DE102008003037B3 - Method for determining physical characteristics of atmospheric particles, involves preparing vertical profile of optical characteristic of particles and determining vertical profile of physical characteristic of particle by inversion method - Google Patents

Abstract

The method involves preparing a vertical profile of an optical characteristic of a particles and determining a vertical profile of a physical characteristic of the particles by an inversion method from the vertical profile of the optical characteristic. The vertical profile of the physical characteristic is determined by a discrepancy parameter profile. Independent claims are included for the following: (1) a device for determining physical characteristics of atmospheric particles; (2) a computer-readable storage medium for storing a program for determining physical characteristics of atmospheric particles; and (3) a computer program element for determining physical characteristics of atmospheric particles.

Description

The The invention relates to a method, a device, a computer-readable Storage medium and a computer program element for determining Vertical profiles of physical (or microphysical) properties atmospheric Particle.

lidar (light detection and ranging), also referred to as light radar, is a method related to radar (radio detection and ranging) Distance and speed measurement of objects. It will be emitted electromagnetic signals and the signal delay and / or the intensity detects the reflected or scattered by the objects radiation. Instead of radio waves as in the case of radar, laser radiation is produced at the lidar used. Lidar methods are also used in atmospheric research used for remote measurement of atmospheric parameters. With help Lidar instruments can be used, for example, for optical properties from atmospheric Examine particles. atmospheric Particles (also called aerosols) are z. Forest fire particles, desert dust and by human activities caused pollution from z. Industry and transport.

1 shows the schematic structure of a lidar measuring arrangement 100 , The Lidar measuring arrangement 100 is set up as a so-called multi-wavelength Raman lidar. There are light pulses 101 at one or more wavelengths of one or more lasers 102 sent out into the atmosphere. At present, Nd: YAG lasers (neodymium-doped yttrium aluminum garnet lasers) are mainly used in lidar research.

such Lasers emit laser radiation at the fundamental wavelength of 1064 nm in the form of short bursts of radiation, several times per second be emitted. By so-called frequency multiplier crystals can laser radiation with twice the frequency, ie half Wavelength (532 nm), and with three times the frequency, that is, the third-order wavelength (355 nm). In principle, higher frequency multiples can be generated become. However, for example, crystals are used to produce the fourfold frequency, so at 266 nm wavelength, currently not atmospheric physical reasons used.

The light pulses 101 be by means of an arrangement 103 optical elements (eg, mirrors, lenses, beam expander, and / or other elements) are emitted into the atmosphere. The particles and molecules present in the atmosphere scatter the emitted laser radiation, or in some cases this radiation is also absorbed. The scattered at an angle of 180 ° portion of the radiation (reference numeral 104 ) is by means of a second arrangement of optical elements (telescope 105 , Mirror, beam splitter 106 and interference filters) and subsequently a signal acquisition unit 107 fed. There, the optical signals are converted into electrical signals and can subsequently be stored, for example, on a computer storage medium (eg, computer RAM, floppy disk, CD / DVD-ROM, hard disk, memory stick, or any other storage medium).

From the raw signals obtained in the manner described above, vertical profiles of optical particle properties are determined in a first step of the data analysis. These quantities can be determined at one or more measurement wavelengths. The Lidargeräte, which are used in aerosol research in the context of micro-physical particle characterization, are usually so-called multi-wavelength Raman lidar instruments, such as in 1 shown arrangement 100 , Such devices make it possible to obtain meaningful results for the microphysical particle properties from the measured optical properties.

With these devices can backscatter and extinction coefficients at several wavelengths become. The particle backscatter coefficient describes the scattering behavior of the particles at exactly 180 degrees. Of the Particle extinction coefficient describes the light extinction (= Sum of light scattering and light absorption capacity). In this context It should be noted that the following always speaks of "light". this includes is in the following electromagnetic radiation in the visible or invisible wavelength range Understood.

2 shows by way of example the results of a typical measurement of optical particle properties at several wavelengths. In the diagram 201 are the backscatter coefficients at the measurement wavelengths 355 nm (curve 201 ), 532 nm (curve 201b ) and 1064 nm (curve 201c ). The backscatter coefficient may, in a first simple approximation, be considered as a measure of the presence and concentration (number, surface or volume) of the particles. The backscatter coefficient 201 shows in this example special This clearly shows the variability of the particle concentration in the individual altitude levels. The particle backscatter coefficient at the individual measuring wavelengths shows a similar structuring of the height distribution of the particles. The differences in the absolute values at the individual wavelengths at the same height levels shows the so-called wavelength dependence of the optical particle properties.

In the diagram 202 are the particle extinction coefficients at the measurement wavelengths 355 nm (curve 202a ) and 532 nm (curve 202b ), which can be determined with the Raman Lidargeräten used, but also in principle with so-called high-spectral resolution lidar devices. Again, there is a wavelength dependence of the extinction coefficient.

There are differences in the wavelength dependence of backscatter and extinction spectra. In this context, one speaks of the backscattering or extinction-related current exponent. The Ångströmexponenten are in the diagram 204 shown. With α-A, the particle extinction-related Ångströmexponent for the measuring wavelengths 355 nm and 532 nm is designated (curve 204a ). With β-A, the particle backscattering Ångströmexponent for the measuring wavelengths 355 nm and 532 nm (curve 204b ) or 532 nm and 1064 nm (curve 204c ) designated. Positive values in the diagram 204 indicate here that the respective coefficient decreases with increasing wavelength, negative correspondingly that the coefficient increases with increasing wavelength. The differences in the wavelength dependence are important information in the inversion process described below, with which the (micro) physical properties can be derived from the optical properties.

From the wavelength dependence of the Ångströmexponenten also follows a wavelength dependence of the quotients of backscatter and extinction coefficients. These quotients of particle extinction and particle backscatter coefficients, which in the present example are calculated at the two wavelengths of 355 nm and 532 nm, are referred to as extinction to backscatter ratio or lidar ratio and are shown in the diagram 203 shown. The curve 203a shows the lidar ratio at 355 nm, and the curve 203b shows the lidar ratio at 532 nm. Also, this information is used to arrive at meaningful results by means of the inversion method.

at a conventional, also referred to as classical application, inversion method become individual height intervals for the Data inversion used. Within these height intervals, the optical Data averaged first before using in data inversion become. As a result, the statistical fluctuations of the optical Averaged data in successive layers. By the averaging should be unacceptably high, physically nonsensical variations in avoided the obtained mikrophysikalisches Teilcheneigenschaf th become. One speaks in this context of the stabilization of the solution space in the individual height layers. A more detailed explanation the term "stabilization" is used in a subsequent section of this application.

It Please note that height ranges are scanned with lidar devices. Depending on the device type, Measuring arrangement and scientific objective can be the vertical resolution between a few meters and many dozen meters. The applicability of the inversion process described in this application is not in principle limited to a specific choice of height resolution. The Averaging of optical data points over height intervals, however, leads to a reduced height resolution of sought-after microphysical parameters.

Out the optical data sets can By means of the inversion process physical particle properties (eg particle size distribution and additionally complex refractive index) are derived. These are the at the different measuring wavelengths directly measured quantities of Particle backscatter coefficient and particle extinction coefficients in the inversion algorithm entered. Various currently in the field of lidar research for the Data evaluation used inversion algorithms (corresponding to the mathematical definition of "inversion") for example, in [1], [2], [3], [4] and [5] described. In the [1], [2] and those described in [3], [4] Procedures are in their execution to be classified as equivalent in principle.

A Prerequisite to get meaningful inversion results is generally that the optical particle properties at at least three wavelengths in the spectral range from 355 nm to 1064 nm, see to z. B. [2], [3]. This wavelength range results for Part of the properties of currently used laser sources, but is without limitation from the theoretical point of view Spectral ranges below 355 nm and extendable above 1064 nm.

To a further part results in the above-mentioned measuring wavelength range from the fact that especially properties of particles in the so-called optically active spectral range to be determined. Below the optically active spectral range is the wavelength range of visible sunlight. In this area is the Climate impact of particles particularly large. Because of this wavelength range In particular, such particles are to be examined in detail, the Particle radius in a size range from about 50 nm to 500 nm.

As already mentioned, comes with multi-wavelength Raman lidar both the particle backscatter coefficient and the particle extinction coefficient. Shed optical Particle properties are used to stabilize the to achieve the inversion process described below.

simulations with synthetically generated data showed that with pure absorbance measurements only the particle size and under circumstances the real part of the complex refractive index can be determined. An access to the imaginary part However, the complex refractive index is not possible.

Further Simulations have shown that the processing of particle backscatter coefficients both information on particle size and the complex refractive index supplies. However, in this case, a strong sensitivity the algorithms used against measurement errors, which in the measure the processing of extinction coefficients does not occur.

In order to This results in a combination of both data forms on the one hand a sensitivity regarding the Particle size and the complex refractive index allows. On the other hand, a relatively high robustness against measurement errors reached.

Classic one-dimensional regularization

in the the following will be the solution finding from optical to microphysical particle properties of the above, referred to as classical application, inversion method (or Inversion algorithm). The inversion method used a so-called classical one-dimensional regularization.

