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本文研究了一种利用三次 B 条样对散射的、噪声数据作多平滑拟合的新方法。确定了最佳平滑函数,以使范数 l_2达到最小。范数是由数据剩余值、一阶和二阶导数构成,它们分别表示数据的总的不适定量(misfit)、波动量和函数的粗糙度。此函数用具有等间隔节点的三次 B 条样的展开式来近似。这种解法可以从三个方面来解释。从随机的观点来看,它是容许函数之间的最大似然估算,这里采用了这样一个先验条件,即由于随机误差(白噪声),一阶、二阶导数在各处皆为零。从物理的角度看,它是张力状态下的一根细棒或一个薄板的横向移动时有限元近似问题。由于弹簧的作用,这个棒或板在横向上被拉到数据点上。从技术观点看,它是一种改进的条样-拟合算法。使导数范数极小化的附加条件可稳定展开系数的线性方程系统。
In this paper, we study a new method for smoothing the scattered and noisy data using cubic B samples. The best smoothing function is determined so that norm l_2 is minimized. The norm is made up of the data residuals, the first and second derivatives, which represent the total misfit, fluctuation, and roughness of the data, respectively. This function is approximated by a cubic B-strip expansion with equally spaced nodes. This solution can be explained from three aspects. From a stochastic point of view, it allows for maximum likelihood estimation between functions. Here a priori conditions are adopted where first and second derivatives are zero everywhere due to random errors (white noise). From a physical point of view, it is a Finite Element Approximation problem when a thin rod or a thin plate moves laterally in tension. Due to the action of the spring, this bar or plate is pulled transversely to the data point. From a technical point of view, it is an improved strip-fitting algorithm. An additional condition that minimizes the derivative norm stabilizes the system of linear equations that expand the coefficients.