基金項(xiàng)目:國(guó)家自然科學(xué)基金項(xiàng)目(41172302,40672196)
摘要:提出了N維陣的概念和常用的幾種定義和運(yùn)算,包括N維陣的加減、方括號(hào)乘法和Hadamard積,給出了其性質(zhì)及相應(yīng)的說(shuō)明;指出立體陣是N維陣的一個(gè)特例。N維陣方法的優(yōu)勢(shì)在于對(duì)多維數(shù)據(jù)表示更加簡(jiǎn)潔,理論分析較方便。最后,通過(guò)江紹拼合帶中西段CuZnAbAgSnAs元素組合異常研究和浙西地區(qū)銅多金屬礦成礦預(yù)測(cè),說(shuō)明了N維陣在實(shí)際問(wèn)題應(yīng)用中的方法及步驟。N維陣在處理地學(xué)多維數(shù)據(jù)方面有著重要的應(yīng)用前景。
關(guān)鍵詞:數(shù)學(xué)地質(zhì);N維陣;方括號(hào)乘法;Hadamard積;立體陣;元素組合;成礦預(yù)測(cè)
中圖分類號(hào):P628;O189.12文獻(xiàn)標(biāo)志碼:A
Study of Ndimensional Matrices and Its Application in Geology
SHEN Wei
(State Key Laboratory of Geological Processes and Mineral Resources, China University of
Geosciences, Beijing 100083, China)
Abstract: A new basic concept of Ndimensional matrices was presented by the cubic matrices conception. The definition and arithmetic (including addsubtract, bracket multiplication and Hadamard product) of Ndimensional matrices were studied and proved. The cubic matrices was a special case of Ndimensional matrices. The multidimensional data were expressed more brief and facility in theoretical analysis by the method of Ndimensional matrices. The geological examples, which included the research on CuZnAbAgSnAs element combination anomalies in the middlewest of JiangshanShaoxing matching belt and the metallogenic prediction of copper polymetallic ore in the western of Zhejiang, were given to illustrate the method and procedure of Ndimensional matrices in geological application.The method of Ndimensional matrices is considered as a good tool in exploration and forecast.
Key words: mathematical geology; Ndimensional matrices; bracket multiplication; Hadamard product; cubic matrices; element combination; metallogenic prediction
0引言