3 shows a flowchart 300 , in which the basic procedure of the data inversion for calculating the mathematical solution space is shown. Shown is the case of finding a solution for an optical data set in a height layer. The optical data set, or the optical particle properties, have been determined experimentally, for. B. by means of a lidar device, for example, the in 1 shown Raman Lidars. The simple classical one-dimensional regularization step mentioned above and described in more detail below is in the field 301 shown. The fields 302a and 302b denote different steps of a data analysis. A more detailed explanation of the importance of this data analysis is given below.

As input to the data inversion process, the optical parameters (backscatter and extinction coefficients) measured with the multi-wavelength Raman lidar are used. In order to arrive at the desired microphysical parameters, the following two integral equations are solved numerically:

These Equations describe the optical properties of particles in height layers Z of the measured profiles. For simplicity, below is first on the designation Z is omitted. The integral equations (1) are also referred to as Fredholm integral equations of first order. The numerical solution equations (1) is called inversion.

The expression g _i (λ _k ) describes the optical data at the measurement wavelengths λ _k at a certain distance Z from the lidar. The index i denotes the particle backscatter coefficients (i = β) or the particle extinction coefficients (i = α). The measurement error is with ε exp i (λ k ) designated. The expression υ (r) describes the volume concentration of the particles per radius interval dr. The terms r _min and r _max denote the lower and upper integration limits, respectively, of the particle radii r considered in the inversion.

The term K _i (r, m, λ _k , s) describes the kernel functions of backscatter and extink, respectively tion. The core functions are intuitively the optical cross section per particle volume. The core functions depend on the radius r of the particles, the complex refractive index m = m _Re + _Im of the particles, the wavelength λ _k of the interacting light, as well as the geometric shape properties of s of the particles. In the case where the particles have a spherical shape, the core functions K _i (r, m, λ _k , s) can be weighted from the corresponding backbone and extinction efficiencies Q _i (r, m, λ _k ) for individual particles the geometric cross section πr ² , calculate. This is z. As described in [6]. In this case: K i (r, m, λ k , s) = (3 / 4r) Q i (r, m, λ k ). (2)

The index p will describe below the type and number of optical data. Thus Eq. (1) be rewritten in the form

Eq. (3) is solved by means of a numerical method for the sought particle size distribution υ (r). This is in itself z. As described in [7], [8], [9], [10]. Characteristic of inversion methods with such types of integral equations is that they are in principle unstable (see, for example, [11]). This means that even the smallest errors (eg a few orders of magnitude smaller than 1%) in the measured optical properties usually lead to unusable results of the physical properties. In addition, with such unstable methods, many possible solutions to an optical data set subject to measurement errors can be found. Furthermore, rounding errors (eg of the order of 10 ^-6 ) can lead to a lack of a reasonable solution. In this context, one also speaks of ill-posed inverse problems.

Around these instabilities to correct, are usually mathematical and / or physical boundary conditions introduced, what is summarized under the generic term regularization. This regularization stabilizes the mathematical process d. h., There is a much improved assignment of optical Characteristics of the underlying microphysical particle properties.

For the solution of the invention, the backscatter or extinction efficiencies Q _i are (r, m, λ _k) for a so-called lattice of the complex refractive index m = m _Re + _Im and for particle sizes r calculated in the range of, for example, 1 nm to 10 microns. For the refractive indices m, it was determined from simulation studies that a range of 1.2 to 2 in the real part m _Re and a range of 0 to 0.5 in the imaginary part m _{Im is} generally sufficient for applications to particles in the atmosphere. However, this range of numerical values for m _Re or m _Im is not binding and is not binding for the functionality of the inversion method.

In a further step of the inversion method, so-called basis functions B _j (r) are calculated. These represent support points within the numerical solution. Such a method is also referred to as a projection method.

The sought particle size distribution υ (r) is discretized by means of a linear combination of the basis functions B _j (r), ie

The w _j designate the development coefficients of the linear combination or the weighting factors with which the individual basis functions B _j (r) in the linear combination are weighted. N _B denotes the number of basic functions used.

The right-hand side of equation (4) contains the mathematical residual error ε ^math (r) which is caused by the approximation of the desired size distribution υ (r) by means of the basis functions B _j (r).

For example, base functions in the form of triangles on a semilogarithmic scale can be used as basis functions B _j (r). The use of triangle functions, which are also referred to as B-spline functions of the first degree, is for. As described in [1]. Alternative to triangle functions is any other form of base function usable (eg, histogram columns (also referred to as zero-degree B-spline functions), cubic functions (ie, second-order B-spline functions), monomodal logarithmic normal distributions, polynomials (n, m) -th degree (where n and m are natural numbers)).

The exact position of the desired particle size distribution along the in Eq. (3) size range (ie, the integration interval [r _min , r _max ]) is unknown in most cases. For this reason, a so-called inversion window can be used, which serves as a filter function. This window has a variable width and a variable position along the size range examined. Details on this are described, for example, in [1]. Within this inversion window, the basis functions are arranged side by side such that their nodes have the same distance from each other on a logarithmic radius scale. It should be noted that if the inversion window does not cover the position of the sought particle size distribution, then no reasonable solutions are usually obtained.

The weighting factors w _j follow from the insertion of Eq. (4) in Eq. (3) and the transformation of these equations into a vector-matrix equation of the form: g = Aw + ε. (5)

In Eq. (5) the optical data are shown as vector g = [g _p ]. The weighting factors are shown as vector w = [w _j ]. The errors are represented as vector ε = [ε _p ]. The expression ε p = ε exp p + ∫K p (r, m) ε math p dr consists of the sum of experimental and mathematical errors. The matrix A = [A _pj ] is called a weighting matrix, see z. For example, [12]. Your elements are written as

The simple solution of Eq. (5) for the weighting factors w leads to the solution w = A -1 g + ε ', (7) where A ^-1 denotes the matrix inverse matrix and ε '= -A ^-1 ε. The solution according to Eq. (7) provides non-physical or unstable solutions. The reason for this is to be seen in the above-mentioned "bad position" of the mathematical problem (see, for example, [12]).

in the The framework of the inversion process therefore becomes a so-called stabilization term introduced. This calculation step is also called "regularization". Illustrative are mathematical or physical constraints introduced for stabilization.

Without the aforementioned regularization can indeed size distributions It can be determined which, if one from these the optical data back calculated very good with the given optical data. However, have these solutions usually no similarity with the sought-after physical size distributions. The reason for this paradoxical behavior arises from the property that bad asked inverse problems a non-identity of the solution space produce.

In the context of regularization, a first step is required to find those solutions for which ε in Eq. (5) is minimum, or more specifically, for which the distance between the vector Aw and the vector g of the optical data is smaller than a predetermined value. This step leads to the so-called minimization concept (also referred to as the minimum distance method). The minimization concept uses a so-called penalty function e ² , which denotes the maximum acceptable distance between Aw and g and is defined by the simple Euclidean norm || · || (see eg [1], [9], [12]) according to: e 2 ≥ || ε || 2 = || Aw - g || 2 , (8th)

The expression δ * = || Aw - g ||, (9) ie the distance between Aw and g is usually called a (classical) discrepancy. The discrepancy represents clearly a measure of how far a solution calculated by inversion deviates from the true solution. With the help of the minimization concept those solutions are selected for which the norm of ε, ie the discrepancy, falls below a certain minimum value> 0. From first principles it is impossible to reproduce the given optical data exactly from the solutions found to the microphysical properties of the particles.

By Introduce further (useful) mathematical or physical boundary conditions The minimization request may continue in a second step reinforced become. In the following, the so-called smoothness condition becomes a boundary condition considered. Alternatively you can Other constraints, such. B. positivity of the solution to be considered.

The smoothness condition is established by introducing an additional penalty term Γ (υ) in Eq. (8), so the new minimization problem is: e 2 ≥ | Aw - g || 2 + γΓ (υ). (10)

) sein. γ ist ein nicht-negativer Regularisierungs-Parameter bzw. Lagrange-Multiplikator.The expression Γ (υ) is a non-negative scalar, which measures the distance of the size distribution υ (r) obtained from the inversion of an a priori selected smoothness. The smoothness can be of the order zero, one, two or π (n ε

) be. γ is a non-negative regularization parameter or Lagrange multiplier.

Denoting by w ^T is the transpose of the vector w, so can the mathematical definition of smoothness condition Γ (υ) written as (see eg. B. [12]) Γ (υ) = w T Hw. (11)

für den Fall, dass fünf Basisfunktionen B_j(r) (j = 1, 2, ..., 5) und ein optischer Datensatz g = [g_p] bestehend aus fünf Datenpunkten (p = 1, 2, ..., 5) verwendet werden. Weitere Beispiele für Glattheitsbedingungen und zugehörige Matrizen H sind z. B. in [12] beschrieben.In other words, the smoothness is defined mathematically by a quadratic combination of the weighting factors w _j and thus by the in Eq. (11) described square shape. The matrix H is usually a simple fast diagonal matrix. For example, in the case of a second-order smoothness condition, the mathematical form of H is

in the case where five basis functions B _j (r) (j = 1, 2, ..., 5) and one optical data set g = [g _p ] consisting of five data points (p = 1, 2, ..., 5) can be used. Further examples of smoothness conditions and associated matrices H are z. As described in [12].

