Revisiting matrix-based inversion of scanning mobility particle sizer (SMPS) and humidified tandem differential mobility analyzer (HTDMA) data

Tikhonov regularization is a tool for reducing noise amplification during data inversion. This work introduces RegularizationTools.jl, a general-purpose software package for applying Tikhonov regularization to data. The package implements well-established numerical algorithms and is suitable for systems of up to ~1000 equations. Included is an abstraction to systematically categorize specific inversion configurations and their associated hyperparameters. A generic interface translates arbitrary linear forward models defined by a computer function into the corresponding design matrix. This obviates the need to explicitly write out and discretize the Fredholm integral equation, thus facilitating fast prototyping of new regularization schemes associated with measurement techniques. Example applications include the inversion involving data from scanning mobility particle sizers (SMPSs) and humidified tandem differential mobility analyzers (HTDMAs). Inversion of SMPS size distributions reported in this work builds upon the freely available software DifferentialMobilityAnalyzers.jl. The speed of inversion is improved by a factor of ~200, now requiring between 2 and 5 ms per SMPS scan when using 120 size bins. Previously reported occasional failure to converge to a valid solution is reduced by switching from the L-curve method to generalized cross-validation as the metric to search for the optimal regularization parameter. Higher-order inversions resulting in smooth, denoised reconstructions of size distributions are now included in DifferentialMobilityAnalyzers.jl. This work also demonstrates that an SMPS-style matrixbased inversion can be applied to find the growth factor frequency distribution from raw HTDMA data while also accounting for multiply charged particles. The outcome of the aerosol-related inversion methods is showcased by inverting multi-week SMPS and HTDMA datasets from ground-based observations, including SMPS data obtained at Bodega Marine Laboratory during the CalWater 2/ACAPEX campaign and co-located SMPS and HTDMA data collected at the US Department of Energy observatory located at the Southern Great Plains site in Oklahoma, USA. Results show that the proposed approaches are suitable for unsupervised, nonparametric inversion of large-scale datasets as well as inversion in real time during data acquisition on low-cost reducedinstruction- set architectures used in single-board computers. The included software implementation of Tikhonov regularization is freely available, general, and domain-independent and thus can be applied to many other inverse problems arising in atmospheric measurement techniques and beyond.

Exploring polynomial based interpolation schemes for photoacoustic tomographic image reconstruction

Photoacoustic tomography (PAT) imaging employing polynomial-based interpolation methods is discussed. Nearest-neighbor, bilinear, bicubic and biquintic algorithms were implemented for the construction of the model matrix, and images were formed using the Tikhonov regularization and total variation (TV) minimization procedures. The performance of the interpolation methods was assessed by comparing the reconstructed images of three numerical and two experimental phantoms. The numerical and experimental studies demonstrate that the performance of the interpolation schemes is nearly equal for large PA sources. The simplest nearest-neighbor technique provides better image reconstruction for a sparse source compared to the others. The nearest-neighbor protocol may be adopted in practice for vascular imaging using PAT.

The method of detection and localization of configuration defects in geodetic networks by means of Tikhonov regularization

In adjusted geodetic networks, cases of local configuration defects (defects in the geometric structure of the network due to missing data or errors in point numbering) can be encountered, which lead to the singularity of the normal equation system in the least-squares procedure. Numbering errors in observation sets cause the computer program to define the network geometry incorrectly. Another cause of a defect may be accidental omission of certain data records, causing local indeterminacy or lowering of local reliability rates in a network. Obviously, the problem of a configuration defect may be easily detectable in networks with a small number of points. However, it becomes a real problem in large networks, where manual checking of all data becomes a very expensive task. The paper presents a new strategy for the detection of configuration defects with the use of the Tikhonov regularization method. The method was implemented in 1992 in the GEONET system (www.geonet.net.pl).

Identification of Matrix Diffusion Coefficient in a Parabolic PDE

An inverse problem of identifying the diffusion coefficient in matrix form in a parabolic PDE is considered. Following the idea of natural linearization, considered by Cao and Pereverzev (2006), the nonlinear inverse problem is transformed into a problem of solving an operator equation where the operator involved is linear. Solving the linear operator equation turns out to be an ill-posed problem. The method of Tikhonov regularization is employed for obtaining stable approximations and its finite-dimensional analysis is done based on the Galerkin method, for which an orthogonal projection on the space of matrices with entries from L 2 ⁢ ( Ω ) L^{2}(\Omega) is defined. Since the error estimates in Tikhonov regularization method rely heavily on the adjoint operator, an explicit representation of adjoint of the linear operator involved is obtained. For choosing the regularizing parameter, the adaptive technique is employed in order to obtain order optimal rate of convergence. For the relaxed noisy data, we describe a procedure for obtaining a smoothed version so as to obtain the error estimates. Numerical experiments are carried out for a few illustrative examples.

Methodology improvements for three-dimensional UAV-based travel-time acoustic atmospheric tomography

This paper describes a method for measuring continuous, three-dimensional temperature and wind velocity patterns in the Atmospheric Surface Layer (ASL) using Unmanned Aerial Vehicle-Based Acoustic Atmospheric Tomography (UBAAT). An Unmanned Aerial Vehicle (UAV) is flown over an array of microphones on the ground. The travel-time for sound rays between the UAV and each microphone is used to reconstruct 3D temperature and wind velocity fields, with the continuous motion of the UAV generating far more ray paths over much greater volumes of atmosphere than can be obtained using static speakers and microphones. Significant improvements over previous UBAAT techniques include the use of a synthetic tone rather than the natural sound generated by the UAV, use of vertical temperature and wind profiles to improve modelling of sharp changes near the ground, normalization of observations to incorporate weighted least squares techniques within Tikhonov regularization, and normalization of the model matrix to reduce bias in estimating modelling parameters when using Tikhonov regularization. This is the first case where UBAAT has been performed in three dimensions and also compared with independent temperature and wind velocity measurements. A summary of the results of simulation studies and trials results is provided, which shows that UBAAT can estimate three-dimensional temperature and wind velocity fields in the ASL with useful accuracy (approximately 1°C for temperature and 1 m/s for wind speed).