Abstract
Consider a system in which the input is described by a set of M parameters m, and the prescribed output of the system by a linear transform on m, A m, where A is an M × N matrix. The objective is to minimize the difference, specified by the 2-norm ‖A m − d‖2, between the expected output from the system and the observed data d (e.g. grey scale values for each cell in a rasterized image). The estimated (least squares) solution is given by: \( {\boldsymbol{m}}_{est}=\frac{1}{{\mathbf{A}}^{\mathrm{T}}\mathbf{A}}{\mathbf{A}}^{\mathrm{T}}\boldsymbol{d} \), where T represents matrix transposition. However, because in practice, real data contains errors, the matrix A is often ill-conditioned, and an optimum solution has to be sought for using the technique of regularization. The objective function is:
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Bibliography
ADRAIN, R. (1818). Investigation of the Figure of the Earth and of gravity in different latitudes. Transactions of the American Philosophical Society, 1, 119–135.
AGOCS, W.B. (1951). Least squares residual anomaly determination. Geophysics, 16, 686–696.
AGTERBERG, F.P. (1974). Automatic contouring of geological maps to detect target areas for mineral exploration. Journal of the International Association for Mathematical Geology, 6, 373–395.
AGTERBERG, F.P. (1989). LOGDIA – FORTRAN77 program for logistic regression with diagnostics. Computers & Geosciences, 15, 599–614.
AGTERBERG, F.P., BONHAM-CARTER, G.F., CHENG, Q. and WRIGHT, D.F. (1993). Weights of evidence modeling and weighted logistic regression for mineral potential mapping. In: DAVIS, J. and HERZFELD, J.C. (eds.). Computers in geology – 25 years of progress. Oxford, Oxford University Press, 13–32.
AHRENS, L.H. (1954a). The lognormal distribution of the elements (A fundamental law of geochemistry and its subsidiary). I. Geochimica et Cosmochimica Acta, 5, 49–73.
AHRENS, L.H. (1954b). The lognormal distribution of the elements (A fundamental law of geochemistry and its subsidiary). II. Geochimica et Cosmochimica Acta, 6, 121–131.
AITCHISON, J. (1981). A new approach to null correlation of proportions. Journal of the International Association for Mathematical Geology, 13, 175–189.
AITCHISON, J. (1982). The statistical analysis of compositional data. Journal of the Royal Statistical Society, ser. B, 44, 139–177.
AITCHISON, J. (1986). The statistical analysis of compositional data. London, Chapman and Hall.
AITCHISON, J. (2003). The statistical analysis of compositional data. 2nd edn., London, Chapman and Hall.
AITCHISON, J. and BROWN, J.A.C. (1957). The lognormal distribution with special reference to its uses in economics. Cambridge, Cambridge University Press.
AITCHISON, J. and SHEN, S.M. (1980). Logistic-normal distributions: some properties and uses. Biometrika, 67 (2), 261–272.
ALABERT, F.G. (1987). The practice of fast conditional simulations through the LU decomposition of the covariance matrix. Mathematical Geology, 19, 369–386.
AL-AHMADI, K., AL-AHMADI, S. and AL-AMRI, A. (2014). Exploring the association between the occurrence of earthquakes and the geologic-tectonic variables in the Red Sea using logistic regression and GIS. Arab Journal of Earth sciences, 7, 3871–3879.
AL-SADI, H.N. (1980). Seismic exploration. Technique and Processing. Astronomisch-geophysikalische Reihe, v. 7. Basel, Birkhliuser Verlag.
ALPER, A.M. and POLDERVAART, A. (1957). Zircons from the Animas stock and associated rocks, New Mexico. Economic Geology, 52, 952–971.
ANALYTICAL METHODS COMMITTEE (1987). Recommendations for the definition, estimation and use of the detection limit. The Analyst, 112, 199–204.
ANALYTICAL METHODS COMMITTEE (2001). Measurement of near zero concentrations: recording and reporting results that fall close to or below the detection limit. The Analyst, 126, 256–259.
ANDREWS, D.F., BICKEL, P.J., HAMPEL, F.R., HUBER, P.J., ROGERS, W.H. and TUKEY, J.W. (1972). Robust estimates of location: Survey and advances. Princeton, NJ, Princeton University Press.
ASTER, R.C., BORCHERS, B. and THURBER, C.H. (2013). Parameter estimation and inverse problems. 2nd edn., Kidlington, Academic Press.
AZZALINI, A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 12, 171–178.
AZZALINI, A., DAL CAPPELLO, T. and KOTZ, S. (2003). Log-skew-normal and log-skew-t distributions as models for family income data. Journal of Income Distribution, 11, 12–20.
BAGNOLD, R.A. and BARNDORFF-NIELSEN, O. (1980). The pattern of natural size distribution. Sedimentology, 27, 199–207.