In a further step of regularization, inequality (10) is written as an equation and Γ (υ) is given by Eq. (11) expressed: e 2 = || Aw - g || 2 + γw T Hw. (13)

Now the solution is sought, for the entire penalty function, ie the right side in Eq. (13), takes a minimum. The solution for the weighting factors w is obtained (see for example [12]): w = (A T A + γH) -1 A T g, (14) where A ^T denotes the matrix transposed to A.

The inverse problem is stabilized by the matrix (A ^T A + γ H) ^-1 A ^T. The influence of the matrix H is determined by the Lagrange multiplier -. For γ = 0, no smoothing is performed. For γ → ∞ we get a perfectly smooth solution υ (r), which is independent of g. The optimal value of γ lies between these two extreme cases. In [1] an overview of various techniques for finding an optimal value of γ is described. One technique that is used in [2], for example, is the so-called generalized cross-validation or generalized cross-validation on).

Another technique, used for example in [3], is based on a modified minimum-discrepancy principle. Here is the so-called modified discrepancy δ = || A | w | - g || (15) considered. The expression | w | in Eq. (15) is understood to mean that each element of the weighting vector w has been converted into its absolute value. The idea of the modified discrepancy principle is to minimize the expression δ as a function of the Lagrange multiplier or regularization parameter γ. The optimal value of the Lagrangian multiplier (γ _opt ) is then the value for which the equation according to Eq. (14) solution of the weight vector w has the expression δ in Eq. (15) minimized.

In a final calculation step, the optimum value of the desired size distribution for an unknown value of the refractive index m is determined. For this purpose, the equation system (1) is investigated for various complex refractive indices, which can be wavelength-dependent or wavelength-independent as well as particle size-dependent or particle size-independent. This property is determined by the kernel functions K _p (r, m) in Eq. (3) introduced. By analyzing the individual solutions found as a function of the respective optimal value of the Lagrange multiplier γ, one then determines the solutions for the complex refractive index m.

With The inversion method just described becomes the so-called mathematical solution space determined. This solution space is a superset of the sought-after physical solution space, and in the inversion process must have a complex data post-processing the physically meaningful solution space be determined.

The reason for this ambiguity or instability of the solution space is the typical property of a badly set inverse problem. Here, the measurement errors also play a role, which are typically greater than 5% in Lidarmessungen and can easily reach more than 50%. These measurement errors depend on z. B. from the atmospheric conditions, the duration of measurement and the spatial (vertical) resolution of the measurements. By means of appropriate averaging times, the errors can be pushed to such an extent that the inversion methods can be successfully applied if regularization is used. From simulation studies it is known that measurement errors should typically be less than 20%. Without regularization, inaccuracies of less than 10 ^{-6 are} sufficient to get ineffective inversion results.

With the so-called regularization is sought in the inversion process, a solution room to determine which meets the following two conditions: First should the solution space to the physical quantities the wanted sizes so close as possible come. On the other hand, those calculated from this solution space are to be counted back optical data reproduce the input data as well as possible. As mentioned above it for fundamental reasons not possible, to find the exact microphysical solutions, d. h., the Solutions, from which reproduce exactly the same time the optical input data to let. This peculiarity of ill-posed inverse problems therefore requires a complex Post-processing of the mathematical solution space. This was a Scheme designed by means of which the mathematical solution space limited to a physically meaningful solution space, which However, associated with a high expenditure of time and personnel is.

The procedure for finding the physical solution space will be described below. Here, one starts from the mathematical solution space, as he, for example, based on the in 3 illustrated solution schemes 300 was determined. In this context, it should be noted that in the method described above, only optical data sets in a vertical layer of the vertical profiles are always evaluated as an input data record.

Of the special approach in the method described above, which is also called "classical one-dimensional "regularization is the default, that one from the optical profiles Height layers selects within which the optical data are averaged to smaller variations in the optical data. The concept of "smaller Fluctuations "is difficult to quantify. Corresponding studies are missing so far, and from the literature, no lasting conclusions can be drawn. You have to be present assume that fluctuations in the means of the optical Parameter Ångström exponent and lidar ratio within 20% are considered quasi-constant.

With another inversion method disclosed in [14] can be an averaging of the optical data in superimposed altitude layers be avoided. According to this Procedures are whole sections of optical vertical profiles (from a minimum height up to a freely selectable maximum height) used as input data. This will be a modified approach introduced in the matrix calculation, which also called "two-dimensional Regularization " becomes. Matrix calculations are a core part of this inversion method.

Of the The invention is based on the problem of providing a method with which the determination of physical or microphysical Characteristics of atmospheric Particles in a simpler way is compared with usual Method.

The Problem is solved by a method, apparatus, computer readable storage medium and a computer program element for determining physical Properties of atmospheric particles with the features according to the independent claims.

exemplary Embodiments of the invention will become apparent from the dependent claims. The further embodiments of the invention, which in connection with are described in the procedure apply mutatis mutandis to the device, the computer-readable Storage medium and the computer program element.

at a method of determining physical properties of atmospheric Particles will have at least one vertical profile of at least one optical Property of the particles provided. Furthermore, at least a vertical profile of at least one physical property of Particles by means of an inversion of the at least one Vertical profile of the at least one optical property of the particles determined wherein in the context of the inversion process one for a plurality of height layers the vertical profiles common discrepancy parameter profile determined and that at least one vertical profile of the at least one physical property using the discrepancy parameter profile is determined.

It There is further provided an apparatus for determining physical Characteristics of atmospheric Particle. The device points to a provision device for providing at least one vertical profile of at least one optical property of the particles. The device also has to a determination device for determining at least one vertical profile at least one physical property of the particles from the at least one vertical profile of the at least one optical property the particle by means of an inversion method, wherein the determining means is set up so that in the context of the inversion process for one Plurality of height layers the vertical profiles common discrepancy parameter profile determined and that at least one vertical profile of the at least one physical property of the particles using the discrepancy parameter profile is determined.

It In addition, a computer-readable storage medium is provided. In the computer-readable storage medium is a program for determining stored by physical properties of atmospheric particles, when executed by a processor, the following method steps comprising: providing at least one vertical profile at least an optical property of the particles; Determine at least one Vertical profiles of at least one physical property of Particles by means of an inversion of the at least one Vertical profile of the at least one optical property of the particles, wherein in the context of the inversion process one for a plurality of height layers the vertical profiles common discrepancy parameter profile determined and the at least one vertical profile of the at least one physical Property of the particles using the discrepancy parameter profile is determined.

Further becomes a computer program element for determining physical Characteristics of atmospheric Provided particles which, when executed by a processor, comprising the following method steps: providing at least one Vertical profiles of at least one optical property of the particles; Determining at least one vertical profile of at least one physical Property of the particles by means of an inversion process the at least one vertical profile of the at least one optical property the particle, wherein in the context of the inversion process one for a plurality of altitude layers the vertical profiles common discrepancy parameter profile determined and that at least one vertical profile of the at least one physical property of the particles using the discrepancy parameter profile is determined.

For the purposes of this application, a vertical profile is to be understood as meaning that a specific property, quantity or parameter for different altitude values is detected or determined and thus dependent on it can be represented by the height. In other words, the property (or size or parameter) is detected or measured vertically resolved. Stated another way, the vertical profile describes the dependence of a certain property, quantity or parameter on altitude. For example, the vertical profile of an optical property of atmospheric particles describes the dependence of this optical property on altitude (in the atmosphere).

Under the term "discrepancy parameter" is in of this application the discrepancy, as described in Eq. (15) is to understand. More general can be under a discrepancy parameter understood an expression that the discrepancy, as z. B. in Eq. (9) is defined. The two terms "discrepancy" and "discrepancy parameters" become used in this context on an equal footing.

Under the term "discrepancy parameter profile" or "common Discrepancy parameter profile " in this application understood that for the discrepancy parameter a optimal value is calculated or calculated, which in all height layers equally valid is. In other words, it becomes a common value of the discrepancy parameter for all elevation levels determined.

According to one embodiment of the invention, the at least one vertical profile of the at least one optical property of the particles is provided by detecting the at least one optical property by means of at least one optical measurement. According to another embodiment, the at least one optical measurement is a lidar measurement. The lidar measurement may be performed using a lidar device, for example, a multi-wavelength lidar device, such as a multi-wavelength Raman lidar. In other words, the optical measurement can be carried out using laser radiation or laser light pulses. For example, the lidar device may be configured as described above 1 has been described.

According to one Embodiment, the at least one optical property at one or more wavelengths be detected or measured, for example, in one or more wavelength in the range of about 355 nm to about 1064 nm. For example, the at least one optical property for at least three different wavelengths be measured, for. At 355 nm, 532 nm and 1064 nm. Alternatively can additional and / or other measuring wavelengths be used in the optical measurement.

According to one another embodiment, the at least one optical property the particle in each one of the plurality of height layers detected.

According to one another embodiment will / will by means of Lidar measurement of Backscatter coefficient and / or the extinction coefficient of the particles at least one optical wavelength as the detected at least one optical property. With others Words become the particle backscatter coefficient and / or the particle extinction coefficient for particles in one or more elevation levels recorded (or measured). The measured coefficients can be considered Input variables in the Inversion method can be used.