在地學(xué)研究中,經(jīng)常遇到多維數(shù)據(jù),例如對(duì)于許多復(fù)雜的地質(zhì)現(xiàn)象,要考慮全局范圍各個(gè)方向的平穩(wěn)性,即區(qū)別各向同性或各向異性分布規(guī)律,同時(shí)它們包含多個(gè)因變量和層次,每個(gè)因變量和層次具有不同的統(tǒng)計(jì)特征,必須用多變量與多個(gè)參數(shù)(即多維數(shù)據(jù)信息)來(lái)描述,才能全面刻畫其特征。在礦床預(yù)測(cè)研究中經(jīng)常遇到以下難題:礦質(zhì)運(yùn)移及其空間展布具有多維性;礦床演變與控礦構(gòu)造具有多層次、多階段性等[19]。目前,對(duì)于地質(zhì)模型研究常用方法有非線性邏輯回歸模型、多維對(duì)數(shù)線性模型、多維標(biāo)度法和數(shù)量化理論等[1011],但這些方法主要用于處理二維數(shù)據(jù),因此,提出和研究處理多維數(shù)據(jù)的新方法是非常必要的[12]。N維陣概念的提出和應(yīng)用,為研究多維數(shù)據(jù)提供了一個(gè)有力的工具。筆者提出了N維陣的概念、常用的幾種定義和運(yùn)算,包括N維陣的加減、方括號(hào)乘法和Hadamard積,同時(shí)研究了其性質(zhì),認(rèn)為N維陣在處理地學(xué)多維數(shù)據(jù)中有重要應(yīng)用前景。
1N維陣的定義
定義1.1稱r1×r2×…×rn的n維數(shù)組S為N維陣,S=(si1i2…in),1≤i1≤r1,1≤i2≤r2,…,1≤in≤rn。
例如,當(dāng)n=4時(shí),i4、i3分別表示橫坐標(biāo)與縱坐標(biāo)(或經(jīng)緯度),i2表示高度(或高程),si1i2i3i4表示第i1種地球化學(xué)元素(如金)在三維坐標(biāo)(i2,i3,i4)處的數(shù)值,i1=1,2,…,r1。
定義1.2設(shè)S和T均為r1×r2×…×rn的N維陣,定義S與T之和(差)為S±T=(si1i2…in±ti1i2…in) ,1≤i1≤r1,1≤i2≤r2,…,1≤in≤rn。
例如,當(dāng)n=4時(shí),si1i2i3i4+ti1i2i3i4表示兩類地質(zhì)變量數(shù)據(jù)(如物探數(shù)據(jù)和化探數(shù)據(jù))si1i2i3i4與ti1i2i3i4在三維坐標(biāo)(i2,i3,i4)處的數(shù)值之和,i1=1,2,…,r1。
定義1.3設(shè)A為m×r1階矩陣,S為r1×r2×…×rn的N維陣,定義A與S的方括號(hào)乘法T=[A][S]為一個(gè)m×r2×…×rn的N維陣,即tsi2i3…in=∑r1 k=1askski2i3…in,1≤s≤m,1≤i2≤r2,1≤i3≤r3,…,1≤in≤rn。
特別地,當(dāng)n=4時(shí),當(dāng)A等于a,為一個(gè)r1維向量時(shí),T=[A][S]為r2×r3×r4的三維陣,即ti2i3i4=∑r1 k=1akski2i3i4,1≤i2≤r2,1≤i3≤r3,1≤i4≤r4。例如,a=(a1,a2,…,ar1),其中,ak是權(quán)數(shù),∑r1 k=1ak=1,ti2i3i4表示r1種地球化學(xué)元素(如金)si1i2i3i4在三維坐標(biāo)(i2,i3,i4)處的加權(quán)數(shù)值之和,i1=1,2,…,r1。
方括號(hào)乘法的基本性質(zhì)為
[A±B][S]=[A][S]±[B][S](1)
[A][S±T]=[A][S]±[A][T](2)
式中:A、B為m×r1階矩陣。
定義1.4設(shè)S和T均為r1×r2×…×rn的N維陣,定義S與T的Hadamard積ST=(si1i2…in·ti1i2…in),1≤i1≤r1,1≤i2≤r2,…,1≤in≤rn。
Hadamard積的基本性質(zhì)為
ST=TS(3)
(ST)U=S(TU)(4)
(S+T)U=SU+TU(5)
式中:S、T、U均為r1×r2×…×rn的N維陣。
定義1.5設(shè)k為一個(gè)常數(shù),S為r1×r2×…×rn的N維陣,定義T=kS=(ksi1i2…in),為r1×r2×…×rn的N維陣,1≤i1≤r1,1≤i2≤r2,…,1≤in≤rn。
2立體陣的定義
立體陣的概念及運(yùn)算最早由Bates等在1980年提出[13],1983年Tsai在博士論文中對(duì)其進(jìn)行了初步整理,1989年中國(guó)學(xué)者韋博成在Tsai博士論文的基礎(chǔ)上進(jìn)行了系統(tǒng)總結(jié)和擴(kuò)充[14]。立體陣在非線性模型的非線性強(qiáng)度度量中有廣泛的應(yīng)用,在估計(jì)非線性模型的固有曲率和參數(shù)效應(yīng)曲率時(shí),起著關(guān)鍵的作用,在非線性回歸分析、搏弈論與經(jīng)濟(jì)學(xué)等中都有廣泛的應(yīng)用前景。實(shí)際上,立體陣是N維陣的一個(gè)特例(n=3)。
