Methods for determining regularization for atmospheric retrieval problems

The atmosphere of Earth has already been investigated by several spaceborne instruments, and several further instruments will be launched, e.g., NASA's Earth Observing System Aura platform and the European Space Agency's Environmental Satellite. To stabilize the results in atmospheric retrievals, constraints are used in the iteration process. Therefore hard constraints (discretization of the retrieval grid) and soft constraints (regularization operators) are included in the retrieval. Tikhonov regularization is often used as a soft constraint. In this study, different types of Tikhonov operator were compared, and several new methods were developed to determine the optimal strength of the constraint operationally. The resulting regularization parameters were applied successfully to an ozone retrieval from simulated nadir sounding spectra like those expected to be measured by the Tropospheric Emission Spectrometer, which is part of the Aura platform. Retrievals were characterized by means of estimated error, averaging kernel, vertical resolution, and degrees of freedom.     

III-posed retrieval of aerosol extinction coefficient profiles from Raman lidar data by regularization

In the analysis of Raman lidar measurements of aerosol extinction, it is necessary to calculate the derivative of the logarithm of the ratio between the atmospheric number density and the range-corrected lidar-received power. The statistical fluctuations of the Raman signal can produce large fluctuations in the derivative and thus in the aerosol extinction profile. To overcome this difficult situation we discuss three methods: Tikhonov regularization, variational, and the sliding best-fit (SBF). Three methods are performed on the profiles taken from the European Aerosol Research Lidar Network lidar database simulated at the Raman shifted wavelengths of 387 and 607 nm associated with the emitted signals at 355 and 532 nm. Our results show that the SBF method does not deliver good results for low fluctuation in the profile. However, Tikhonov regularization and the variational method yield very good aerosol extinction coefficient profiles for our examples. With regard to, e.g., the 532 nm wavelength, the L2 errors of the aerosol extinction coefficient profile by using the SBF, Tikhonov, and variational methods with respect to synthetic noisy data are 0.0015(0.0024), 0.00049(0.00086), and 0.00048(0.00082), respectively. Moreover, the L2 errors by using the Tikhonov and variational methods with respect to a more realistic noisy profile are 0.0014(0.0016) and 0.0012(0.0016), respectively. In both cases the L2 error given in parentheses concerns the second example.     

Inversion with regularization for the retrieval of tropospheric aerosol parameters from multiwavelength lidar sounding

We present an inversion algorithm for the retrieval of particle size distribution parameters, i.e., mean (effective) radius, number, surface area, and volume concentration, and complex refractive index from multiwavelength lidar data. In contrast to the classical Tikhonov method, which accepts only that solution for which the discrepancy reaches its global minimum, in our algorithm we perform the averaging of solutions in the vicinity of this minimum. This averaging stabilizes the underlying ill-posed inverse problem, particularly with respect to the retrieval of number concentration. Results show that, for typical tropospheric particles and 10% error in the optical data, the mean radius could be retrieved to better than 20% from a lidar on the basis of a Nd:YAG laser, which provides a combination of backscatter coefficients at 355, 532, and 1064 nm and extinction coefficients at 355 and 532 nm. The accuracy is improved if the lidar is also equipped with a hydrogen Raman shifter. In this case two additional backscatter coefficients at 416 and 683 nm are available. The combination of two extinction coefficients and five backscatter coefficients then allows one to retrieve not only averaged aerosol parameters but also the size distribution function. There was acceptable agreement between physical particle properties obtained from the evaluation of multiwavelength lidar data taken during the Lindenberg Aerosol Characterization Experiment in 1998 (LACE 98) and in situ data, which were taken aboard aircraft.     

探测低空大气温度分布的转动拉曼激光雷达

提出了一种新的探测对流层低层大气温度的转动拉曼激光雷达方法,通过测量N2和O2的后向散射的纯转动拉曼谱的强度,计算它们的比值来确定大气温度的垂直分布,并对其性能进行了数值模拟。转动拉曼激光雷达的光源是一个调Q的Nd:YAG激光器,经扩束器后输出能量200mJ;采用双光栅单色仪提取所需要的氮气和氧气的转动拉曼谱;接收机采用光电倍增管和双通道光子计数器,量子效率是10%(48000个脉冲累加)。夜晚它对近地面10.2km高度内的探测信噪比在10:1以上,白天它对近地面3.6km高度内的探测信噪比在10:1以上,计算的温度与模拟用的温度真值阔线相差约0.3K。     

Hybrid regularization method for the ill-posed inversion of multiwavelength lidar data in the retrieval of aerosol size distributions

