Linear approximation

About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.

A high-resolution algorithm for complex spectrum search

Abstract: The principal aim of this work is to estimate, or to approximate, the complex k-space spectrum of the wave field arriving on a linear array. First, using linear approximation, the location-dependent effect of the wave field magnitudes is modeled as an extra “loss” factor in the complex spectral variable. This complex spectrum model may provide a better description of the physical process and require less sensor elements than the real spectrum model because of the additional degree of freedom provided by the “loss” factor. A high-resolution algorithm combining the singular value decomposition method and the eigen-matrix pencil method is then employed to find the complex spectra representing the incoming real spectrum and the location dependent factors of multipath and multimode arrivals. Five key features (noise immunity, robustness, resolution, accuracy, and physical insight) of the proposed algorithm are studied using numerical examples.

Accuracy of stochastic perturbuation methods: the case of asset pricing models

Abstract: This paper investigates the accuracy of a perturbation method in approximating the solution to stochastic equilibrium models under rational expectations. As a benchmark model, we use a version of asset pricing models proposed by Burnside [1988] which admits a closed-form solution while not making the assumptions of certainty equivalence. We then check the accuracy of perturbation methods -extended to a stochastic environment- against the closed form solution. Second an especially fourth order expansions are then found to be more efficient than standard linear approximation, as they are able to account for higher order moments of the distribution.

Statistical bias and variance for the regularized-inverse problem: application to Space-based atmospheric CO2 retrievals

Abstract: Remote sensing of the atmosphere is typically achieved through measurements that are high-resolution radiance spectra. In this article, our goal is to characterize the first-moment and second-moment properties of the errors obtained when solving the regularized inverse problem associated with space-based atmospheric CO2 retrievals, specifically for the dry air mole fraction of CO2 in a column of the atmosphere. The problem of estimating (or retrieving) state variables is usually ill posed, leading to a solution based on regularization that is often called Optimal Estimation (OE). The difference between the estimated state and the true state is defined to be the retrieval error; error analysis for OE uses a linear approximation to the forward model, resulting in a calculation where the first moment of the retrieval error (the bias) is identically zero. This is inherently unrealistic and not seen in real or simulated retrievals. Nonzero bias is expected since the forward model of radiative transfer is strongly nonlinear in the atmospheric state. In this article, we extend and improve OE's error analysis based on a first-order, multivariate Taylor series expansion, by inducing the second-order terms in the expansion. Specifically, we approximate the bias through the second derivative of the forward model, which results in a formula involving the Hessian array. We propose a stable estimate of it, from which we obtain a second-order expression for the bias and the mean square prediction error of the retrieval.

Initial Velocity Models for Full Waveform Inversion

Abstract: Full Waveform Inversion of seismic data requires the solution of a strongly non-linear optimization problem. Usually iterative gradient based methods are employed for the solution. These methods is based on the Born approximation and assumes that the derivative of the error function with respect to the velocity can be adequately represented by a linear approximation in each iteration step. This usually requires that the initial velocity model is close to the initial model to avoid cycle-skipping which cannot be described by the Born approximation. A common approach has been to use an initial velocity model based on ray tomography and depth migration which produces smooth models with kinematic properties similar to the true model. We suggest to replace ray tomography with Wave Equation Migration Velocity Analysis (WEMVA) based on a differential semblance error function. This method yields smooth velocity models comparable with models obtained from ray tomography, and in many cases requires only relatively simple one-dimensional models as a starting point. The advantage of this approach is that models obtained with WEMVA can then be used as initial models for Full Waveform Inversion to obtain a potentially automatic two-stage work flow for estimating accurate velocity models.