According to one In another embodiment, the inversion method has at least one Regula rationsschritt, and determining the discrepancy parameter profile takes place within the scope of the at least one regularization step. In other words, regularization occurs as part of the data inversion. Under regularization is understood as described above, that suitable boundary conditions are introduced in the inversion process, to stabilize the inversion process.

According to one other embodiment than the at least one physical Property of particles the mean size (eg effective radius or average volume radius) of particle size distributions and / or the complex refractive index of the particles and / or the shape of the particles determined or determined. In other words, the vertical profile of the average particle size and / or the vertical profile of the complex refractive index of the particles and / or the vertical profile of the particle shape can be determined. alternative or additionally can Vertical profiles of the number concentration and / or the surface concentration and / or the volume concentration are determined.

According to one embodiment of the invention, vertical profiles of physical properties of atmospheric particles (eg particle size distribution, mean particle size, complex refractive index and integral parameters derived from the distribution such as number, surface and volume concentration) can be determined. The determination of size distributions may, for example, be in the range of about 50 nm to about 5 μm particle radius, e.g. B. in the event that measurement wavelengths in Be range from 355 nm to 1064 nm can be used to detect the optical particle properties.

According to some Embodiments of the invention will be the derivation of particle size distributions and / or complex refractive index of the particles of optical properties allows the particle.

According to one Another embodiment of the invention is the provision device arranged so that the at least one optical property of Particles detected by at least one optical measurement and can be provided.

According to one Another embodiment is the device for determining the physical Properties set up as Lidar measuring device, z. B. as a Raman lidar measuring device, for example, as a multi-wavelength Raman lidar.

According to one Embodiment of the invention is an inversion method as an inversion method used with reformulated one-dimensional regularization.

According to one In another embodiment, the inversion method is an inversion method used with two-dimensional regularization.

in the The following are the reformulated one-dimensional regularization (also referred to as modified one-dimensional regularization and subsequently the two-dimensional regularization is described.

Reformulated one-dimensional inverse Problem: "height direction"

In a first step of regularization will be the removal again Z between the location of the optical scattering process and the receiving unit of the lidar device introduced.

The optical data for l = 1, ..., N _L different height layers are provided. The optical data may, for example, by means of one or more optical measurements, for. B. Lidar measurements, as described above, have been recorded or measured. Eq. (3) is rewritten to solve the inverse problem not only for one elevation layer but for multiple elevation layers at the same time:

This set of linear equations is re-formulated into a vector-matrix relationship, ie, one writes G (L) = A (L) w (L) + ε (L) , l = 1, ..., N L (17) or (by combining the N _L equation systems from Eq. (17)) g ~ = ω ~ + ε ~. (18)

The matrix à and the vectors w ~, g ~ and ε ~ in Eq. (18) have the following structure:

The dimension of the matrix and vectors is determined by the expressions N _O , N _B and N _L. The term N _O denotes the number of optical data points, ie the backscatter and extinction coefficients measured at the different wavelengths. The term N _B denotes the number of basis functions B _j (r), and the term N _L denotes the number of height layers. The set of linear equations in Eq. (18) is solved simultaneously, and analogous to the steps of Eq. (5) led to the regularized equation (14), from Eq. (18) the regularized solution: w ~ = (Ã T Ã + γH ~) -1 Ã T g ~. (23)

und der Lagrange-Multiplikator γ ist eine N_BN_L×N_BN_L-Matrix mit diagonaler Form, d. h.The matrix H ~ in Eq. (23) has the following form:

and the Lagrange multiplier γ is an N _B N _L × N _B N _L matrix of diagonal shape, ie

The solutions for the weighting vector w ^(l) in the N _L height layers of Eq. (18) are independent of each other. Solving this system of equations is similar to the case of the classic one-dimensional regularization described above, but with one particular difference: in this reformulated form of inversion, the choice of the discrepancy parameter for each individual layer of height must be optimized simultaneously. In this case, the principle of the discrepancy parameter according to Eq. (15) can be used. This step is important, also in the context of the two-dimensional regularization described below, since this optimization approach for selecting the discrepancy parameter ultimately uses the correlation of optical or physical particle properties in successive height layers.

According to one Embodiment of the invention, a method is provided with An optimum profile of the discrepancy parameter can be determined can. Determining this optimal discrepancy parameter profile according to one embodiment will be described below. In this context, it should be noted that from the choice of the discrepancy parameter ultimately the profile of the Lagrange multiplier follows. In other words follows the optimum profile of the discrepancy parameter is also the optimal one Profile of the Lagrange Multiplier. In other words the profile of the Lagrange multiplier is implicit the profile of the discrepancy parameter. Therefore, below the description of the determination of the discrepancy parameter profile renounced an explicit separation to the term Lagrange multiplier.

Two-dimensional regularization

in the Frame of inversion with two-dimensional regularization becomes a smoothness condition along the z-direction by means of another Matrix K introduced. In this way successive height layers are linked together. This represents a physically meaningful boundary condition, since the particle properties within a particle layer only gradually with the height inside change this particle layer. Strong changes the particle properties can However, at the border between two layers, which are different Particle types exist, occur.

The particle properties along the z-direction are described by the following functions ben:

The integration of Eq. (16) over z ε [z _min , z _max ] leads to the two-dimensional Fredholm integral equation of the first kind:

Substitution of the investigated particle size distribution υ (r) by Eq. (4) results in

The following condition applies here:

so dass man folgenden Ausdruck erhält:

The sum in Eq. (27) can be split by using the parameter φ _j with

so that one receives the following expression:

Eq. (29) describes a set of Fredholm integral equations of first type, but now along the z-direction in analogy to the sentence the Gln. (16).

Hierbei wurde zunächst das Integral in Gl. (29) als Summe geschrieben mit insgesamt N_L Summanden der Form A_pj(m(z_l))·w_j(z_l)·Δz, wobei l = 1, ..., N_L und

und anschließend wurde die Gleichung durch Δz geteilt und das Gleichungssystem in Matrix-Form geschrieben. Dabei beschreibt Δz die geometrische Dicke jeder einzelnen Höhenschicht. In diesem beschriebenen Verfahren wird der Einfachheit halber angenommen, dass die geometrische Dicke jeder Höhenschicht gleich ist, d. h. man schreibt Δz_j = Δz. Prinzipiell ist das Verfahren aber nicht von der Wahl einer einheitlichen Schichtdicke abhängig.By reformulating Eq. (29) you get:

Here, the integral in Eq. (29) is written as a sum with a total of N _L summands of the form A _pj (m (z _l )) · w _j (z _l ) · Δz, where l = 1, ..., N _L and

and then the equation was divided by Δz and the equation system written in matrix form. Δz describes the geometric thickness of each layer of elevation. In this described method, for the sake of simplicity, it is assumed that the geometric thickness of each height layer is the same, ie, Δz _j = Δz. In principle, however, the method does not depend on the choice of a uniform layer thickness.

The vectors g - and ε - in Eq. (30) are given by

und p = 1, ..., l = 1, ..., N_L.The A _j are N _O × N _L matrices with the entries A _pj (m (z _l )), and the w _j are N _L × 1 vectors with the entries w _j (z _l ), where

and p = 1, ..., l = 1, ..., N _L.

wobei

Das erweiterte System von Lineargleichungen (Gl. (33)) lässt sich wieder mit Inversion durch Regularisierung lösen. Man erhält: w - = (A -TA - + κH -)–1A -TG -. (38) By summarizing the N _B equations from Eq. (30) to a matrix-vector equation we finally obtain: G - = A -w - + E -, (33) with the vectors

in which

The extended system of linear equations (Eq. (33)) can be solved again with inversion by regularization. You get: w - = (A - T A - + κH -) -1 A - T G -. (38)

The matrix H - and the vector w - in Eq. (38) have the following structure:

The vectors w _j describe the j-th weighting coefficient in the approximation, which is given by Eq. (4) is given in the direction z.

The vector w - differs from the vector w ~ from Eq. (23) only by the order of the corresponding elements (or lines). The rearrangement of the same rows of the matrix H, which is given by Eq. (39) given such that vector w - → w ~, allows the construction of the matrix K, which for Eq. (23) is appropriate.

The Rearranging the elements will be described below. The expression x - denote the given matrices and vectors, i. e., A -, H -, and w -, and the expression x denote the rearranged form of these matrices / vectors. Further denote τ - the lines of x - and τ denote the lines of x.

The rearrangement of the elements of A and W is performed using the following formulas: τ = (1-1) N B + j, j = 1, 2, ..., N B , l = 1, 2, ..., N L , (41) τ - = (j - 1) N L + 1, j = 1, 2, ..., N B , l = 1, 2, ..., N L , (42)

For this purpose, the Eelier equations (see, for example, [15]) for the Tikhonov functions described by equations (18) and (33) are considered: Ã T Ãw ~ + γH ~ w ~ - Ã T g ~ = 0, (43) A T Aw ~ + κKw ~ - A T G ~ = 0. (44)

The matrix A and the vector w - result from the rearrangement of A - and w - according to Eq. (41) and Eq. (42). The two Eelier equations can be summed to the following equation: (Ã T Ã + A T A) w ~ + (γH ~ + κK) w ~ = Ã T g ~ + A T G ~. (45)

Forming gives the expression w ~ = (Ã T Ã + A T A + γH ~ + κK) -1 (Ã T g ~ + A T G ~). (46)

folgt die Lösung der zweidimensionalen Regularisierung aus Gl. (46) in der folgenden Weise: w ~ = (ÃTÃ + γH ~ + κK)–1ÃTg ~ (47) If there is no additional information, for example about the factors φ - ₁ , ...,

follows the solution of the two-dimensional regularization from Eq. (46) in the following way: w ~ = (Ã T Ã + γH ~ + κK) -1 Ã T g ~ (47)

Of the Parameter γ (Lagrange multiplier) describes the strength the smoothing along the grid points of the sought particle size distribution (see [1], [3]). The value of the Lagrange multiplier can be optimized, the functionality of the inversion process described above in principle not on an explicit choice of method for finding the optimal one Value for the Lagrange multiplier is limited.