定義2.1定義n×p×q的三維數(shù)組X=(xkij)為立體陣[13],1≤k≤n,1≤i≤p,1≤j≤q。其表達(dá)式為
X=X1
X2
Xn(6)
其中Xk=xk11 … xk1q
xkp1 … xkpq
例如,j、i分別表示橫坐標(biāo)與縱坐標(biāo)(或經(jīng)緯度),xkij表示第k種地球化學(xué)元素(如金)在坐標(biāo)(i,j)處的數(shù)值,k=1,2,…,n。
定義2.2設(shè)X和Y均為n×p×q立體陣,定義X與Y之和(差)為X±Y=(xkij±ykij)[15] ,1≤k≤n,1≤i≤p,1≤j≤q。
例如,xkij+ykij表示兩類地質(zhì)變量數(shù)據(jù)(如物探數(shù)據(jù)和化探數(shù)據(jù))xkij與ykij在坐標(biāo)(i,j)處的數(shù)值之和,k=1,2,…,n。
定義2.3設(shè)A為m×n階矩陣,X為n×p×q立體陣,定義A與X的方括號(hào)乘法Y=[A][X][15],為一個(gè)m×p×q的立體陣,即ysij=∑n k=1askxkij,1≤s≤m,1≤i≤p,1≤j≤q。
特別地,當(dāng)A=a,為一個(gè)n維向量時(shí),Y=[a][X]為p×q階矩陣,即yij=∑n k=1akxkij,1≤i≤p,1≤j≤q。例如,a=(a1,a2,…,an),其中ak是權(quán)數(shù),∑n k=1ak=1,yij表示n種地球化學(xué)元素(xkij)在坐標(biāo)(i,j)處的加權(quán)數(shù)值之和,k=1,2,…,n,1≤i≤p,1≤j≤q。
方括號(hào)乘法的基本性質(zhì)(立體陣)
[C±D][X]=[C][X]±[D][X](7)
[C][X±Y]=[C][X]±[C][Y](8)
式中:X、Y為2個(gè)n×p×q立體陣;C、D為m×n階矩陣。
定義2.4設(shè)X和Y均為n×p×q立體陣,定義X與Y的Hadamard積XY=(xkijykij) [16],1≤k≤n,1≤i≤p,1≤j≤q。
Hadamard積的基本性質(zhì)(立體陣)
XY= YX(9)
(XY)Z=X(YZ)(10)
(X+Y)Z=XZ+YZ(11)
式中:Z為n×p×q立體陣。
定義2.5設(shè)k為一個(gè)常數(shù),X為n×p×q立體陣,定義Y=kX=kX1
kX2
kXn為n×p×q立體陣。
3應(yīng)用實(shí)例
3.1江紹拼合帶中西段CuZnAbAgSnAs元素組合異常研究
江紹拼合帶中西段1∶200 000地球化學(xué)元素Cu、Zn、Ab、Ag、Sn和As的數(shù)據(jù)(每個(gè)元素的數(shù)據(jù)量為5 544個(gè))可以組成6×56×99的三維數(shù)組X=(xkij),即立體陣,1≤k≤6,1≤i≤56,1≤j≤99。通過(guò)成礦及伴生元素共生組合規(guī)律研究,確定相關(guān)性較強(qiáng)的元素組合。以相關(guān)性較強(qiáng)元素的地球化學(xué)觀測(cè)數(shù)據(jù)為基礎(chǔ),計(jì)算多元素的累加指數(shù)值,定量表示和研究地球化學(xué)元素組合異常及其空間分布規(guī)律。最常用的方法是通過(guò)多元相關(guān)分析方法,確定成礦元素組合,然后用元素累加指數(shù)制作元素組合異常等值線圖(圖1)。
累加指數(shù)計(jì)算公式為
(zij)=∑6 k=1xkij ∑56 i=1∑99 j=1xkij/(56×99)=[a][X](12)
其中a=∑56 i=1∑99 j=1x1ij/(56×99)
∑56 i=1∑99 j=1x2ij/(56×99)
∑56 i=1∑99 j=1x6ij/(56×99),
1≤i≤56,1≤j≤99。
3.2浙西地區(qū)銅多金屬礦成礦預(yù)測(cè)
根據(jù)浙西地區(qū)1∶200 000地質(zhì)圖和相關(guān)礦種分布圖,在對(duì)區(qū)域地質(zhì)背景深入分析的基礎(chǔ)上,針對(duì)
圖1CuZnPbAgSnAs元素組合異常等值線
Fig.1Contour Map of CuZnAbAgSnAs Element Combination Anomalies