A specially developed method is proposed to retrieve the particle volume distribution, the mean refractive index, and other important physical parameters, e.g., the effective radius, volume, surface area, and number concentrations of tropospheric and stratospheric aerosols, from optical data by use of multiple wavelengths. This algorithm requires neither a priori knowledge of the analytical shape of the distribution nor an initial guess of the distribution. As a result, even bimodal and multimodal distributions can be retrieved without any advance knowledge of the number of modes. The nonlinear ill-posed inversion is achieved by means of a hybrid method combining regularization by discretization, variable higher-order B-spline functions and a truncated singular-value decomposition. The method can be used to handle different lidar devices that work with various values and numbers of wavelengths. It is shown, to my knowledge for the first time, that only one extinction and three backscatter coefficients are sufficient for the solution. Moreover, measurement errors up to 20% are allowed. This result could be achieved by a judicious fusion of different properties of three suitable regularization parameters. Finally, numerical results with an additional unknown refractive index show the possibility of successfully recovering both unknowns simultaneously from the lidar data: the aerosol volume distribution and the refractive index.     

超声逆散射成像问题中正则化参数研究

研究了二维理想情况下,基于精确场描述的超声逆散射成像问题,先用矩量法将波动方程化为离散形式,分别用BI和DBI算法进行迭代重建。影响整个算法的一个关键因素是散射场方程的正则化求解,具有明显的不适定性。文章基于L曲线法,提出以解的范数和残差变化量的加权形式作为确定正则化参数的依据,在迭代过程根据问题不适定性程度,自适应地调整搜索范围。仿真结果表明,该算法可快速地找到最优正则化参数。     

A three parameter equation of state for gases

This paper presents an equation of state having the following features. It holds for many gases for temperatures from near critical to about five times critical, and for densities from zero to a little above critical, with the exception of the immediate vicinity of the critical point. It also holds for mixtures. Its accuracy is excellent, in most cases within experimental error. Exceptions are H₂O, CO₂, H₂ at high density. In the best cases the rms deviations in the compressibility factor are a few parts in ten thousand. In relatively poor cases the deviations are a few parts per thousand.The equation has a single reduced form for all gases, and requires only three characteristic parameters to identify each gas: a reducing parameter, T_B; a reducing density, d_o; and a dimensionless parameter, k_B. These three parameters are readily (though often not uniquely) evaluated from experimental data. Besides determining the PVT behaviour of a gas, including its virial coefficients, these three parameters are directly related to those in the common three-parameter intermolecular potentials.     

Apodization effects in the retrieval of volume mixing ratio profiles

In remote sensing applications, spectra measured by Fourier-transform spectrometers are routinely apodized. A rigorous analysis approach would explicitly account for correlations induced in the covariance matrix by apodization, but these correlations are often ignored to simplify and speed up the processing. Using spectra measured by the Atmospheric Trace Molecule Spectroscopy missions, we investigated the effect of apodization on the retrieval of volume mixing ratio profiles for the case in which these correlations are ignored. Minor discrepancies occur between results for apodized and unapodized spectra, particularly when lines with a low signal-to-noise ratio are fitted. A set of microwindows is reported for O3 in the range of 1550-3350 cm(-1).     

Regularizing active set method for retrieval of the atmospheric aerosol particle size distribution function

The determination of the aerosol particle size distribution function by using the particle spectrum extinction equation is an ill-posed integral equation of the first kind [S. Twomey, J. Comput. Phys.18, 188 (1975);Y. F. Wang, Computational Methods for Inverse Problems and Their Applications (Higher Education Press, 2007)], since we are often faced with limited or insufficient observations in remote sensing and the observations are contaminated. To overcome the ill-posed nature of the problem, regularization techniques were developed. However, most of the literature focuses on the application of Phillips-Twomey regularization and its variants, which are unstable in several cases. As is known, the particle size distribution is always nonnegative, and we are often faced with incomplete data. Therefore, we study the active set method and propose a regularizing active set algorithm for ill-posed particle size distribution function retrieval and for enforcing nonnegativity in computation. Our numerical tests are based on synthetic data for theoretical simulations and the field data obtained with a CE 318 Sun photometer for the Po Yang lake region of Jiang Xi Province, China, and are performed to show the efficiency and feasibility of the proposed algorithms.     

A lattice parameter method for the investigation of solid state precipitation

Over strictly limited composition ranges, the relationship between lattice parameter and solid solution composition can often be taken as linear even in multicomponent solid solution systems. The constants in the assumed linear equations relating the lattice parameter to the atomic percentage of each component may be calculated from lattice parameters measured for solution heat-treated alloys of known compositions. The subsequent changes in lattice parameters which occur during ageing of these alloys are shown to yield useful information about composition changes and the precipitate phases which occur during ageing. In particular, if the composition of the precipitate is known, then the linear equations may be used to calculate the composition of the remaining solid solution alloys. Even if both the composition of the precipitate phase and the remaining solid solution are unknown, changes in lattice parameter, combined with known lattice parameter versus composition functions, may be used to determine what average precipitate compositions are not allowed. These methods are illustrated in the case of Al-Cu, Cu-Ni-Al and Cu-Zn-Ni-Al alloys.     