Of the Parameter κ describes the strenght the smoothing along the height layers. It should be noted that the value of γ may change if the additional Regularization parameter κ used becomes.

The definition of the regularization parameter κ can be done, for example, by means of the following criterion:

For this purpose, first the sequence κ ₁ , κ ₂ , ..., κ _max and the sequence γ ₁ , γ ₂ ,..., Γ _{max are} defined. These sequences are, for example, geometric sequences (ie κ _{i + 1} / κ _i = q) with a geometric ratio q> 1. Large values for κ _max and γ _max lead to an excessive smoothing of the investigated particle size distribution. The terms κ ₁ and γ ₁ have a value close to zero.

Eq. (47) is solved by means of the modified discrepancy principle for each individual value κ ₁ , κ ₂ , ..., κ _max . Thus, a new sequence S (κ ₁ ), S (κ ₂ ), ..., S (κ _max ) is calculated, the individual sequence terms S (κ _i ) being defined as:

The optimum value κ _{opt of} the regularization parameter is assumed to be the value which corresponds to the minimum of the sequence S (κ ₁ ), S (κ ₂ ), ..., S (κ _max ). In other words, S (κ _opt ) ≦ S (κ _i ).

For example, if three elevation layers (N _L = 3) are chosen, and five basis functions (N _B = 5) to reconstruct the particle size distribution being examined, and if the identity matrix is assumed (K _j = E, j = 1, ... N _B ), we obtain H - for the smoothing matrix:

), und jeder dieser Faktoren hat seine Wichtungsmatrix K₁, K₂, ...,

in der z-Richtung (Höhenachse). Die Matrix K_j = E hat die Dimension N_L × N_L, da die Anzahl der Höhenschichten entlang der z-Achse gleich N_L ist. Die Anzahl der Matrizen K_j = E ist gleich N_B, da die Anzahl der Wichtungsfaktoren (oder Gitterpunkte (grid bins)) gleich N_B ist.It should be noted that the matrix K _j uses the weighting factors w _j. Each weighting factor w _j must be determined for all height layers 1, 2, ..., N _L (for this reason, the weighting factors have the second index 1, ie w (l) / j). There are N _B weighting factors (w ₁ , w ₂ ,.

), and each of these factors has its weighting matrix K ₁ , K ₂ ,.

in the z-direction (altitude axis). The matrix K _j = E has the dimension N _L × N _L , since the number of height layers along the z-axis is equal to N _L. The number of matrices K _j = E is equal to N _B because the number of weighting factors (or gridbins) is equal to N _B.

The rearrangement of the elements of the matrix H - from Eq. (50) in the equation given in Eq. (41) and Eq. (42), the matrix K ₀ for zero-order smoothing gives:

In the case of second-order smoothing, the matrix H is written as:

The elements of this matrix can in turn be rearranged, and the final shape of the smoothing matrix K ₂ for the second order smoothing is obtained:

According to one embodiment of the invention, the basic procedure of data inversion with reformulated one-dimensional regularization or with two-dimensional regularization can be carried out in a similar manner as in connection with FIG 3 described in which 3 shown regularization step 301 represents the reformulated one-dimensional regularization or the two-dimensional regularization. Furthermore, the time spent in the in 3 shown data post-analysis steps 302a . 302b in principle significantly reduced, or can be completely eliminated (depending on the given, to be analyzed data set).

in the Comparison with the classical approach (that is, the classical one-dimensional Regularization) is dropped in the modified approach (i.e., two-dimensional regularization) a disadvantage of the classical regularization method, namely the Need to decide what a quasi-constant record is. By doing so leaves avoid a subjective factor in the data analysis, namely the Decision of the data-evaluating person. In addition, affect in in the modified method, fluctuations in the optical data in successive height layers lower on the stability of the inversion process. In the last consequence is the procedure with two - dimensional regularization to no particular measurement resolution of lidar limited.

The subjectivity in the evaluation methodology in the classical method with one-dimensional Regularization lies in the wealth of experience or lack Experience justified, which a person must contribute to the data analysis. So must the data-evaluating person has extensive knowledge in the field of Particle properties and knowledge of the impact of meteorological Conditions (eg ambient humidity) on particle properties have. In addition, a deeper insight into the device technology to be available. This allows mathematical evaluation in the data evaluation solutions of physical solutions separate. However, such a level of knowledge is often not given.

One Advantage of the method with two-dimensional regularization is the increased Stabilization of the solution process. In the process of classical one-dimensional regularization the stabilization takes place by extensive reworking of the mathematical solution space in the context of sensitivity studies and consistency tests.

As examples of consistency tests and postprocessing methods in classical one-dimensional regularization, the following points are mentioned. To limit the solution space, the classical one-dimensional regularization is used to examine parameters which are generated by the inversion method during the data inversion. These parameters are the discrepancy value following the inversion (see eg [12]). Furthermore, the solutions of size distributions at their lower and upper particle radius limits are investigated. Furthermore, the spectral profile of the Ångströmexponenten and Lidarverhältnisse are detected. Furthermore, it should be ensured that the solutions obtained from the solutions found (particle size ßenverteilung and complex refractive index) recalculated optical data with the optical input data within the given measurement errors match.

in the As part of the data analysis, a strict consistency test is performed. It is (usefully) assumed that microphysical Particle properties can not change significantly, provided the optical Data (Ångström exponent and lidar ratio) in superimposed elevation levels also change only insignificantly. It is therefore from assuming that the underlying mathematical process a linearization of the solution space allows. The fact is, however, that badly posed inverse problems highly nonlinear behavior. That means it usually applies in the found mathematical solution space, that a variation of the optical parameters does not have a linear relationship with the change the microphysical parameter is linked.

Consequently the following scenario emerges: If the consistency test is negative fails, a new set of search parameters is chosen to the solution space limit. Subsequently will be rechecked whether with the same parameters in all height layers a consistency achieved in the obtained microphysical properties. This procedure is performed until consistency in all examined height layers is reached. This type of data post is usually very time consuming.

In contrast, will in the method with two-dimensional regularization an optimal average search parameter space, with which the mean solution can be found. An advantage of this is that the user has one substantially limited possibility has data postprocessing. That is, the number of degrees of freedom in terms of the one to select Parameter is due to the underlying mathematical formalism severely limited, whereby the initially mentioned subjectivity is significantly reduced in the data evaluation.

One Another advantage of the method with two-dimensional regularization consists in the faster data evaluation time. This is the Possibility, an operational version of the algorithm with two-dimensional To use regularization. Such an operational approach is not in the case of classical one-dimensional regularization or only with difficulty. The robustness of the method with two-dimensional regularization across from Measurement errors in the optical data is also increased, causing another advantage of the process is given.

The inversion method described above with two-dimensional regularization is not on measurements with lidar devices limited, but lets generally apply to the evaluation of data using measurement methods be determined, which have either spatial or temporal resolution.

If, as in the case of lidar measurements, both spatial and temporal resolution given or possible is, the process can easily be on a multidimensional Regularization be converted. In other words, according to a Embodiment of the invention in the case of a spatially and temporally resolved measurement an inversion method with a multi-dimensional regularization be used.

The Applicability of the inversion method described in this application is not limited to a specific choice of height resolution. The However, averaging of optical data points over height intervals can be with a reduced height resolution of desired microphysical parameters.

in the The following describes a method for determining a discrepancy parameter profile according to one embodiment the invention. By means of the method can be an optimal profile be determined by the discrepancy parameter, which in the context of a Inversion with regularization, for example in the frame of the above described inversion method with reformulated one-dimensional Regularization or with two-dimensional regularization, in addition can be used, an improved solution set for the physical to be determined To determine particle properties.

Under the term "discrepancy parameter profile" is according to this embodiment the calculation of the expression in Eq. (15). It will be a optimal value for calculates the discrepancy parameter, which in all elevation layers equally valid is. In this context, it should be noted that the choice of a common value for the discrepancy parameter has different values for the Lagrange multiplier in the individual height layers can (but does not have to, since this is to be examined by the Data record depends).