需要預(yù)測(cè)礦種與地質(zhì)背景的關(guān)系,提取以下信息作為地質(zhì)預(yù)測(cè)變量(圖層):有利地層面積百分比(X1);地層組合熵(X2);同熔型及重熔型花崗巖存在與否(X3);燕山期巖體存在與否(X4);斷裂等密度(X5);斷裂交點(diǎn)數(shù)(X6);銀礦點(diǎn)數(shù)(X7);金礦點(diǎn)數(shù)(X8);鉛鋅礦點(diǎn)數(shù)(X9);鐵礦點(diǎn)數(shù)(X10);成礦元素簇團(tuán)因子異常得分(X11)。
將研究區(qū)劃分為5 km×5 km的網(wǎng)格單元,共1 755個(gè)單元,每個(gè)單元可以看作平面上的一個(gè)點(diǎn),計(jì)算11個(gè)地質(zhì)預(yù)測(cè)變量(圖層)的每個(gè)單元成礦有利度(即銅多金屬礦發(fā)生的條件概率),其中安徽、江西單元部分(共301個(gè))的有利度設(shè)為0。這樣,11個(gè)地質(zhì)預(yù)測(cè)變量(圖層)Xk的成礦有利度可以組成11×39×45的三維數(shù)組X=(xkij),即立體陣,1≤k≤11,1≤i≤39,1≤j≤45。其中,i、j表示單元的坐標(biāo)。將11個(gè)地質(zhì)預(yù)測(cè)變量(圖層)的成礦有利度加權(quán)平均,得到浙西地區(qū)銅多金屬礦成礦有利度等值線圖(圖2)。
成礦有利度加權(quán)平均值為
(yij)=∑11 k=1akxkij=[a][X](13)
其中X=X1
X2
X11,1≤i≤39,1≤j≤45
a=(a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11)=(012,0.08,0.05,0.09,0.06,0.01,0.19,021,0.01,0.05,0.13)
式中:ak為相應(yīng)的地質(zhì)預(yù)測(cè)變量(圖層)權(quán)數(shù),∑11 k=1ak=1。
4結(jié)語(yǔ)
(1)提出了N維陣的概念、常用的幾種定義和
圖2銅多金屬礦成礦有利度等值線
Fig.2Contour Map of Oreforming Favorability of Copper Polymetallic Deposit
運(yùn)算,包括N維陣的加減、方括號(hào)乘法和Hadamard積,并給出了其性質(zhì)及相應(yīng)的說(shuō)明。
(2)通過(guò)江紹拼合帶中西段CuZnAbAgSnAs元素組合異常研究與浙西地區(qū)銅多金屬礦成礦預(yù)測(cè),說(shuō)明N維陣在實(shí)際問(wèn)題中應(yīng)用的方法及步驟,認(rèn)為N維陣在處理地學(xué)多維數(shù)據(jù)中有著重要的應(yīng)用前景。
(3)N維陣方法的優(yōu)勢(shì)在于對(duì)多維數(shù)據(jù)表示與分析更加簡(jiǎn)潔與方便。N維陣也是處理空間和時(shí)間地質(zhì)空間多維數(shù)據(jù)的有效工具。
參考文獻(xiàn):
References:
[1]趙鵬大.成礦定量預(yù)測(cè)與深部找礦[J].地學(xué)前緣,2007,14(5):110.
ZHAO Pengda.Quantitative Mineral Prediction and Deep Mineral Exploration[J].Earth Science Frontiers,2007,14(5):110.
[2]葉天竺,薛建玲.金屬礦床深部找礦中的地質(zhì)研究[J].中國(guó)地質(zhì),2007,34(5):855869.
YE Tianzhu,XUE Jianling.Geological Study in Search of Metallic Ore Deposits at Depth[J].Geology in China,2007,34(5):855869.
[3]翟裕生,鄧軍,王建平,等.深部找礦研究問(wèn)題[J].礦床地質(zhì),2004,23(2):142149.
ZHAI Yusheng,DENG Jun,WANG Jianping,et al.Researches on Deep Ore Prospecting[J].Mineral Deposits,2004,23(2):142149.
[4]申維.分形混沌與礦產(chǎn)預(yù)測(cè)[M].北京:地質(zhì)出版社,2002.
