, so are in the usual least-squares method (see. 4 ) determines the n model parameters x by minimizing the error squares:

In the use of the orthogonal distance regression ODR according to the invention, errors δ are also taken into account in the measurement variable t and the minimization condition of the ordinary least squares method given in equation (7) becomes appropriate 5 replaced by:

welches ein Standard–Minimalisierungsproblem ohne Nebenbedingungen darstellt. Die Lösung dieses Problems erfolgt in technisch bekannter Weise durch Verfahren der numerischen linearen Algebra, wobei die besondere Struktur des Problems ausge nutzt wird, um eine effiziente und stabile Implementierung zu gewährleisten. Dazu wird mit Hilfe der Definitionen

und dem m + n-Vektor

die Gleichung (9) in ein gewöhnliches Least-Squares-Verfahren

für m + n Parameter und 2m Gleichungen umgeschrieben. Für die Effizienz der Implementierung ist es entscheidend, dass die Block-Struktur der erweiterten Ableitungsmatrix (Jacobi–Matrix)

berücksichtigt wird mit

I ∊ Rm×n Einheitsmatrix. (16) By substituting the constraint condition into the minimization condition, one obtains

which is a standard minimization problem without constraints. The solution of this problem is done in a technically known manner by methods of numerical linear algebra, the particular structure of the problem is exploited to ensure an efficient and stable implementation. This is done using the definitions

and the m + n vector

the equation (9) in a usual least squares method

rewritten for m + n parameters and 2m equations. For the efficiency of the implementation, it is crucial that the block structure of the extended derivative matrix (Jacobi matrix)

is considered with

I ε R m × n Identity matrix. (16)

In the method operating according to the invention, the orthogonal distance regression ODR is now advantageously applied to the evaluation of horizontal sounding measurements. The measured spectra I ^(obs) correspond to the erroneous measured quantities y, and the modeling of the spectra according to equation (1) corresponds to f. The discretized representation of the atmospheric profile to be determined (temperature, gas concentration, etc.) is symbolized by x as in the earlier explanations. In the simplest case of horizontal sounding, in which only one spectral point is measured for all tangent heights (eg the integral total intensity of a spectral line or spectral band of the gas to be determined), the independent variable t in equation (8) et seq faulty tangent heights h _t (or the observation angle).

und der Raum der y durch deren Abbildung auf Intensitätswerte I(ν, h_t) aufgespannt wird,

In the general case of m measured intensity values per tangent height h _t is in the invent In accordance with this method, the possibility of the orthogonal distance regression ODR is exploited to evaluate multidimensional data. The independent variable t and the dependent variable y thus correspond to measured variables from two-dimensional spaces, wherein the space of t is spanned by wavenumbers ν and tangent heights h _t ,

and the space of y is spanned by their mapping to intensity values I (ν, h _t ),

In order to are advantageously the previously mentioned facts considered, which the connections between the individual spectra.

In the implementation of the orthogonal distance regression ODR for the evaluation of a sequence of limb spectra, it is additionally to be considered that the dependent quantities y, f (here intensities) and independent quantities t, δ (tangent heights or their errors) are physical quantities of different dimensions are. The formulation of the least squares problem derived from the orthogonal distance regression ODR according to equation (9) can be independent of the choice of the representation of the dependent quantities y, f (here intensities) and independent variables t, δ (here tangent heights or their errors) (for example, intensity in W / (cm ² sr cm ^-1 ) or erg / (s cm ² sr cm ^-1 ) and tangent heights in kilometers or meters), by referring the quantities to their assumed measurement uncertainties. This can generally be done by weighting with the covariance matrices of y and t. (If only one estimated value is known, for example, from the standard deviation, then the corresponding covariance matrix can be used as the product of this variance with a unit matrix of suitable dimensions.) By pre-whitening (multiplication of the measurement vector y and the vector f of the model function f (x , t) with the root of the covariance matrix) the problem is reduced to the standard form according to equation (9).

Furthermore, the conditioning of the least squares problem according to equation (7) or the ODR problem according to equation (8) essentially depends on the variation of the order of magnitude of the parameters x and t. Thus, for gases whose volume mixing ratio in the atmosphere decreases by several orders of magnitude with increasing altitude, it is better to consider the particle density n = VMR * n _air as unknown x. In general, by introducing diagonal weighting matrices for x and t, these parameters can be transformed into the same order of magnitude. The different physical dimensions and the different accuracies of the relevant quantities can thus be taken into account by a weighted orthogonal distance regression ODR.

