三維平面回歸分析圖在氬氬定年法之應用

Abstract

理想的鉀氬同位素衰變狀態下，良好封閉的氬同位素系統內36Ar、39Ar、40Ar成份關係可以用平面關係式（40Ar=α36Ar+β39Ar）表示，每個資料點均為三維座標系統(x=36Ar,y=39Ar,z= 40Ar)上的一個點，將其回歸平面，回歸參數α、β可分別代表同位素初始值與衰變源參數值。回歸平面的方法，首先考慮所有資料點的變異數與相關係數為已知，在假設隨機誤差的機率分布是屬於常態誤差分布的情況下，依照標準的統計方法，以最大概似估計建構回歸模型(maximum likelihood)，並藉由牛頓羅夫森法(Netwton-Raphson method) 迭代反覆求解找出最佳化的回歸參數，進一步推估可靠的年代值；另一方面，以氬氬定年法經常使用的MSWD 統計值(mean square weighted deviate)進行評估回歸適合度的檢測。總而言之，當坐落在相同平面上的資料點群，則代表一個完整封閉之同位素系統。 為了使運算過程有效率且容易使用，本研究工作利用Matlab程式語言處理矩陣代數的數值運算，透過較為精簡的程式語言來表達廣大的聯立方程組與資料陣列，並且建立一套兼顧視覺與計算之互動式使者用者圖形介面，命名為ArArPLOT，這個介面以讓資料處理過程更加簡便為出發點，提供下列三個功能：第一、提供許多三維展現與選擇資料點的工具，可清楚地分辨氬同位素的空間關係，分辨標本內可能存在同位素系統，並求得其端源成份，第二、處理三維平面回歸的運算，提供年代與同位素初始值之估計，第三、繪製氬氬定年相關之作圖。為驗證此一介面的實用性，本研究工作分析處理印度東北Rajmahal -Sylhet火山岩體標本定年，並藉以比較三維平面回歸作圖法與二維線性回歸作圖法之優劣，所得結果顯示Rajmahal -Sylhet火山岩體的大規模噴發年代介於119-116Ma，屬Kerguelen Plume演化早期之產物。 有別於二維同位素對比法在分析處理含有三個變數值時，所進行的投影技術與計算參數誤差傳遞之煩瑣，結合三維平面回歸分析方法，與使用ArArPLOT處理分析過程提供了以下好處：第一、以同位素含量作圖與運算，避免了同位素對比法中誤差的傳遞累積，因此提高了誤差計算之精確度；第二、允許明顯呈現資料點群的空間關係，了解標本內氬同位素系統的變異；第三、提高解析礦物岩石標本內部的不同氬同位素系統分布情況與性質之可信度。換言之，本研究的方法在解釋氬氬定年分析結果更加容易，因此，對於利用同位素分析、探討歷史複雜岩體的研究工作，提供一項有利的解析方法。

In an ideal K-Ar isotopic systematics, the measured concentrations of three argon isotope (36Ar, 39Ar, 40Ar) would satisfy a plane equation: 40Ar =α36Ar + β39Ar. In other words, the sample data should lie close to a plane in three-dimensional, XYZ space. If analytical errors are responsible for the scatter of data points on the 36Ar-39Ar-40Ar isotope correlation plot, the regression plane can be fitted using two steps. First, we employ matrix algebra to lay out the method of maximum likelihood for fitting a plane equation between any number of variables in the equation, all subject to analytical errors with known variances and covariances, although these may vary among the data points. In addition, the well-known Newton-Raphson method can subsequently be adopted in optimization. Secondly, in order to examine the maximum likelihood method of regression and to determine the standard errors of the parameters of a best-fit plane, the efficient-accurate methods can access standard errors, goodness-of-fit parameter, and MSWD (mean square weighted deviate) value. If the data are well fitted to the regression plane, the estimates of regression parameters α and β, will lead to estimates of the trapped argon composition and the radiometric age for sample. To publish the above efficient algorithms and easy-to-use method, the present project designs a user-friendly Matlab-based graphical program, namely ArArPLOT, to perform the methods in age calculation and their corresponding diagrams (isotope abundance diagrams, isotope correlations diagrams) for 40Ar/39Ar geochronological studies. The program is portable between computer operating systems supported by Matlab. The major benefit of this program is capable of clarifying whether the data points from a single sample representing different isotopic systematics, using plane fitting methods in XYZ space, as mentioned above. In contrast to 2-D isotope correlation diagrams which utilize projection technique, this 3-D plane fitting program exhibits the following advantages: (1) it allows to clearly display the spatial relation of data points, and to improve the precision of error calculations; (2) it can be used as means in clarifying different isotopic systematics in sample; (3) the interpretation of isotope abundance diagrams becomes much easier because of better visualization of data distribution in 3-D space. In summary, the current project successfully designs a user-friendly program which is powerful in data analysis in 40Ar/39Ar geochronology and is potentially useful to data analyses of other isotopes in compositional space.