SHEN Wei.Fractal and Chaos with Application in Mineral Resource Prediction[M].Beijing:Geological Publishing House,2002.
[5]SHEN W,ZHAO P D.Multidimensional Selfaffine Distribution with Application in Geochemistry[J].Mathematical Geology,2002,34(2):109123.
[6]申維,房叢卉,張德會(huì).地球磁極倒轉(zhuǎn)的分形混沌研究[J].地學(xué)前緣,2009,16(5):201206.
SHEN Wei,FANG Conghui,ZHANG Dehui.Fractal and Chaos Research of Geomagnetic Polarity Reversal[J].Earth Science Frontiers,2009,16(5):201206.
[7]趙鵬大.數(shù)字地質(zhì)與礦產(chǎn)資源評(píng)價(jià)[J].地質(zhì)學(xué)刊,2012,36(3):225228.
ZHAO Pengda.Digital Geology and Mineral Resources Evaluation[J].Journal of Geology,2012,36(3):225228.
[8]趙鵬大,李桂范,張金川.基于地質(zhì)異常理論的頁(yè)巖氣有利區(qū)塊圈定與定量評(píng)價(jià)[J].天然氣工業(yè),2012,32(6):18.
ZHAO Pengda,LI Guifan,ZHANG Jinchuan.Shale Gas Favorable Blocks Delineation and Quantitative Evaluation Based on the Geological Anomaly Theory[J].Natural Gas Industry,2012,32(6):18.
[9]趙鵬大.找礦理念:從定性到定量[J].地質(zhì)通報(bào),2011,30(5):625629.
ZHAO Pengda.Prospecting Idea:From Qualification to Quantification[J].Geological Bulletin of China,2011,30(5):625629.
[10]王世稱,陳永良,夏立顯.綜合信息礦產(chǎn)預(yù)測(cè)理論與方法[M].北京:科學(xué)出版社,2000.
WANG Shicheng,CHEN Yongliang,XIA Lixian.Theory and Method of Synthetic Information Mineral Resources Prognosis[M].Beijing:Science Press,2000.
[11]王世稱.綜合信息礦產(chǎn)預(yù)測(cè)理論與方法體系新進(jìn)展[J].地質(zhì)通報(bào),2010,29(10):13991403.
WANG Shicheng.The New Development of Theory and Method of Synthetic Information Mineral Resources Prognosis[J].Geological Bulletin of China,2010,29(10):13991403.
[12]申維.深部找礦非線性定量理論與技術(shù)方法研究進(jìn)展綜述[J].地學(xué)前緣,2010,17(5):278288.
SHEN Wei.Progress in Nonlinear Quantitative Theory,Technology and Methods of Deep Exploration[J].Earth Science Frontiers,2010,17(5):278288.
[13]BATES D M,WATTS D G.Relative Curvature Measures of Nonlinearity[J].Journal of the Royal Statistical Society Series B:Methodological,1980,42(1):125.
[14]韋博成.近代非線性回歸分析[M].南京:東南大學(xué)出版社,1989.
WEI Bocheng.Analysis of Modern Nonlinear Regression[M].Nanjing:Southeast University Press,1989.
[15]王新洲.非線性模型參數(shù)估計(jì)理論與應(yīng)用[M].武漢:武漢大學(xué)出版社,2002.
WANG Xinzhou.Theory and Application of Nonlinear Models Parameter Estimation[M].Wuhan:Wuhan University Press,2002.
[16]倪國(guó)熙.常用的矩陣?yán)碚摵头椒ǎ跰].上海:上??茖W(xué)技術(shù)出版社,1984.
NI Guoxi.Theory and Method of Common Matrix[M].Shanghai:Shanghai Science and Technology Press,1984.第36卷第2期 2014年6月地球科學(xué)與環(huán)境學(xué)報(bào) Journal of Earth Sciences and EnvironmentVol36No2 June 2014
[3]翟裕生,鄧軍,王建平,等.深部找礦研究問(wèn)題[J].礦床地質(zhì),2004,23(2):142149.
ZHAI Yusheng,DENG Jun,WANG Jianping,et al.Researches on Deep Ore Prospecting[J].Mineral Deposits,2004,23(2):142149.
