Iterative Solution of Weighted Linear Least Squares Problems

In this report we show that the iterated regularization scheme due to Riley and Golub, sometimes also called the iterated Tikhonov regularization, can be generalized to damped least squares problems where the weights matrix D is not necessarily the identity but a general symmetric and positive definite matrix. We show that the iterative scheme approaches the same point as the unique solutions of the regularized problem, when the regularization parameter goes to 0. Furthermore this point can be characterized as the solution of a weighted minimum Euclidean norm problem. Finally several numerical experiments were performed in the field of rigid multibody dynamics supporting the theoretical claims.

Determination of Groundwater Surface using Damped Least-Squares Inversion in the Bekasap Field, Riau

Groundwater is a primary water source for the sustainability of human life. Groundwater is located in the subsurface area in the saturated zone called aquifer. The presence of an aquifer can be identified through a geophysical survey by determining the upper boundary of the aquifer called the groundwater table. DC Resistivity geoelectrical method is one of the geophysical measurements which is effective to be used to determine the depth of the water table. Measurements were performed using the Wenner electrode configuration in Bekasap to attain preferable depth resolution. The process of measurement data modelling yields rms error. In order to reduce the rms error, damped least-squares is applied into the inversion solution. This process will improve the model parameter iteratively until the minimum rms error is obtained. The damped least-squares modeling was tested on three synthetic models which have Resistivity variation. Furthermore, the damped least-squares was applied on the observed data at Bekasap. From the processing and modeling using damped least-squares, the depth of the groundwater table and aquifer can be obtained.

Inversion of Bottom Hole Temperatures for Gradient Determination by the Damped Least Squares Method for Noise Attenuation

RESUMONeste trabalho foram utilizados dados BHT - Bottom Hole Temperature ou temperatura de fundo de poço na inversão de dados com o objetivo de obter a distribuição 1-D do gradiente geotérmico. Antes da inversão propriamente dita, foi utilizado o método de correção de Horner, para determinar a temperatura correta da formação. A inversão foi realizada em um modelo sintético inspirado em dados reais do Campo de Pineview (Utah, EUA), no caso, com o objetivo de obter gradientes geotérmicos de nove formações utilizando dados BHT de 32 poços. A matriz do problema geotérmico contém os elementos , ou seja, a espessura da i-ésima camada perfilada no j-ésimo poço. O método dos mínimos quadrados foi utilizado, e devido à existência de ruído foi necessário o amortecimento. A implementação numérica da inversão, ou seja, a determinação do operador inverso ou foi através da decomposição em valores singulares. As inversões iniciais não geraram resultados satisfatórios, melhorando bastante com a introdução do amortecimento. A melhoria dos resultados é explicada quantitativamente pelo fato do número de condição da matriz a ser inversa reduziu bastante com a utilização do amortecimento. Por seu turno, o amortecimento demanda a escolha de um parâmetro ótimo, sendo que foi utilizada a curva L para esse fim. Palavras-chaves: problemas inversos; gradiente geotérmico, bottom hole temperature. ABSTRACTThis study consists in obtain the 1-D distribution of the geothermal gradient from the inversion of Bottom Hole Temperature (BHT) data. Before the inversion procedure, Horner correction method was used to determine the correct formation temperature. The inversion was performed in a synthetic model based on real data from Pineview Field (Utah, USA), in this case, to obtain geothermal gradients from nine formations using BHT data from 32 wells. The matrix of the geothermal problem contains the elements , i.e., the thickness of the i-th layer logged in the j-th well. The least squares method was used, and, because of the occurrence of noise, damping was required. The numerical implementation of the inversion, i.e., the determination of the inverse operator or was performed by singular value decomposition. Initial inversions did not produce satisfactory results, but they significantly improved with the introduction of damping. The improvement of the results is quantitatively explained by the fact that the condition number of the matrix to be inverted greatly reduced with the use of the damping. In turn, damping requires the choice of an optimal parameter, and the L-curve was used for this purpose.Keywords: inverse problems; geothermal gradient, bottom hole temperature.