The problem of deriving a complete set of aerosol optical properties from Sun and sky radiance measurements is discussed. Algorithm development is focused on improving aerosol retrievals by means of including a detailed statistical optimization of the influence of noise in the inversion procedure. The methodological aspects of such an optimization are discussed in detail and revised according to both modern findings in inversion theory and practical experience in remote sensing. Accordingly, the proposed inversion algorithm is built on the principles of statistical estimation: the spectral radiances and various a priori constraints on aerosol characteristics are considered as multisource data that are known with predetermined accuracy. The inversion is designed as a search for the best fit of all input data by a theoretical model that takes into account the different levels of accuracy of the fitted data. The algorithm allows a choice of normal or lognormal noise assumptions. The multivariable fitting is implemented by a stable numerical procedure combining matrix inversion and univariant relaxation. The theoretical inversion scheme has been realized in the advanced algorithm retrieving aerosol size distribution together with complex refractive index from the spectral measurements of direct and diffuse radiation. The aerosol particles are modeled as homogeneous spheres. The atmospheric radiative transfer modeling is implemented with well-established publicly available radiative transfer codes. The retrieved refractive indices can be wavelength dependent; however, the extended smoothness constraints are applied to its spectral dependence (and indirectly through smoothness constraints on retrieved size distributions). The positive effects of statistical optimization on the retrieval results as well as the importance of applying a priori constraints are discussed in detail for the retrieval of both aerosol size distribution and complex refractive index. The developed algorithm is adapted for the retrieval of aerosol properties from measurements made by ground-based Sun-sky scanning radiometers used in the Aerosol Robotic Network (AERONET). The results of numerical tests together with examples of experimental data inversions are presented.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2000JD900282

A flexible inversion algorithm for retrieval of aerosol optical properties from Sun and sky radiance measurements

Let M be a mesh consisting of $n^2 $ squares called elements, formed by subdividing the unit square $(0,1) \times (0,1)$ into $n^2 $ small squares of side ${1 / h}$, and having a node at each of the $(n + 1)^2 $ grid points. With M we associate the $N \times N$ symmetric positive definite system $Ax = b$, where $N = (n + 1)^2 $, each $x_i $ is associated with a node of M, and $A_{ij} \ne 0$ if and only if $x_i $ and $x_j $ are associated with nodes of the same element. If we solve the equations via the standard symmetric factorization of A, then $O(n^4 )$ arithmetic operations are required if the usual row by row (banded) numbering scheme is used, and the storage required is $O(n^3 )$. In this paper we present an unusual numbering of the mesh (unknowns) and show that if we avoid operating on zeros, the $LDL^T $ factorization of A can be computed using the same standard algorithm in $O(n^3 )$ arithmetic operations. Furthermore, the storage required is only $O(n^2 \log _2 n)$. Finally, we prove that all ord...

Nested Dissection of a Regular Finite Element Mesh

The equations of motion of a 180° domain wall in an infinite uniaxially anisotropic medium which is exposed to an instantaneously applied uniform dc magnetic field H0 have been integrated numerically. Below the critical field Hc =2παM0 (α is the Gilbert loss parameter and M0 the saturation magnetization), where a steady‐state solution is known to exist, it is shown that the wall motion tends smoothly to this solution. Above Hc, the magnetization precesses about the field and a periodic component appears in the forward motion of the wall. Analytic solutions for the wall motion have been found based upon approximations suggested by the computed behavior; these reproduce the computer results very accurately.

The motion of 180° domain walls in uniform dc magnetic fields

A general formulation of alternating direction methods

Census enumerations are usually packaged in irregularly shaped geographical regions. Interior values can be interpolated for such regions, without specification of “control points,” by using an analogy to elliptical partial differential equations. A solution procedure is suggested, using finite difference methods with classical boundary conditions. In order to estimate densities, an additional nonnegativity condition is required. Smooth contour maps, which satisfy the volume preserving and nonnegativity constraints, illustrate the method using actual geographical data. It is suggested that the procedure may be used to convert observations from one bureaucratic partitioning of a geographical area to another.

/pdf/smooth-pycnophylactic-interpolation-for-geographical-regions-3qu6uhm4ew.pdf

Smooth Pycnophylactic Interpolation for Geographical Regions

/pdf/finite-difference-methods-for-partial-differential-equations-41umnob08z.pdf

William Nachbar

Papers

Finite-Difference Methods for Partial Differential Equations