[4]申維.分形混沌與礦產(chǎn)預(yù)測(cè)[M].北京:地質(zhì)出版社,2002.
SHEN Wei.Fractal and Chaos with Application in Mineral Resource Prediction[M].Beijing:Geological Publishing House,2002.
[5]SHEN W,ZHAO P D.Multidimensional Selfaffine Distribution with Application in Geochemistry[J].Mathematical Geology,2002,34(2):109123.
[6]申維,房叢卉,張德會(huì).地球磁極倒轉(zhuǎn)的分形混沌研究[J].地學(xué)前緣,2009,16(5):201206.
SHEN Wei,FANG Conghui,ZHANG Dehui.Fractal and Chaos Research of Geomagnetic Polarity Reversal[J].Earth Science Frontiers,2009,16(5):201206.
[7]趙鵬大.數(shù)字地質(zhì)與礦產(chǎn)資源評(píng)價(jià)[J].地質(zhì)學(xué)刊,2012,36(3):225228.
ZHAO Pengda.Digital Geology and Mineral Resources Evaluation[J].Journal of Geology,2012,36(3):225228.
[8]趙鵬大,李桂范,張金川.基于地質(zhì)異常理論的頁(yè)巖氣有利區(qū)塊圈定與定量評(píng)價(jià)[J].天然氣工業(yè),2012,32(6):18.
ZHAO Pengda,LI Guifan,ZHANG Jinchuan.Shale Gas Favorable Blocks Delineation and Quantitative Evaluation Based on the Geological Anomaly Theory[J].Natural Gas Industry,2012,32(6):18.
[9]趙鵬大.找礦理念:從定性到定量[J].地質(zhì)通報(bào),2011,30(5):625629.
ZHAO Pengda.Prospecting Idea:From Qualification to Quantification[J].Geological Bulletin of China,2011,30(5):625629.
[10]王世稱,陳永良,夏立顯.綜合信息礦產(chǎn)預(yù)測(cè)理論與方法[M].北京:科學(xué)出版社,2000.
WANG Shicheng,CHEN Yongliang,XIA Lixian.Theory and Method of Synthetic Information Mineral Resources Prognosis[M].Beijing:Science Press,2000.
[11]王世稱.綜合信息礦產(chǎn)預(yù)測(cè)理論與方法體系新進(jìn)展[J].地質(zhì)通報(bào),2010,29(10):13991403.
WANG Shicheng.The New Development of Theory and Method of Synthetic Information Mineral Resources Prognosis[J].Geological Bulletin of China,2010,29(10):13991403.
[12]申維.深部找礦非線性定量理論與技術(shù)方法研究進(jìn)展綜述[J].地學(xué)前緣,2010,17(5):278288.
SHEN Wei.Progress in Nonlinear Quantitative Theory,Technology and Methods of Deep Exploration[J].Earth Science Frontiers,2010,17(5):278288.
[13]BATES D M,WATTS D G.Relative Curvature Measures of Nonlinearity[J].Journal of the Royal Statistical Society Series B:Methodological,1980,42(1):125.
[14]韋博成.近代非線性回歸分析[M].南京:東南大學(xué)出版社,1989.
WEI Bocheng.Analysis of Modern Nonlinear Regression[M].Nanjing:Southeast University Press,1989.
[15]王新洲.非線性模型參數(shù)估計(jì)理論與應(yīng)用[M].武漢:武漢大學(xué)出版社,2002.
WANG Xinzhou.Theory and Application of Nonlinear Models Parameter Estimation[M].Wuhan:Wuhan University Press,2002.
[16]倪國(guó)熙.常用的矩陣?yán)碚摵头椒ǎ跰].上海:上??茖W(xué)技術(shù)出版社,1984.
NI Guoxi.Theory and Method of Common Matrix[M].Shanghai:Shanghai Science and Technology Press,1984.第36卷第2期 2014年6月地球科學(xué)與環(huán)境學(xué)報(bào) Journal of Earth Sciences and EnvironmentVol36No2 June 2014
[3]翟裕生,鄧軍,王建平,等.深部找礦研究問(wèn)題[J].礦床地質(zhì),2004,23(2):142149.
