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In this paper, we present a modified version of the Pereyra–Rosen algorithm to the solution of ill-conditioned linear and nonlinear inverse problems arising in gravimetry. We perform a sensitivity analysis of the solution to the two main tuning parameters of this algorithm, comparing its solution with LSQR and the truncated SVD. First, we show the application to a general-purpose linear system whose system matrix has been created by conditional simulation and whose solution is known and to a synthetic 1D gravimetric problem for two different geological set-ups (smooth Gaussian and blocky geophysical anomalies) in the noise-free and noisy cases. The Pereyra–Rosen algorithm provides very good results using a reduced number of column vectors of the system matrix. We finally show the fast inversion of a real gravity profile in the Atacama Desert (north Chile). The algorithm is well suited for the solution of large-scale ill-conditioned linear problems as the ones encountered in the fine discretization of continuous linear inverse problems in several dimensions and in the iterative linearization of nonlinear inverse problems in several fields of geosciences.

We present the mathematical deduction and properties of the mean traveltime curves for homogeneous elliptical anisotropic media. These curves generalize their isotropic counterparts which have been introduced in the past as a simple data quality analysis technique at the pre-inversion stage for 2D transmission experiments, allowing the inference of prior velocity models to gain stability at the tomographic inversion. Also, the anisotropy parameters (maximum velocity, anisotropic direction and ratio) are shown to affect the shape of these curves. The degree of asymmetry of the anisotropic mean traveltime curves (displacement of the mean time and standard deviation minima from the middle of the gathering line) is related to the direction of anisotropy which can then be visually estimated. Least squares’ fitting of the anisotropic theoretical models to their experimental counterparts is an effective method to estimate at the pre-inversion stage a macroscopic elliptical anisotropic velocity model, valid at the scale of the experiment, and able to match the experimental mean traveltime distribution.

Transmission tomography methods show a great sensibility to data variability, which eventually includes data errors, often present in field experiments. Local optimization methods, traditionally used to solve this inverse problem, are very sensitive to these difficulties, failing to converge properly in the presence of spurious data. Regularization methods partially cope with these weaknesses, damping the instabilities.

In this article we present a geostatistical approach to the transmission tomographic inverse problem, which is based on consideration of the inverse problem variables (velocity and traveltime errors) as regionalized variables (R.V.). Their structural analysis provides us with a new method to study the geophysical anisotropy of the rock, an important source of a priori information in order to design the anisotropic corrections. The underlying idea is that the geophysical structure can be deduced from the spatial structure of the regionalized variables which result from solving the tomographic problem with an isotropic algorithm. Also, the application of the structural analysis technique to the anisotropic corrected velocity field allows us to characterize the reliability of these corrections (model quality analysis). Geostatistical formalism also provides us with different techniques (parametric and non-parametric) to estimate and even simulate the velocity in the areas where this field has been considered anomalous based on field studies and on geophysical and statistical criteria. The kriging acts as a low-pass smoothing filter for the anomalous model parameters (velocities), but is not a substitute for an adequate filtering of the outliers before the inversion. This methodology opens the possibility of considering the inverse problem variables as stochastic processes, an important feature in cases where the tomogram is to be used as a tool of assessment to quantify the rock heterogeneities.

On site concrete strength assessment via non-destructive techniques (NDT) is one of the main challenges that NDT is faced with. Along the past 50 years a huge number of both laboratory and onsite experimental programs have been carried out and a variety of models have been proposed in order to correlate concrete strength with rebound hammer measurements. It is commonly agreed that none of these models is able to accurately predict the concrete strength with enough accuracy for using the assessed value for further structural computations. The large variety of concrete compositions and the fact that other influent factors may exist are the major arguments advance in order to explain the failure of modeling and assessment strategies. Based on (a) an in-depth literature review enabling a meta-analysis of existing models and, (b) Monte-Carlo simulations used to simulate what happens when rebound measurements are carried on onsite, and used to assess the concrete strength after a calibration stage, we will point first the previously unnoticed consistency between existing models. This consistency will be then explained by the trade-off which exists between the unknown model parameters that must be identified. In the context of real measurements, i.e., those concerning heterogeneous materials with uncontrolled factors and error measurements, it will be shown that this trade-off systematically exists. This fact is inherent to any inverse or identification problem. Its magnitude depends on several factors, among which the quality of the NDT measurement and the number of control cores used for model calibration play a key role. Its consequences are (a) a limited ability to assess strength and, (b) the impossibility to use the model in different context. Some proposals are finally presented in order to obtain a more robust assessment for the concrete strength. They combine guidelines about the measurement error estimation, a calibration approach which will be less dependent on the information provided by a limited number of cores, and the combination of several measurements in order to reduce the effect of uncontrolled factors.