The For example, choosing the discrepancy parameter profile allows one Time savings in data processing, as not for every single layer, as in the classical method, a separate discrepancy parameter determined must become. It can for example every height layer To be avoided repetitive steps. The choice of a Profiles involves the application of the correlation of the underlying microphysical parameters in successive height layers, for example, to stabilize the solution space along the vertical. Dimension contributes. Furthermore, it can also stabilize the solution space within the individual elevation levels be achieved. The profile of the discrepancy parameter can also be used be further evaluated, for example, to statements about the Goodness of found solutions to get. From the application of the discrepancy principle in classical one-dimensional regularization is known to be the values of the Discrepancy parameters, for the solution space is accepted, a qualitative measure of the accuracy of the found solutions represents.

in the Frame of the method for determining the discrepancy parameter profile If the above-described modified discrepancy δ according to Eq. (15) considered.

The expression δ _u denotes the ordered sequence of the discrepancy parameters, where u = 1, 2,..., U _max . In this context, "ordered" is to be understood as meaning that the discrepancy parameters δ _u are sorted by size. Furthermore, the expression t _u denotes the ordered sequence of the acceptable solutions, ie those solutions whose discrepancy value δ (according to equation (15)) satisfies the condition δ ≦ δ _u . By "ordered" in this context is meant that the solutions are sorted by the value of their associated discrepancy.

The expression δ ~ (L) (t u ) denote the discrepancy of the average size distributions of the t _u solutions in the height layer l, where l = 1, ..., N _L. The discrepancy δ ~ ^(l) (t _u ) is calculated according to: δ ~ (L) (t u ) = || g (L) - g (L) ave ||. (54)

A detailed explanation of this type of discrepancy is given in [4], where the expression δ ~ (L) (t u ) is called ρ ^ave . The term g (l) / ave denotes the optical data calculated from the averaged particle size distribution as per se described in [3], for example. This averaged solution is calculated from all the solutions of the particle microphysical parameters found for the corresponding height layer 1 in the so-called averaging interval δ _u .

The significance of the averaging interval is clear that all solutions found up to the optimal value of the discrepancy value belong to this interval of solutions. These solutions are then averaged subsequently. In other words, an averaging interval belongs to each discrepancy parameter δ _u , and all solutions whose individual discrepancy values are smaller than the value δ _u are averaged. A further explanation of the meaning of the averaging interval is given below in connection with the in 7 shown embodiment.

The calculation of the optical properties is carried out using a Mie-scattering algorithm, as per se, for example, described in [6]. The averaging interval contains all solutions (according to equation (23)) as a function of the Lagrange multiplier γ, which is varied within a predetermined number range from 0 to γ _max . This averaging interval follows from the choice of the optimal value of the discrepancy parameter, ie, δ _u = δ _opt .

In the case of inversion with classical one-dimensional regularization, δ _opt must be estimated for each individual height layer l. This determination of δ _opt is very time consuming if the number N _{L of} the height layers is large. A further disadvantage is that this optimum value δ _{opt of} the averaging interval has to be estimated individually and independently of the δ _opt in all other height layers.

According to the method described here, the determination of the optimum value of δ _{u is} as follows.

The term t _u denotes the ordered sequence of acceptable solutions, ie those solutions used in the averaging procedure for the determination of the final solution space (see above). The expression δ ~ (L) (t u ) denotes the discrepancy which describes the average size distribution from the t _u individual solutions in the layer 1 with l = 1,..., N _L. Furthermore, let δ ~ (dead) (t u ) the discrepancy, which the average size distribution from the t _{u .} Describes solutions according to the entire profile.

The discrepancy δ ~ (dead) (t u ) can be estimated from the expression:

Jede Zahl t_u ist definiert durch die Bedingung δ(l)(t) ≤ δu l = 1, ..., NL, t = 1, 2, ... tu, (56)und δ^(l)(t = t_u + 1) > δ_u, bei einem bestimmten l = 1, ..., N_L.According to the method described here, the sequence δ ~ ^(tot) (t _u ) is generated as a function of the number of averaged solutions, ie, t ₁ , t ₂ ,.

Every number _u is defined by the condition δ (L) (t) ≤ δ u l = 1, ..., N L , t = 1, 2, ... t u , (56) and δ ^(l) (t = t _u + 1)> δ _u , for a given l = 1, ..., N _L.

da jeder Wert t_u mit dem vorab definierten Diskrepanzparameter δ_u übereinstimmt.Considering all u _max solutions, the function δ ~ ^(tot) (δ _u ) can be represented graphically as a sequence δ ~ (dead) (t 1 ), δ ~ (dead) (t 2 ) ...

since each value t _u coincides with the predefined discrepancy parameter δ _u .

An advantage of the function δ ~ (dead) (δ u ) is that it describes the set of mean solutions of the examined profile depending on the discrepancy parameter. By analysis of the function δ ~ (dead) (δ u ) At the same time you will find the optimal solution set for all layers.

In general, results are searched that are within the range of the discrepancy parameter δ _opt , where, on the one hand, the function δ ~ (dead) (δ u ) is stable and furthermore a later increase of the discrepancy parameter does not result in a strong change of the examined microphysical parameters. That is, the values of the parameters of interest remain largely invariable.

embodiments The invention is illustrated in the figures and will be described below explained in more detail.

In show the figures:

1 a schematic structure of a lidar measuring arrangement;

2 Results of Measurement of Particle Optical Properties at Multiple Wavelengths;

3 a flowchart for illustrating the basic procedure of a data inversion method;

4 synthetically generated vertical profiles of optical particle properties used in a method of determining physical particle properties according to an embodiment of the invention;

5 with a statistical error noisy vertical profiles, which consist of the in 4 obtained profiles were obtained;

6 Vertical profiles of micro-particle physical properties, which were determined by means of a method for determining physical particle properties according to an embodiment of the invention;

7 a diagram illustrating the determination of a discrepancy parameter profile according to an embodiment of the invention.

The following is with reference to 4 to 6 a method for determining physical properties of atmospheric particles according to an embodiment of the invention. According to the described embodiment, an inversion method with reformulated one-dimensional regularization or two-dimensional regularization, as described above, is used in the method, wherein a synthetically generated optical data set serves as the output data record for the data inversion. In other words, synthetically generated optical parameter data is provided as input to the inversion process. In this way, a determination of the functionality of the algorithm used is made possible, since in this case the solutions are known and thus can be used for a test of the solution of the task. The method described here is self-evident is not limited to the use of synthetically generated data, but can be applied equally to data sets acquired or measured experimentally (eg, by one or more lidar measurements).

4 shows the synthetic output data set, which is processed in the inversion algorithm.

According to this described embodiment a so-called two-layer structure is used, with others Words a structure with two on top of each other arranged particle layers. Such structures typically become in the atmosphere found. Multilayer structures are also possible, i. H. Structures with more than two layers. The procedure can be apply in an analogous manner to such multi-layer structures.

The Profiles of particle backscatter and particle extinction coefficients are calculated in this example from synthetic particle size distributions with a so-called Mie spreading algorithm. The use of a Mie scattering algorithm is in itself z. In [6] described. In the simulations, the backscatter coefficients in the wavelength 355 nm, 532 nm and 1064 nm and the extinction coefficients at the wavelengths Considered 355 nm and 532 nm. The process is not up, though the said wavelengths and Combinations of backscattering and extinction coefficients limited. In alternative embodiments can other wavelengths or other combinations of optical parameters.

The optical particle properties are calculated from given logarithmic normal distributions (see eg [13]), which describe naturally occurring particle size distributions sufficiently well (in this case, too, the method is not limited to a specific form of the particle size distribution):

The term dn (r) denotes the number concentration of the particles in the size interval [lnr; lnr + dlnr], n _t is the total number concentration, r _{mod, N} is the mode radius with respect to the number concentration, and σ denotes the geometric standard deviation (mode width). Eq. (57) describes a monomodal distribution, ie a distribution with a maximum. In alternative embodiments, the method may also be extended to multi-modal distributions.

The optical profiles (in other words, the vertical profiles of the backscatter and extinction coefficients) become the inversion algorithm and the profiles of the integral properties of the particle size distributions, d. H. Number, surface and volume concentration are determined. The mean particle radius, or me the effective radius, can be derived from these sizes become. The complex refractive index also follows directly from this Inversion process.

4 shows the example of the two-layer structure. Shown are the synthetically generated profiles of the optical particle properties, ie the particle backscatter coefficients (Diagram 401 ) and the particle extinction coefficient (Diagram 402 ). The backscatter coefficients are shown for the 355 nm, 532 nm, and 1064 nm wavelengths, and the particle extinction coefficients are shown for the 355 nm and 532 nm wavelengths 4 It can be seen that the transition or the boundary between the two particle layers of the two-layer structure takes place at a height of approximately 1.5 km. The two-layer system is split into so-called sublayers (or sublayers), represented by the individual symbols. In other words, every symbol in the diagram represents 401 the value of the backscatter coefficient (for each wavelength) in a given sublayer, and correspondingly, each symbol in the graph represents 402 the value of the extinction coefficient in a given sublayer.

A Such subdivision can be related to the height resolution of a Lidarmessung. In the example shown, a height resolution of 200 m chosen. However, the procedure is not limited to the concrete choice of Height resolution.

In the 4 shown data are error free. In addition, a statistical error in the optical data in each altitude layer is additionally selected. In other words, the synthetically generated, error-free data turns out 4 noisy with a statistical error.