For the iterative solution of the orthogonal distance regression ODR or the non-linear least-squares problem derived therefrom, the derivatives are required according to the atmospheric profile to be determined (represented by the vector x) and the error-prone tangent heights h _t according to equation (13). Since these derivatives can only be calculated analytically with great effort, automatic differentiation methods are used in the method operating according to the invention. Since the differentiation of mathematical functions, in contrast to integration, can be carried out by applying simple rules, this set of rules can also be applied to the differentiation of the functions implemented by programming languages such as C or Fortran. Thus, to calculate the derivatives of the radiation intensity described by equations (1) and (2), the corresponding radiative transfer computation program is transformed by a pre-processor into a program which, in addition to the function (intensity), simultaneously calculates its derivative.

The solution of the non-linear least-squares problem derived from equation (8) according to equation (12) (or the usual non-linear least-squares problem according to equation (7)) is given in tech known manner iteratively, z. By a Gauss-Newton or Levenberg-Marquardt method. Even with weighted orthogonal distance regression ODR, the derivative matrix (Jacobi matrix) can be approximately singular (high condition number), so that in the calculation of the parameter update vector δη by QR factorization of the Jacobi matrix a prior scaling usually with the norm of Columns of the Jacobi matrix is important.

wobei Ω die Regularisierungsmatrix und λ der Regularisierungsparameter ist. Die Bestimmung der Regularisierungsgrößen und die Lösung von Gleichung (19) erfolgen in technisch bekannter Weise, z. B. mittels L-Kurven-Verfahren oder iterativ regularisierter Gauß-Newton-Verfahren. Diese Regularisierung vermittelt physikalisch sinnvolle Lösungen des im Allgemeinen schlechtgestellten inversen Problems.The introduction of additional information for the stabilization of the solution (regularization) is carried out in a technically known manner by extending the least-squares problem according to equation (7) or the ODR problem according to equation (8) by quadratic secondary conditions (smoothness) and / or inequality secondary conditions , z. B.

where Ω is the regularization matrix and λ is the regularization parameter. The determination of the regularization quantities and the solution of equation (19) are carried out in a technically known manner, for. Example by L-curve method or iteratively regularized Gauss-Newton method. This regularization provides physically meaningful solutions to the generally ill-posed inverse problem.

Claims (5)

A method for determining concentration, pressure and temperature profiles in any, preferably gaseous media by means of non-contact spectrometric radiation measurement, wherein a sequence of spectra in any spectral range measured and evaluated simultaneously in terms of a "global fit" in a compensation calculation method by the den be concatenated to individual spectra corresponding measurement vectors to form an overall measurement vector, characterized in that the sought profile function or the sought profile functions is converted by a Discretisierungsvorschrift in a state vector or are that the models used for the compensation calculation of the spectra in terms of a generally non-linear Least Squares-fit method by varying the state vector (model parameter) to the corresponding measurement spectra are adjusted to match in the context of noise, that in the compensation calculation the Different accuracies of the independent variables, namely the observation angle or tangent height on the one hand and the wavenumber on the other hand, by orthogonal distance regression instead of standard least squares are taken into account and that derivatives of the known equation for radiative transfer

For example, where I is the radiance (intensity) observed by an observer at location s, τ is the transmission as a function of wavenumber ν, B (v, T) is the Planck function to temperature T, p is the pressure, k _g and ρ _{g is} the absorption cross section and the particle density of the gth gas, s 'is the path coordinate along the line of sight, I _{0 is} the source intensity, and s _{0 is} the source value of the path coordinate s'. Method according to claim 1, characterized in that that in the formulation of the evaluation of the sequence measured Spectra for determining the profile function or the profile functions in the sense of an orthogonal distance regression, the measurement uncertainties considered by pre-whitening become. Method according to claim 1 or 2, characterized that in the iterative solution a weighted orthogonal distance regression by Gauss-Newtonian or Levenberg-Marquardt method the conditioning of the Jacobi matrix forming a derivative matrix is improved by column normalization. Method according to one of the preceding claims, characterized in that in the so-called inverse problem representing determining the profile or profiles from the spectroscopic measurements a numerically stable and physically meaningful solution by additional regularization by barriers, z. B. positivity, and / or by means of quadratic constraints, z. B. smoothness, is determined. Method according to one of the preceding claims for the evaluation a sequence of limb spectra in the context of an atmospheric Remote Sensing and analogous manner for the evaluation a sequence of spectra for a measurement of engine exhaust.