Parâmetro de regularização em problemas inversos: estudo numérico com a transformada de Radon

In general, an inverse problem corresponds to find a value of an element x in a suitable vector space, given a vector y measuring it, in some sense. When we discretize the problem, it usually boils down to solve an equation system f(x) = y, where f : U Rm ! Rn represents the step function in any domain U of the appropriate Rm. As a general rule, we arrive to an ill-posed problem. The resolution of inverse problems has been widely researched along the last decades, because many problems in science and industry consist in determining unknowns that we try to know, by observing its effects under certain indirect measures. Our general subject of this dissertation is the choice of Tykhonov´s regulaziration parameter of a poorly conditioned linear problem, as we are going to discuss on chapter 1 of this dissertation, focusing on the three most popular methods in nowadays literature of the area. Our more specific focus in this dissertation consists in the simulations reported on chapter 2, aiming to compare the performance of the three methods in the recuperation of images measured with the Radon transform, perturbed by the addition of gaussian i.i.d. noise. We choosed a difference operator as regularizer of the problem. The contribution we try to make, in this dissertation, mainly consists on the discussion of numerical simulations we execute, as is exposed in Chapter 2. We understand that the meaning of this dissertation lays much more on the questions which it raises than on saying something definitive about the subject. Partly, for beeing based on numerical experiments with no new mathematical results associated to it, partly for being about numerical experiments made with a single operator. On the other hand, we got some observations which seemed to us interesting on the simulations performed, considered the literature of the area. In special, we highlight observations we resume, at the conclusion of this work, about the different vocations of methods like GCV and L-curve and, also, about the optimal parameters tendency observed in the L-curve method of grouping themselves in a small gap, strongly correlated with the behavior of the generalized singular value decomposition curve of the involved operators, under reasonably broad regularity conditions in the images to be recovered

Estudo da atividade sísmica em São Caetano-PE em 2007

In this dissertation we studied the seismic activity in the São Caetano county, Pernambuco State, Northeastern Brazil, located near the Pernambuco Lineament. The Pernambuco Lineament is a one of Neoproterozoic continental-scale shear zones that deforms the Borborema province. The seismicity estudied occurred in a NE trending branch of Pernambuco Lineament. The seismic activity in São Caetano started in 2006 and in May 20th, 2006 a 4,0 mb earthquake hit there. This was the largest earthquake ever reported in Pernambuco State. This dissertation is the result of a campaign done in the period from Februay 1th 2007 to July 31 th 2007. In this campaign up to nine three-component digital seismographic stations were deployed and the collected data was used to determine hypocenters and focal mechanism. A total of 214 earthquakes, recorded at least by three stations, were analyzed. To determine hypocenters and time origin the HYPO71 program was used assuming a half-space model with parameters : VP (P-wave velocity) equal to 5.90 km/s and the ratio VP/VS 1.70, where VS is the S-wave velocity. The earthquakes hypocentral distribution was approximately 4 km long and agrees with the NE-SW direction of the Pernambuco Lineamento branch. Hypocentres depth range from 2 to 8 km. The composed focal mechanism was made from a group of 14 selected earthquakes. We try firstly to find the fault plane solution matching the polarity distribution at stations, using the FPFIT program. The result was 43 deg ± 15 deg for strike, 59 deg ± 9 deg for dip and -142 deg ± 15 deg for rake. We also fitted a plane using the hypocentral distribution to obtain the dip and azimuth of the hypocentral distribution. The results obtained by this fit were 58 deg for the azimuth, 55 deg for the dip and -155 deg for rake. This result shows a mechanism of a strike-slip dextral fault with a normal component. This dissertation shows, once more, that there is a good correlation between the seismic activity and geological features in the region near the Pernambuco Lineament and its NE branches

Estudoda sismicidade na região de Sobral - CE em 2008

The town of Sobral is located at the northwest part of the Ceará State, 250 km away from its capital, Fortaleza. In January 2008, an intense seismic activity began near Sobral with one event with magnitude 4,2mb on May 21. Since the start of its seismic activity, all events were recorded by the SBBR station (located on EMBRAPA Caprinos Farm), which operates in the region since August 2007. After this event, monitoring the seismic activity was carried out with the deployment of a local three component digital seismographic network, from June 5, 2008 until September 24. Initially, this network was composed of six seismographic stations. Later additional five stations were deployed until August 26 2008. This local network detected approximately 2,800 earthquakes. In this study we analyzed 581 earthquakes recorded by at least three stations for hypocentral and focal mechanism determination, and to contribute to a better explanation of the seismicity which in this region. To determine the hypocenters, we used a half-space model, with vP = 6,00 km/s and vP/vS = 1,71. From the hypocentral determination, it was revealed an active seismic zone with depth ranging between 1 and 8 km, 6 km long in E - W direction. The determination of fault planes and focal mechanism was obtained using the programs FPFIT and PLAN, which allowed comparison between their respective results in order to obtain more accurate results. A set of 24 earthquakes were selected to determine fault using PLAN planes and focal mechanisms using FPFIT. With the aid of detailed map of hypocenters this set, it was possible to identify three structures. Therefore, the set of 24 earthquakes were divided into three subsets. The type of mechanism was predominantly strike-slip with a dextral direction. Although the region has two tectonic structures near the site of the study area: the Café- Ipueiras Fault (normal fault) and the Sobral-Pedro II Lineament (dextral strike-slip fault) it was not possible to correlate the seismicity founded with those structures