BANKS, R. (1979). The use of linear programming in the analysis of petrological mixing problems. Contributions to Mineralogy and Petrology, 70, 237–244.
BARNARD, G.A. (1949). Statistical inference. Journal of the Royal Statistical Society, ser. B, 11, 115–149.
BARNES, J.A. and MOCKLER, R.C. (1960). The power spectrum and its importance in precise frequency measurements. IRE Transactions on Instrumentation, I-9 (2), 149–155.
BARNES, S.J. (1988). Automated plotting of geochemical data using the LOTUS Symphony package. Computers & Geosciences, 14, 409–411.
BARNETT, C.T. and WILLIAMS, P.M. (2006). Mineral exploration using modern data mining techniques. In: DOGGETT, M.E. and PARRY, J.R. (eds.). Wealth creation in the minerals industry: Integrating science, business and education. Society of Economic Geologists Special Publication 12. Littleton, CO, Society of Economic Geologists, 295–310.
BARTELS, R.H. and GOLUB, G.H. (1969). The simplex method of linear programming using LU decomposition. Communications of the ACM, 12, 266–268.
BARTLETT, M.S. (1946). On the theoretical specification and sampling properties of autocorrelated time-series. Journal of the Royal Statistical Society, Supplement, 8, 27–41, 85–97.
BARTLETT, M.S. (1947). The use of transformations. Biometrics, 3, 39–52.
BELSLEY, D.A., KUH, E. and WELSCH, R.E. (1980). Regression diagnostics. New York, NY, John Wiley & Sons
BENEDEK, G. and GREYTAK, T. (1965). Brillouin scattering in liquids. Proceedings of the IEEE, 53, 1623–1629.
BERKSON, J. (1944). Application of the logistic function to bio-assay. Journal of the American Statistical Association, 39, 357–365.
BICKEL, P.J. and BÜHLMANN, P. (1996). What is a linear process? Proceedings of the National Academy of Sciences of the United States of America, 93, 12128–12131.
BIRCH, F. and BANCROFT, D. (1938). Elasticity and internal friction in a long column of granite. Bulletin of the Seismological Society of America, 28, 243-254.
BIRD, D.N. (1982). A linear programming approach to time-term analysis. Bulletin of the Seismological Society of America, 72, 2171-2180.
BIRKS, H.J.B. (1995). Quantitative palaeoenvironmental reconstructions. In: MADDY, D. and BREW, J.S. (eds.). Statistical modelling of Quaternary Science data. Technical Guide 5. Cambridge, Quaternary Research Association, 161-254.
BIVAND, R.S., PEBESMA, E. and GÓMEZ-RUBIO, V. (2013). Applied spatial data analysis with R. 2nd edn., New York, NY, Springer-Verlag.
BLACKMAN, R.B. and TUKEY, J.W. (1958). The measurement of power spectra from the point of view of communications engineering. Bell System Technical Journal, 37, 185–282, 485–569.
BLOOMFIELD, P. (1976). Fourier analysis of time series: An introduction. New York, John Wiley & Sons.
BODE, H.W. (1934). A general theory of electric wave filters. Journal of Mathematical Physics, 13, 275–362.
BOGERT, B.P., HEALY, M.J.R. and TUKEY, J.W. (1963). The quefrency alanysis of time series for echoes: cepstrum, pseudo-autocovariance, cross-cepstrum and saphe-cracking. In: ROSENBLATT, M. (ed.). Proceedings of the symposium on time series analysis. New York, John Wiley & Sons, 209–243.
BOOLE, G. (1843). Exposition of a general theory of linear transformations, Parts I and II. Cambridge mathematical Journal, 3, 1–20, 106–119.
BOOLE, G. (1844). On a general method in analysis. Philosophical Transaction of the Royal Society, London, 134, 225–282.
BORN, M. (1936). On the linearization of the energy density of the electromagnetic field. Mathematical Proceedings of the Cambridge Philosophical Society, 32, 102–107.
BOS, H.J.M. (1974). Differentials, higher-order differentials and the derivative in Leibnizian calculus. Archive for the History of Exact Sciences, 14, 1–90.
BOWKER, A.H. (1947). On the norm of a matrix. Annals of Mathematical Statistics, 18, 285–288.
BRIANCHON, C.-J. (1817). Mémoire sur les lignes du second ordre [Memoir on lines of the second order]. Paris, Bachelier.
BRIGGS, H. (1617). Logarithmorum chilias prima [The first thousand logarithms]. London, Unknown.
BRIGGS, H. (1624). Arithmética logarithmica [Logarithmical arithmetic]. London, William Jones.
