On relaxation-spectrum estimation for decades of data: accuracy and sampling-localization considerations
On relaxation-spectrum estimation for decades of data: accuracy and sampling-localization considerations
复制标题
关于数十年数据的松弛谱估计：准确性和采样定位考虑因素
DOI：
10.1088/0266-5611/16/5/324
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发表时间：
2000
期刊：
影响因子：
2.1
通讯作者：
J. Macdonald
中科院分区：
文献类型：
--
作者：
J. Macdonald
在使用电学和流变学现象和过程的弛豫谱模型进行数据反演时，一个重要的目标是获得与所研究的现象或过程相关的松弛时间(DRT)分布的全面和准确的近似。对这样的DRT的估计提出了理论和实验上的挑战。在后一种情况下，通常需要测量几十年的频率响应数据，然后才能收集足够的信息，以便对所需频谱的结构进行适当的估计。在理论层面上，人们必须解决弛豫谱估计的逆问题，以及经常需要处理数十年的数据。本文考察了这两个方面的一些后果。为了解决这一问题，使用具有可变β自由参数和可变正交权重的复非线性最小二乘方法对从KWW连续DRT函数(其τ0参数的值为0.5时)获得的准确(或噪声)宽范围的频率响应数据集进行了数值反演，当不需要为了提高分辨率而牺牲反演精度时，该方法优于当前类型数据的Tikhonov正则化方法。对于从通常的KWW DRT和这种在其低τ端突然被切断的DRT得出的频率响应数据的不同频率范围，研究了所得到的DRT点估计的相对误差。虽然反演是不适定的，但在可察觉的τ范围内，动态响应时间误差被发现是非常小的，但是动态响应时间范围和有限频率范围的截断的影响导致在动态响应时间τ范围末端的相对误差迅速增加，并允许分离和量化这两个限制的影响。用本方法和Tikhonov正则化方法对精确数据和含噪数据的反演结果表明了重要的分辨率差异。最后，根据频率响应反演DRT估计非常简单地计算了时间响应，并与KWW频散模型的精确拉伸指数响应函数值进行了比较。发现了极小的相对误差，但它们在短时间和长时间都显示了上述限制的影响。与傅立叶变换相比，对于大范围数据，到这里所示的时间域的变换要简单得多，更准确，也更方便。
In the inversion of data using the relaxation-spectrum model of electrical and rheological phenomena and processes, an important goal is to obtain a comprehensive and accurate approximation of the distribution of relaxation times (DRT) associated with the phenomenon or process under investigation. The estimation of such a DRT poses theoretical as well as experimental challenges. In the latter, it is often necessary to measure many decades of frequency-response data before sufficient information has been collected to allow an adequate estimate of the structure of the required spectrum to be calculated. At the theoretical level, among other things, one must solve the inverse problem of the relaxation-spectrum estimation, along with the frequent need to work with many decades of data. Some consequences of these two aspects of the problem are examined in this paper. In order to address this problem, accurate (or noisy), wide-range frequency-response data sets derived from the Kohlrausch-Williams-Watts (KWW) continuous DRT function (with a value of its β0 parameter of 0.5) are inverted numerically using a complex nonlinear least-squares method with variable τ free parameters and variable quadrature weighting, a method superior to Tikhonov regularization for data of the present type when inversion accuracy need not be sacrificed for increased resolution. The relative errors in the resulting DRT point estimates are investigated for various frequency ranges for both frequency-response data derived from the usual KWW DRT and from such a DRT which is abruptly cut off at its low τ end. Although the inversions are ill-posed, DRT errors were found to be very small over appreciable τ ranges, but the effects of cutoff of the range of the DRT and of a limited frequency range led to rapid relative-error increases at the ends of the DRT τ range, and allowed separation and quantification of the effects of these two limitations. Important resolution differences are illustrated between the results of inversions of accurate and of noisy data by the present method and by Tikhonov regularization. Finally, the temporal response was calculated very simply from frequency-response inversion DRT estimates and compared with exact stretched-exponential response function values associated with the KWW dispersion model. Extremely small relative errors were found, but they nevertheless showed the effects of the above limitations at short and long times. Transformation to the time domain of the kind illustrated here for wide-range data is far simpler, more accurate, and more convenient than Fourier transformation.