S-system parameter estimation for noisy metabolic profiles using newton-flow analysis

Biochemical systems are commonly modelled by systems of ordinary differential equations (ODEs). A particular class of such models called S-systems have recently gained popularity in biochemical system modelling. The parameters of an S-system are usually estimated from time-course profiles. However, finding these estimates is a difficult computational problem. Moreover, although several methods have been recently proposed to solve this problem for ideal profiles, relatively little progress has been reported for noisy profiles. We describe a special feature of a Newton-flow optimisation problem associated with S-system parameter estimation. This enables us to significantly reduce the search space, and also lends itself to parameter estimation for noisy data. We illustrate the applicability of our method by applying it to noisy time-course data synthetically produced from previously published 4- and 30-dimensional S-systems. In addition, we propose an extension of our method that allows the detection of network topologies for small S-systems. We introduce a new method for estimating S-system parameters from time-course profiles. We show that the performance of this method compares favorably with competing methods for ideal profiles, and that it also allows the determination of parameters for noisy profiles.     

Infrared atmospheric sounding interferometer correlation interferometry for the retrieval of atmospheric gases: the case of H2O and CO2

Correlation interferometry is a particular application of Fourier transform spectroscopy with partially scanned interferograms. Basically, it is a technique to obtain the difference between the spectra of atmospheric radiance at two diverse spectral resolutions. Although the technique could be exploited to design an appropriate correlation interferometer, in this paper we are concerned with the analytical aspects of the method and its application to high-spectral-resolution infrared observations in order to separate the emission of a given atmospheric gas from a spectral signal dominated by surface emission, such as in the case of satellite spectrometers operated in the nadir looking mode. The tool will be used to address some basic questions concerning the vertical spatial resolution of H2O and to develop an algorithm to retrieve the columnar amount of CO2. An application to complete interferograms from the Infrared Atmospheric Sounding Interferometer will be presented and discussed. For H2O, we have concluded that the vertical spatial resolution in the lower troposphere mostly depends on broad features associated with the spectrum, whereas for CO2, we have derived a technique capable of retrieving a CO2 columnar amount with accuracy of ≈±7 parts per million by volume at the level of each single field of view.     

Optimal eigenanalysis for the treatment of aerosols in the retrieval of atmospheric composition from transmission measurements

The separation of the individual contributions of aerosol and gases to the total attenuation of radiation through the atmosphere has been the subject of much scientific investigation since remote sensing experiments first began. We describe a new scheme to account for the spectral variation of the aerosol extinction in the inversion of transmission data from occultation measurements. Because the spectral variation of the aerosol extinction is generally unknown,the inversion problem is underdetermined and cannot be solved without a reduction in the number of unknowns in the set of equations used to describe the attenuation at each wavelength. This reduction can be accomplished by a variety of methods, including use of a priori information, the parameterization of the aerosol spectral attenuation, and the specification of the form of the aerosol size distribution. We have developed and implemented a parameterization scheme based on existing empirical and modeled information about the microphysical properties of aerosols. This scheme employs the eigenvectors from an extensive set of simulations to parameterize the aerosol extinction coefficient for incorporation into the inversion algorithm. We examine the accuracy of our method using data sets containing over 24,000 extinction spectra and compare it with that of another scheme that is currently implemented in the Polar Ozone and Aerosol Measurement (POAM) satellite experiment. In simulations using 80 wavelengths in the UV-visible-near-IR spectral range of the Stratospheric Aerosol and Gas Experiment III (SAGE) instrument, we show that, for our optimal parameterization, errors below 1% are observed in 80% of cases, whereas only approximately 20% of all cases are as accurate as this in a quadratic parameterization employing the logarithm of the wavelength.     

Radiative transfer codes for atmospheric correction and aerosol retrieval: intercomparison study

Results are summarized for a scientific project devoted to the comparison of four atmospheric radiative transfer codes incorporated into different satellite data processing algorithms, namely, 6SV1.1 (second simulation of a satellite signal in the solar spectrum, vector, version 1.1), RT3 (radiative transfer), MODTRAN (moderate resolution atmospheric transmittance and radiance code), and SHARM (spherical harmonics). The performance of the codes is tested against well-known benchmarks, such as Coulson's tabulated values and a Monte Carlo code. The influence of revealed differences on aerosol optical thickness and surface reflectance retrieval is estimated theoretically by using a simple mathematical approach. All information about the project can be found at http://rtcodes.ltdri.org.     