ZHAI Yusheng,DENG Jun,WANG Jianping,et al.Researches on Deep Ore Prospecting[J].Mineral Deposits,2004,23(2):142149.
[4]申維.分形混沌與礦產(chǎn)預(yù)測(cè)[M].北京:地質(zhì)出版社,2002.
SHEN Wei.Fractal and Chaos with Application in Mineral Resource Prediction[M].Beijing:Geological Publishing House,2002.
[5]SHEN W,ZHAO P D.Multidimensional Selfaffine Distribution with Application in Geochemistry[J].Mathematical Geology,2002,34(2):109123.
[6]申維,房叢卉,張德會(huì).地球磁極倒轉(zhuǎn)的分形混沌研究[J].地學(xué)前緣,2009,16(5):201206.
SHEN Wei,FANG Conghui,ZHANG Dehui.Fractal and Chaos Research of Geomagnetic Polarity Reversal[J].Earth Science Frontiers,2009,16(5):201206.
[7]趙鵬大.數(shù)字地質(zhì)與礦產(chǎn)資源評(píng)價(jià)[J].地質(zhì)學(xué)刊,2012,36(3):225228.
ZHAO Pengda.Digital Geology and Mineral Resources Evaluation[J].Journal of Geology,2012,36(3):225228.
[8]趙鵬大,李桂范,張金川.基于地質(zhì)異常理論的頁(yè)巖氣有利區(qū)塊圈定與定量評(píng)價(jià)[J].天然氣工業(yè),2012,32(6):18.
ZHAO Pengda,LI Guifan,ZHANG Jinchuan.Shale Gas Favorable Blocks Delineation and Quantitative Evaluation Based on the Geological Anomaly Theory[J].Natural Gas Industry,2012,32(6):18.
[9]趙鵬大.找礦理念:從定性到定量[J].地質(zhì)通報(bào),2011,30(5):625629.
ZHAO Pengda.Prospecting Idea:From Qualification to Quantification[J].Geological Bulletin of China,2011,30(5):625629.
[10]王世稱,陳永良,夏立顯.綜合信息礦產(chǎn)預(yù)測(cè)理論與方法[M].北京:科學(xué)出版社,2000.
WANG Shicheng,CHEN Yongliang,XIA Lixian.Theory and Method of Synthetic Information Mineral Resources Prognosis[M].Beijing:Science Press,2000.
[11]王世稱.綜合信息礦產(chǎn)預(yù)測(cè)理論與方法體系新進(jìn)展[J].地質(zhì)通報(bào),2010,29(10):13991403.
WANG Shicheng.The New Development of Theory and Method of Synthetic Information Mineral Resources Prognosis[J].Geological Bulletin of China,2010,29(10):13991403.
[12]申維.深部找礦非線性定量理論與技術(shù)方法研究進(jìn)展綜述[J].地學(xué)前緣,2010,17(5):278288.
SHEN Wei.Progress in Nonlinear Quantitative Theory,Technology and Methods of Deep Exploration[J].Earth Science Frontiers,2010,17(5):278288.
[13]BATES D M,WATTS D G.Relative Curvature Measures of Nonlinearity[J].Journal of the Royal Statistical Society Series B:Methodological,1980,42(1):125.
[14]韋博成.近代非線性回歸分析[M].南京:東南大學(xué)出版社,1989.
WEI Bocheng.Analysis of Modern Nonlinear Regression[M].Nanjing:Southeast University Press,1989.
[15]王新洲.非線性模型參數(shù)估計(jì)理論與應(yīng)用[M].武漢:武漢大學(xué)出版社,2002.
WANG Xinzhou.Theory and Application of Nonlinear Models Parameter Estimation[M].Wuhan:Wuhan University Press,2002.
[16]倪國(guó)熙.常用的矩陣?yán)碚摵头椒ǎ跰].上海:上??茖W(xué)技術(shù)出版社,1984.
NI Guoxi.Theory and Method of Common Matrix[M].Shanghai:Shanghai Science and Technology Press,1984.第36卷第2期 2014年6月地球科學(xué)與環(huán)境學(xué)報(bào) Journal of Earth Sciences and EnvironmentVol36No2 June 2014