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Full waveform inversion (FWI) is a tool for the inversion of seismic data. There are several sources of uncertainty in the results provided by FWI. The quantification of such uncertainties has been studied through the resolution matrix (Res), which rests on a quadratic approximation that interprets the Hessian matrix as the posterior covariance matrix. Despite efforts in the use of Res, there is no published analysis of the uncertainties contained in the full correlation matrix, (R). Our approach leads to build the full R matrix, which, at the end of the day, is the final quantity that includes all the information associated with uncertainties. We focused on uncertainties related to variation in the starting models of the FWI, and thus propose a method to study the full R matrix, which is-called the Density of Correlation Map, D. By using the D map, we found that the highest uncertainty zones in the FWI inverted model are near the sources, the model boundaries, and the interfaces. We argue that D can be a complement for the study and estimation of uncertainties in FWI. RESUMEN a inversión por campo de onda completa (FWI) es una herramienta para la inversión de datos sísmicos. Hay varias fuentes de incertidumbre en los resultados proporcionados por FWI. La cuantificación de tales incertidumbres se ha estudiado a través de la matriz de resolución (Res), que se basa en una aproximación cuadrática que interpreta la matriz Hessiana como la matriz de covarianza posterior. A pesar de los esfuerzos en el uso de Res, el análisis de las incertidumbres contenidas en la matriz de correlación completa (R) no existe hasta ahora. Nosotros empleamos varias realizaciones para estimar las incertidumbres de los resultados de FWI. Nuestro enfoque lleva a construir la matriz R completa, que es al final, la cantidad que encierra toda la información asociada a las incertidumbres. Nos enfocamos en las incertidumbres relacionadas con las variaciones en los modelos iniciales de FWI, y proponemos un método para estudiar la matriz R completa, denominado Mapa de Densidad de Correlación, D. Al utilizar el mapa D, encontramos que las zonas de más alta incertidumbre en los modelos invertidos están cerca de las fuentes sísmicas, de las interfaces y de las fronteras artificiales. Destacamos que D puede servir como un complemento para el estudio y la estimación de incertidumbres en FWI.

In this work we present a formalism that intends to solve the problem of modeling wave propagation in the context of seismic inversion. The formalism is based on the linear perturbation theory of Cauchy's equations. Based on the foregoing, we derived an equivalent Helmholtz equation for the propagation of waves in a variable density media. Then, we defined a solution, by using the boundary conditions on a half plane. This solution is of an integral nature and resembles expansion in a Neumann series. We implemented the solution of the first terms in the series, considering only the incident wavefield and neglecting the reflections. We show how this approximation works in different media that include lateral in homogeneities in the velocity. The method presented hereunder is intended as a first step in the modelling process for the full wavefield, to be used in seismic inversion methods, Full Waveform Inversion, for example. RESUMEN En esta investigación presentamos un formalismo que pretende contribuir al modelado de la propagación de ondas en el contexto de la inversión sísmica. El formalismo está basado en la teoría de perturbaciones lineales a las ecuaciones de Cauchy. Basados en este procedimiento derivamos una versión de la ecuación de Helmholtz que describe la propagación de ondas en un medio con densidad variable. Luego hallamos una solución en la cual se emplean condiciones de frontera de un plano semi infinito. Tal solución es expresada en forma de integral y recuerda la expansión en series de Neumann. Nosotros implementamos la solución del primer término de la serie, que considera únicamente el campo de onda incidente, sin considerar las reflexiones de onda. Mostramos que esta aproximación funciona en diferentes medios que incluyen variaciones in-homogeneidades laterales en el perfil de velocidad. Este método es presentado como un primer paso en el proceso de modelado del campo de onda completo el cual puede ser usado en métodos de inversión sísmica tales como "Inversión de onda completa", Full Waveform Inversion, (FWI).

The primary difficulty in measuring dynamical masses of galaxy clusters from galaxy data lies in the separation between true cluster members from interloping galaxies along the line of sight. We study the impact of membership contamination and incompleteness on cluster mass estimates obtained with 25 commonly used techniques applied to nearly 1000 mock clusters. We show that all methods overestimate or underestimate cluster masses when applied to contaminated or incomplete galaxy samples respectively. This appears to be the main source of the intrinsic scatter in the mass scaling relation. Applying corrections based on a prior knowledge of contamination and incompleteness can reduce the scatter to the level of shot noise expected for poorly sampled clusters. We establish an empirical model quantifying the effect of imperfect membership on cluster mass estimation and discuss its universal and method-dependent features. We find that both imperfect membership and the response of the mass estimators depend on cluster mass, effectively causing a flattening of the estimated - true mass relation. Imperfect membership thus alters cluster counts determined from spectroscopic surveys, hence the cosmological parameters that depend on such counts., Comment: 18 pages, 11 figures, 3tables; accepted for publication in MNRAS

This article is the second in a series in which we perform an extensive comparison of various galaxy-based cluster mass estimation techniques that utilise the positions, velocities and colours of galaxies. Our aim is to quantify the scatter, systematic bias and completeness of cluster masses derived from a diverse set of 25 galaxy-based methods using two contrasting mock galaxy catalogues based on a sophisticated halo occupation model and a semi-analytic model. Analysing 968 clusters, we find a wide range in the RMS errors in log M200c delivered by the different methods (0.18 to 1.08 dex, i.e., a factor of ~1.5 to 12), with abundance matching and richness methods providing the best results, irrespective of the input model assumptions. In addition, certain methods produce a significant number of catastrophic cases where the mass is under- or over-estimated by a factor greater than 10. Given the steeply falling high-mass end of the cluster mass function, we recommend that richness or abundance matching-based methods are used in conjunction with these methods as a sanity check for studies selecting high mass clusters. We see a stronger correlation of the recovered to input number of galaxies for both catalogues in comparison with the group/cluster mass, however, this does not guarantee that the correct member galaxies are being selected. We do not observe significantly higher scatter for either mock galaxy catalogues. Our results have implications for cosmological analyses that utilise the masses, richnesses, or abundances of clusters, which have different uncertainties when different methods are used., 25 pages, 19 figures, 7 tables. Accepted for publication in MNRAS

This paper is the first in a series in which we perform an extensive comparison of various galaxy-based cluster mass estimation techniques that utilise the positions, velocities and colours of galaxies. Our primary aim is to test the performance of these cluster mass estimation techniques on a diverse set of models that will increase in complexity. We begin by providing participating methods with data from a simple model that delivers idealised clusters, enabling us to quantify the underlying scatter intrinsic to these mass estimation techniques. The mock catalogue is based on a Halo Occupation Distribution (HOD) model that assumes spherical Navarro, Frenk and White (NFW) haloes truncated at R_200, with no substructure nor colour segregation, and with isotropic, isothermal Maxwellian velocities. We find that, above 10^14 M_solar, recovered cluster masses are correlated with the true underlying cluster mass with an intrinsic scatter of typically a factor of two. Below 10^14 M_solar, the scatter rises as the number of member galaxies drops and rapidly approaches an order of magnitude. We find that richness-based methods deliver the lowest scatter, but it is not clear whether such accuracy may simply be the result of using an over-simplistic model to populate the galaxies in their haloes. Even when given the true cluster membership, large scatter is observed for the majority non-richness-based approaches, suggesting that mass reconstruction with a low number of dynamical tracers is inherently problematic., 25 pages, 15 figures, 5 tables. Accepted for publication in MNRAS