5 shows the noisy statistical profiles for the particle backscatter with the statistical error coefficients (diagram 501 ) and the particle extinction coefficient (Diagram 502 ) at 532 nm wavelength. The data at 355 nm and 1064 nm are treated in the same way, giving similar representations. In the embodiment shown, the data was noisy with a statistical error of 10%. Alternatively, other values may be used for the statistical error.

Finally, the inversion is repeated ten times for the noisy data set (ie for the noisy optical profiles at all wavelengths) and once for the error-free data set ( 4 ) carried out.

6 shows the mean profiles of the investigated microphysical parameters and their standard deviation for the reformulated one-dimensional regularization (represented by the circle symbols in the diagrams 601 to 606 ) and the two-dimensional regularization (represented by the square symbols in the diagrams 601 to 606 ). The profiles of the particle microphysical properties represent the mean of all eleven inversion runs. Shown are the profiles of the number concentration (Diagram 601 ), Surface concentration (diagram 602 ) and volume concentration (Diagram 603 ), the effective radius (Diagram 604 ) as well as the real part (diagram 605 ) and imaginary part (Diagram 606 ) of the complex refractive index. The symbols indicate the height layers in which the microphysical parameters were determined, cf. 4 , The results are shown for the reformulated one-dimensional regularization and the two-dimensional regularization. The theoretical values are indicated by solid lines. For the calculations, monomodal logarithmic normal distributions were assumed, as described above. The mode radii in the lower and the upper layer were assumed to be 0.15 μm and 0.1 μm, respectively. The geometric standard deviations (mode widths) are 1.8 in the lower layer and 1.5 in the upper layer.

6 shows that the method captures the transition point between the two particle layers in all parameters sought. Another advantage of the methodology is that without major postprocessing methods, unlike the case of classic one-dimensional regularization, the errors can be kept well below 50%. This error limit is typically taken in order to prove a principle applicability of an evaluation method in aerosol characterization with lidar.

Tab. 1 summarizes the average shift values obtained by means of different methods Table 1. parameter theory Classic 1-Dim. inversion Reformulated 1-Dim. inversion 2-dim. inversion 0-1300 m altitude range r _eff (μm) 12:28 0.30 ± 0.06 0.30 ± 0.06 0.29 ± 0.06 n _t (cm ^-3 ) 1 1.6 ± 1 1.7 ± 1 1.6 ± 0.8 υ _t (μm ³ cm ^-3 ) 0078 0.1 ± 0.03 0.1 ± 0.03 0.091 ± 0.020 α _t (μm ² cm ^-3 ) 0.85 1.0 ± 0.2 1.0 ± 0.2 0.97 ± 0.13 _Re 1:35 1.34 ± 0.05 1.34 ± 0.05 1.34 ± 0.05 _In m 0005 0.011 ± 0.01 0.013 ± 0.011 0.013 ± 0.011 1700-2900 m altitude range r _eff (μm) 12:22 0.26 ± 0.06 0.26 ± 0.06 0.23 ± 0.04 n _t (cm ^-3 ) 0.5 0.74 ± 0.39 0.74 ± 0.38 0.67 ± 0.29 υ _t (μm ³ cm ^-3 ) 12:02 0.029 ± 0.008 0.029 ± 0.008 0.025 ± 0.006 α _t (μm ² cm ^-3 ) 12:27 0.34 ± 0.06 0.34 ± 0.06 0.32 ± 0.05 _Re 1.4 1.35 ± 0.06 1.35 ± 0.06 1.36 ± 0.06 _In m 12:02 0.017 ± 0.013 0.017 ± 0.013 0.017 ± 0.014 the, together. The table shows the microphysical particle parameters (mean ± standard deviation) derived from the classical one-dimensional regularization method (shown in the third table column), the reformulated one-dimensional regularization method (shown in the fourth column of the table) and the two-dimensional regularization method (shown in the fifth tables column) for the example 4 , In the second table column, the theoretical values of the respective parameters are given for comparison. The inversions were performed for the undisturbed data set and ten times for optical data that was noisy with a statistical error of 10%. The results of all eleven inversion runs were averaged. The numbers were calculated as the mean of all elevation layers within the displayed elevation ranges.

The result data sets were obtained by smoothening second order on the examined data sets. In the case of two-dimensional regularization, second-order smoothing was also applied along the z-direction, ie in successive height layers. The inversion with the classical one-dimensional regularization was investigated with the same set of input parameters as in the case of the reformulated one-dimensional regularization and the two-dimensional regularization. The expression r _eff denotes the effective radius. The terms n _t , α _t and υ _t denote the number, and surface and volume concentration. The real part of the complex refractive index is denoted by m _Re . The imaginary part is designated m _Im .

Out Tab. 1 shows the smaller deviations of the results obtained from the given theoretical values in the case of the reformulated one-dimensional regularization compared to classical one-dimensional Regularization. The two-dimensional regularization delivers again improved results, especially in the case of the number concentration.

7 shows a diagram 700 to illustrate the determination of the discrepancy parameter profile for the case of inversion with reformulated one-dimensional regularization, according to an embodiment of the invention. The modified discrepancy parameter δ _{u of} the averaged solutions is calculated for the entire profile, and not just for a certain height layer δ ^ave, as is the case with classical one-dimensional regularization inversion. The in the diagram 700 values shown were using the above related to 4 to 6 determined two-layer structure data set determined.

The diagram 700 shows the deviations (in percent) of the particle parameters effective particle radius r _eff (represented by the squares), number concentration n _t (represented by the triangles), surface concentration α _t (represented by the circles) and volume concentration υ _t (represented by the diamonds) of the respective true values as a function of the value of the discrepancy parameter δ _u , which was used for the averaging of the individual solutions. I have the caption of the x-axis in 7 changed to discrepancy parameter. Furthermore, the discrepancy calculated for the overall profile is δ ~ (dead) depending on the Diskrepanzparameter δ _u plotted (shown by the black dots in the graph 700 ).

In the diagram 700 is shown that an area exists in which the discrepancy δ ~ (dead) increases relatively strongly. This sharp increase marks the area from which the optimum value δ _{opt is} selected, which in the example shown corresponds to δ _opt = 10% (represented by the vertical dashed line). In other words, the dashed line marks the limit (10%) up to which all solutions are accepted. At the same time, the mean values of the effective particle radius, the number concentration, the surface concentration and the volume concentration deviate less from the true values, and further, the variation of these deviations is also comparatively small, more importantly. In principle, the optimum averaging interval can be extended even further, for example up to approximately δ _opt = 20%. It should be noted that the deviation of the profile of the average number concentration increases faster than the deviations of the profiles of the other parameters decrease.

following become further properties and effects of exemplary embodiments of the invention.

According to one Embodiment of the invention is a method for determining Vertical profiles of physical properties (or microphysical Characteristics) more atmospheric Particles provided. Their method is lidar measurements of vertical profiles of optical properties of the particles as input. The physical properties are, for example, the average size (eg. effective radius, mean volume radius) of particle size distributions, the complex refractive index of the particles, as well as their shape (z. B. spherical or not spherical). These parameters are essential input variables in the investigation of the Climate change through particle pollution caused by human Activities caused becomes.

According to one embodiment of the invention is an algorithm or a method for performing ei provided a data inversion. This method can be used in a method for determining physical (or microphysical) properties of particles, e.g. B. atmospheric particles.

According to one In another embodiment, a method is provided for determining an optimal profile of a Lagrangian multiplier, which for a Regularization step used in the context of data inversion can be.

According to one Another embodiment of the invention is an inversion method provided for the determination of vertical profiles of physical Characteristics of atmospheric Particle. Lidar measurements of vertical profiles are used in the procedure of optical properties of the particles as input. The physical Properties are medium size. from Particle size distributions, integral parameters such as the number, surface and volume concentration, the complex refractive index of the particles, as well as their shape (spherical or not spherical). The method is not limited to these parameters.

According to one other embodiment can optical properties measured vertically resolved directly be entered into the algorithm used. This will be a special methodology of matrix calculation in the field of telemetering and telecommunications annotation methodology. The procedure is called two-dimensional Regularization denotes and can easily to a multi-dimensional regularization be extended. With the method can be an improved determination the physical properties of atmospheric particles take place. reason for This improvement is the so-called stabilization of the used mathematical method of data inversion. Such methods are characterized by extreme instability, ambiguity as well as the incompleteness of the solution space obtained. An effect of the two-dimensional regularization can be seen therein be that elaborate Data postprocessing procedures that are conventionally performed out of the mathematical solution space the physical solution space to identify, are not required.

According to one Another embodiment of the invention is a method for determining of vertical profiles of microphysical properties of atmospheric Particles provided in which the profiles within a computing step be determined. In contrast, the conventional Method based on classic one-dimensional regularization lying equation system for every height layer the predetermined optical profiles solved individually. This process of the separate finding solutions requires more extensive knowledge of the data-evaluating person in the field of aerosol properties and knowledge of equipment technology and the associated sources of error to be determined optical profiles. In the method according to this embodiment is a two-dimensional regularization (generally a multidimensional Regularization). With the method of multidimensional and In particular, the two-dimensional regularization can be time-consuming wrap-up procedures the solutions found to a big one Part avoided, in contrast to the conventional method of classical one-dimensional regularization. An effect of the method according to this Design can be seen in that through this technique implement in the data evaluation an applicability on an industrial scale leaves. Furthermore, the method also provides the possibility of data evaluation at the operational level.

