Abstract:In 2005, Felus and Schaffrin discussed the problem of a Structured Errors-in-Variables (EIV) Model in the context of a parameter adjustment for a classical similarity transformation. Their proposal, however, to perform a Total Least-Squares (TLS) adjustment, followed by a Cadzow step to imprint the proper structure, would not always guarantee the identity of this solution with the optimal Structured TLS solution, particularly in view of the residuals. Here, an attempt will be made to modify the Cadzow step in order to generate the optimal solution with the desired structure as it would, for instance, also result from a traditional LS-adjustment within an iteratively linearized GaussHelmert Model (GHM). Incidentally, this solution coincides with the (properly) Weighted TLS solution which does not need a Cadzow step.
SUMMARY The geological setting of southwestern Oklahoma and northeastern Texas is an ideal example of an aulacogen, the result of the tectonic evolution of a failed rift of the North American continent during the Palaeozoic era (540–360 Ma). The Wichita Province forms the uplifted basement portion of this Southern Oklahoma Aulacogen (SOA). The major fault zones to its north and south are clearly evident in gravity gradient maps produced by the recently constructed Earth Gravitational Model 2008 (EGM2008). Fault parameters, such as the dip angle, location and density contrasts have been estimated from profiles of seismic data and local gravimetry in the 1990s. On the other hand, gravitational gradients that are derived from EGM2008 and then combined to form the differential field curvature are particularly indicative of linear structures such as dip-slip faults. They are used here exclusively, that is, without additional geophysical constraints, in an optimal, least-squares estimation based on the Monte Carlo technique of simulated annealing to determine dip angle and location parameters of the major faults that border the Wichita Uplift region. Results show that these faults have small dip angles, in basic agreement with the low-angle faults inferred from seismic studies. The EGM2008 gradients also appear in some cases to provide an improved map of the major faults in the region, thus offering a strong constraint on their location.
The inverse problem of estimating parameters (e.g., size, depth) of subsurface structures can be considered as an optimization problem where the parameters of a constructed forward model are estimated from observations collected on or above the Earth's surface by minimizing the difference between the predicted model and the observations. Traditional solutions based on gradient-based approaches applied to nonlinear and non-unique problems basically depend on the initial conditions and may not always converge to the global minimum of the cost function if the starting model is far away from the true model. Alternatives to these straightforward approaches are innovative methods such as random search techniques that operate directly on the nonlinear models. This study compares a Monte-Carlo optimization method called Simulated Annealing (SA) to the Least-Squares Solution (LESS) within the general Gauss-Helmert formulation to estimate the parameters of a dip-slip fault from gravity gradient measurements as might be collected on profiles surveyed by an airborne system. It is shown that the SA algorithm is a more robust technique with respect to initial conditions in that it proceeds more comprehensively in parameter space and converges to their true values and thus the global minimum of the cost function. The SA algorithm is able to estimate the parameters of the fault as well as or better than LESS, and in the presence of significant background geologic and observation noise.K e y w o r d s :
In recent years, ozone is a disinfectant frequently used for enhancing water quality. However, since its short half-life and relatively low solubility in water, it is not possible to ensure its presence continuously in distribution system. Hence, ozone can only be used as a primary disinfectant. After ozonization, in order to complete the disinfection process in the distribution system, a secondary disinfectant like chlorine, chlorodioxide and monochloramine must be used. As well as odor in water, and inorganic materials such as iron and manganese, ozone effectively removes organic matter, drug residues and pesticides mixed into the water for any reason by oxidation from media. When used appropriately and in the correct amount ozone is an efficient chemical and microbiological disinfectant. However, when ozone is used in unfavorable conditions and/or in large quantities, it can cause the production of unwanted by-products.Ozone shows its disinfection effect through oxidation properties and high reactivity. The effect mechanism of ozone in water is depended on many parameters, mainly on pH. While ozone effects in molecular form at acidic conditions, hydroxyl radical forms are dominant in much higher pH values. By-products formed by both of the mechanisms are different from each other. ÖZET
Reliability has been quantified in a simple Gauss–Markov model (GMM) by Baarda (1968) for the application to geodetic networks as the potential to detect outliers – with a specified significance and power – by testing the least-squares residuals for their zero expectation property after an adjustment assuming “no outliers”. It was shown that, under homoscedastic conditions, the so-called “redundancy numbers” could very well serve as indicators for the “local reliability” of an (individual) observation. In contrast, the maximum effect of any undetectible outlier on the estimated parameters would indicate “global reliability”. This concept had been extended successfully to the case of correlated observations by Schaffrin (1997) quite a while ago. However, no attempt has been made so far to extend Baarda’s results to the (homoscedastic) errors-in-variables (EIV) model for which Golub and van Loan (1980) had found their – now famous – algorithm to generate the total least-squares (TLS) solution, together with all the residuals. More recently, this algorithm has been generalized by Schaffrin and Wieser (2008) to the case where a truly – not just elementwise –weighted TLS solution can be computed when the covariance matrix has the structure of a Kronecker–Zehfuss product. Here, an attempt will be made to define reliability measures within such an EIV-model, in analogy to Baarda’s original approach.
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