Constrained profile retrieval applied to the observation mode of the michelson interferometer for passive atmospheric sounding

To investigate the atmosphere of Earth and to detect changes in its environment, the Environmental Satellite will be launched by the European Space Agency in a polar orbit in October 2001. One of its payload instruments is a Fourier spectrometer, the Michelson Interferometer for Passive Atmospheric Sounding, designed to measure the spectral thermal emission of molecules in the atmosphere in a limb-viewing mode. The goal of this experiment is to derive operationally vertical profiles of pressure and temperature as well as of trace gases O(3), H(2)O, CH(4), N(2)O, NO(2), and HNO(3) from spectra on a global scale. A major topic in the analysis of the computational methodology for obtaining the profiles is how available a priori knowledge can be used and how this a priori knowledge affects corresponding results. Retrieval methods were compared and it was shown that an optimal estimation formalism can be used in a highly flexible way for this kind of data analysis. Beyond this, diagnostic tools, such as estimated standard deviation, vertical resolution, or degrees of freedom, have been used to characterize the results. Optimized regularization parameters have been determined, and a great effect from the choice of regularization and discretization on the results was demonstrated. In particular, we show that the optimal estimation formalism can be used to emulate purely smoothing constraints.     

变形镜参数变化对湍流像差校正效果的影响

利用61单元自适应光学系统湍流校正过程仿真模型,分析计算了变形镜的交连值、高斯 指数与自适应光学系统校正能力的关系。发现当交连值的稳定性变差时,引起校正效果的波动程度与高斯指数有关。高斯指数取1.9~2.1时,波前位相差的校正效果受交连值变化的影响最大,原因是校正算法的前提为各个驱动器的影响函数是线性叠加的。     

Variational method for the retrieval of the optical thickness and the backscatter coefficient from multiangle lidar profiles

A variational method for retrieving the aerosol optical thickness and backscatter coefficient profiles from multiangle lidar measurements is presented and discussed. A monostatic single-wavelength low-energy lidar system was operated at different zenith angles during the Indian Ocean Experiment (INDOEX) campaign in 1999 to characterize the aerosol plumes in the Indian monsoon. The variational method was applied to lidar data to retrieve profiles of optical thickness and the backscatter coefficient for nighttime and daytime measurements. Results are obtained with an uncertainty of 10% below 3 km (nighttime) and 2.8 km (daytime) and a bias of less than 0.01. During daytime the retrieval of optical parameters is indeed limited to a lower altitude owing to the sky background signal and the atmospheric inhomogeneity. In both cases the total aerosol optical thickness is consistent (+/- 10%) with the integrated value derived from sunphotometer measurements. Backscatter-to-extinction ratios estimated in different regions by two distinct methods compared well, which proves the capability of the method to assess optical measurements and account for the altitude dependence of the phase function.     

Tomographic retrieval of atmospheric parameters from infrared limb emission observations

Typical inversion of limb-sounding measurements assumes local horizontal homogeneity of the atmosphere. This simplification corresponds to spectral radiance errors that can exceed the noise level of a typical infrared instrument by a factor of 10 and causes errors in retrieved state parameters. To avoid these errors and to take the horizontal structure of the atmosphere into account, a two-dimensional (2D) tomographic sequential estimation approach is described. Application to temperature retrievals from simulated measurements yields typical retrieval errors of the order of 1 K, and a one-dimensional retrieval with the same synthetic measurements shows differences to the true values up to 10 K in regions with strong horizontal inhomogeneities. The horizontal resolution of the 2D retrieval is even better (up to 40 km) than the horizontal tangent point spacing.     

Error analysis for retrieval of urban atmospheric aerosol properties from downwelling infrared radiation spectra

The possibility of retrieval of urban aerosol physical properties from downwelling atmospheric infrared radiation spectra between 700 and 1400 cm(-1) with 0.24-cm(-1) spectral resolution, which can be obtained from the tropospheric infrared interferometric sounder developed by the Central Research Institute of Electric Power Industry, was estimated from error analysis of the least-squares fit method. The error analysis for retrieval of the aerosol extinction coefficient spectra in three atmospheric layers (boundary, free troposphere, and stratosphere) showed the retrievability only of the boundary layer. Based on this result, we propose the retrieval for particle number density of each aerosol component, which is one of the parameters for the aerosol size distribution function, using the boundary aerosol extinction coefficient spectra. We assume that aerosols in urban areas consist of three types of component, namely, water soluble, soot, and dustlike. Under this assumption, we estimated the error of the retrieved volume density for each aerosol component. For the estimation we used the least-squares fit of Mie-generated spectral extinction coefficients. The estimated error shows that the volume density of each aerosol component in an urban boundary layer is equivalent to the retrieval target. We also show that the aerosol properties can be retrieved with higher accuracy when the effects of multiple scattering by aerosols are included in the retrieval procedure.