According to one Design, the determination of the solution space by means of the above in the section 'Two-dimensional Regularization 'described Formalism'. there is considered the basis of the functionality of the formalism 'that the above in the section 'Reformulated one-dimensional problem 'described Formalism is used. In this formalism, the coupling takes place the optical data sets from individual height layers the optical profiles.

It It should be noted that the inversion methodology described in this application not on the use of optical data (eg backscatter, Extinction, absorption, scattering) is limited. Further, the method also not limited on the use of other data such. B. Measurement of spectrally resolved particle mass distributions, as well as the volume, surface, Number concentration (so-called moments of the particle size distribution). Furthermore the data combination is not for a basic applicability of the described method is necessary. From a mathematical point of view, the methodology is without restriction applicable.

• [1] D. Müller, U. Wandinger, and A. Ansmann, ”Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: Theory,” Appl. Opt. 38, 2346–2357 (1999).
• [2] D. Müller, U. Wandinger, and A. Ansmann, ”Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: Simulation,” Appl. Opt. 38, 2358–368 (1999).
• [3] I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, U. Wandinger, and D. N. Whiteman, ”Inversion with regularization for the retrieval of tropospheric aerosol parameters from multiwavelength lidar sounding,” Appl. Opt. 41, 3685–3699 (2002).
• [4] I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, K. Franke, and D. N. Whiteman, ”Inversion of multiwavelength Raman lidar data for retrieval of bimodal aerosol size distribution,” Appl. Opt. 43, 1180–1195 (2004).
• [5] C. Böckmann, I. Miranova, D. Müller, L. Scheidenbach, and R. Nessler, ”Microphysical aerosol parameters from multiwavelength lidar,” J. Opt. Soc. America-A 22, 518–528 (2005).
• [6] C. F. Bohren and D. R. Huffman, eds., Absorption and scattering of light by small particles (John Wiley, Hoboken, NJ, 1983), Seiten 477–483.
• [7] D. L. Phillips, ”A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
• [8] S. Twomey, ”The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements,” J. Franklin Inst. 279, 95–109 (1965).
• [9] A. N. Tikhonov and V. Y. Arsenin, eds., Solutions of Ill-posed Problems (John Wiley, New York, 1977).
• [10] V. E. Zuev and I. E. Naats, eds., Inverse problems of lidar sensing of the atmosphere (Springer, Berlin, 1983).
• [11] J. Hadamard, ”Sur les problèmes aux derivées partielles et leur signification physique,” Bull. Univ. of Princeton 13, 49–52 (1902).
• [12] S. Twomey, ed., Introduction to the mathematics of inversion in remote sensing and indirect measurements (Elsevier, Amsterdam, 1977), Seiten 115–149.
• [13] W. C. Rinds, ed., Aerosol technology (John Wiley, New York, New York, 1982), Seiten 75–110.
• [14] A. Kolgotin and D. Müller, ”2-Dimensional Regularization for the Retrieval of Profiles of Microphysical Aerosol Properties from Multiwavelength Raman Lidar”, in: Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, 24–28 July 2006, Nara, Japan, Part I, Editors: C. Nagasawa, N. Sugimoto, pp. 291–294.
• [15] V. S. Buslaev, ed., Calculus of Variations (Leningrad University named by A. A. Zhdanov, 1980).

This document cites the following publications:

• [1] D. Müller, U. Wandinger, and A. Ansmann, "Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: theory," Appl. Opt. 38, 2346-2357 (1999).
• [2] D. Müller, U. Wandinger, and A. Ansmann, "Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: simulation," Appl. Opt. 38, 2358-368 (1999).
• [3] I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, U. Wandinger, and DN Whiteman, "Inversion with regularization for the retrieval of tropospheric aerosol parameters from multiwavelength lidar sounding," Appl. Opt. 41, 3685-3699 (2002).
• [4] I. Veselovskii, A. Kolgotin, V. Griaznov, D. Müller, K. Franke, and DN Whiteman, "Inversion of multi-wavelength Raman lidar data for retrieval of bimodal aerosol size distribution," Appl. Opt. 43, 1180-1195 (2004).
• [5] C. Bockmann, I. Miranova, D. Muller, L. Scheidenbach, and R. Nessler, "Microphysical aerosol parameters from multi-wavelength lidar," J. Opt. Soc. America-A 22, 518-528 (2005).
• [6] CF Drilling and DR Huffman, eds., Absorption and scattering of light by small particles (John Wiley, Hoboken, NJ, 1983), pages 477-483.
• [7] DL Phillips, "A technique for the numerical solution of certain integrative equations of the first child," J. Assoc. Comput. Mach. 9, 84-97 (1962).
• [8] S. Twomey, "The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements," J. Franklin Inst. 279, 95-109 (1965).
• [9] AN Tikhonov and VY Arsenin, eds., Solutions of Ill-posed Problems (John Wiley, New York, 1977).
• [10] VE Zuev and IE Naats, eds., Inverse problems of lidar sensing of the atmosphere (Springer, Berlin, 1983).
• [11] J. Hadamard, "Sur les problèmes aux derivées partial et leur signification physique," Bull. Univ. of Princeton 13, 49-52 (1902).
• [12] S. Twomey, ed., Introduction to the mathematics of inversion in remote sensing and indirect measurements (Elsevier, Amsterdam, 1977), pages 115-149.
• [13] WC Rinds, ed., Aerosol technology (John Wiley, New York, New York, 1982), pp. 75-110.
• [14] A. Kolgotin and D. Müller, "2-Dimensional Regularization for the Retrieval of Profiles of Microphysical Aerosol Properties from Multiwavelength Raman Lidar", in: Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, 24-28 July 2006, Nara, Japan, Part I, Editors: C. Nagasawa, N. Sugimoto, pp. 291-294.
• [15] VS Buslaev, ed., Calculus of Variations (Leningrad University named by AA Zhdanov, 1980).

Claims (14)

Method for determining physical properties atmospheric Particles in which: - at least a vertical profile of at least one optical property of the particles provided; - at least a vertical profile of at least one physical property of Particles by means of an inversion of the at least one Vertical profile of the at least one optical property of the particles is determined - in which in the context of the inversion method one for a plurality of height layers the vertical profiles common discrepancy parameter profile determined and that at least one vertical profile of the at least one physical property using the discrepancy parameter profile is determined. Method according to claim 1, wherein the at least one vertical profile of the at least one optical property of the particles is provided by the at least one optical property by means of at least one optical Measurement is detected. Method according to claim 2, wherein the at least one optical property of the particles in Each of the plurality of height layers is detected. Method according to claim 3, wherein the at least one optical measurement is a lidar measurement is. Method according to claim 4, wherein by means of the Lidar measurement of the backscatter coefficient and / or the extinction coefficient of the particles at least one optical wavelength as the at least one optical property are detected. Method according to claim 5, wherein the backscatter coefficient and / or the extinction coefficient for a plurality of optical wavelength be recorded. Method according to one the claims 1 to 6, wherein the inversion process at least one regularization step and wherein determining the discrepancy parameter profile within the framework of the at least one regularization step. Method according to claim 7, wherein the inversion method is one of the following methods is: - one Inversion method with reformulated one-dimensional regularization; - an inversion process with two-dimensional regularization; - an inversion process with multidimensional regularization. Method according to one the claims 1 to 8, where the at least one physical property the particle has one or more of the following properties: - the middle Particle size; - the number concentration the particle; - the surface concentration the particle: - the Volume concentration of the particles: - the complex refractive index the particle; - the Shape of the particles. Device for determining physical properties atmospheric Particles comprising: - one Provision device for providing at least one vertical profile at least one optical property of the particles; - a detection device for determining at least one vertical profile of at least one physical property of the particles from the at least one Vertical profile of the at least one optical property of the particles by means of an inversion method, - wherein the determination device is set up so that in the context of the inversion process for a majority of altitude layers the vertical profiles common discrepancy parameter profile determined and that at least one vertical profile of the at least one physical property of the particles using the discrepancy parameter profile is determined. Device according to claim 10, wherein the providing means is arranged so that the at least one optical property of the particles by means of at least an optical measurement can be detected and provided. Device according to claim 10, set up as a lidar measuring device. Computer-readable storage medium containing a program for determining physical properties of atmospheric Particle is stored which, when executed by a processor, comprising the following method steps: - Provide at least a vertical profile of at least one optical property of Particle; - Determine at least one vertical profile of at least one physical Property of the particles by means of an inversion process the at least one vertical profile of the at least one optical Property of the particles, - in which in the context of the inversion method one for a plurality of height layers the vertical profiles common discrepancy parameter profile determined and that at least one vertical profile of the at least one physical property of the particles using the discrepancy parameter profile is determined. Computer program element for determining physical Characteristics of atmospheric Particle that, when executed by a processor, comprising the following method steps: - Provide at least a vertical profile of at least one optical property of Particle; - Determine at least one vertical profile of at least one physical Property of the particles by means of an inversion process the at least one vertical profile of the at least one optical Property of the particles, - in which in the context of the inversion method one for a plurality of height layers the vertical profiles common discrepancy parameter profile determined and that at least one vertical profile of the at least one physical property of the particles using the discrepancy parameter profile is determined.