BRIGGS, H. (1631). Logarithmicall arithmetike. Or tables of logarithmes for absolute numbers from an unite to 100,000: as also for sines, tangentes and secantes for every minute of a quadrant with a plaine description of their use in arithmetike, geometrie, geographie, &c. London, George Miller.
BRUCE, I. (2012). John Napier: Mirifici Logarithmorum Canonis Descriptio – & Constructio [translated and annotated by I. Bruce; online: http://www.17centurymaths.com/contents/napiercontents.html].
BRUTSAERT, W. (1968). The permeability of a porous medium determined from certain probability laws for pore size distribution. Water Resources Research, 4, 425–434.
BUCCIANTI, A., MATEU-FIGUERAS, G. and PAWLOWSKY-GLAHN, V. (eds.) (2006). Compositional data analysis in the geosciences: From theory to practice. London, The Geological Society.
BURG, J.P. (1967). Maximum entropy spectral analysis. Proceedings of the 37th Meeting of the Society of Exploration Geophysicists, Oklahoma City, Oklahoma, 31 October 1967, pp. 34–41. In: CHILDERS, D.G. (ed.). (1978). Modern spectrum analysis. New York, NY, IEEE Press, 34–39.
BURG, J.P. (1968). A new analysis technique for time series data. Paper given at: NATO Advanced Study Institute on signal processing with emphasis on underwater acoustics, 12–23 August 1968, Twente Institute of Technology, Enschede, The Netherlands. In: CHILDERS, D.G. (ed.). (1978). Modern spectrum analysis. New York, NY, IEEE Press, 42–48.
BURG, J.P. (1975). Maximum entropy spectral analysis. Doctoral dissertation. Stanford Exploration Project Report no. 6, Stanford, CA, Stanford Exploration Project, Stanford University (online: http://sepwww.stanford.edu/data/media/public/oldreports/sep06/).
BURK, F. (1988). Lebesgue measure and integration. An introduction. New York, NY, John Wiley & Sons.
BURK, F.E. (2007). A garden of integrals. The Dolciani Mathematical Expositions v. 31. Washington, DC, The Mathematical Association of America.
BUTTKUS, B. (1991). Spektralanalyse und Filtertheorie in der angewandten Geophysik. Berlin, Springer-Verlag.
BUTTKUS, B. (2000). Spectral analysis and filter theory in applied geophysics [translated by C NEWCOMB].. Berlin, Springer-Verlag.
CALTENCO, J.H., LÓPEZ-BONILLA, J., MORALES, J. and PÉREZ-TERUEL, G.R. (2014). Polynomial solutions of differential equations. The SciTech, International Journal of Engineering Sciences, 2, 73–79.
CAMERON, G.W., ELLIOTT, B.E. and RICHARDSON, K.A. (1976). Effects of line spacing on contoured airborne Gamma-ray spectrometry data. In: Exploration for uranium ore deposits. Proceedings of a symposium: Vienna, 29 March–2 April, 1976. Vienna, International Atomic Energy Agency, 81–92.
CAMINA, A.R. and JANACEK, G.J. (1984). Mathematics for seismic data processing and interpretation. London, Graham and Trotman.
CAMPBELL, G.A. (1922). Physical theory of the electric wave-filter. Bell System Technical Journal, 1 (2), 1–32.
CAUCHY, A.-L. (1843). Rapport sur un memoir de M. Laurent qui a pour titre: Extension du théorème de M. Cauchy relatif à la convergence du développement d'une fonction suivant les puissances ascendantes de la variable x [Report on a memoir by M. Laurent]. Comptes rendus hebdomadaires des Séances de l’Académie des Sciences, Paris, 17, 938–940.
CHATTERJEE, S. and HADI, A.S. (1986). Influential observations, high leverage points, and outliers in regression. Statistical Science, 1, 379–416.
CHENG, Q. (1997). Multifractal modelling and lacunarity analysis. Mathematical Geology, 29, 919–932.
CHILD, J.M. (1920). The early mathematical manuscripts of Leibniz, translated from the Latin texts published by Carl Immanuel Gerhardt with critical and historical notes. London, Open Court Publishing.
CHORK, C.Y. (1991). An assessment of Least Median Squares regression in exploration geochemistry. Journal of Geochemical Exploration, 41, 325–340.
CHORK, C.Y. and ROUSSEEUW, P.J. (1992). Integrating a high-breakdown option into discriminant analysis in exploration geochemistry. Journal of Geochemical Exploration, 43, 191–203.
CHRISTIANSEN, C., BLAESILD, P. and DALSGAARD, K. (1984). Re-interpreting ‘segmented’ grain-size curves. Geological Magazine, 121, 47–51.
CLAERBOUT, J.F. (1986). A canonical program library. Stanford Exploration Project Report SEP-50, Stanford, CA, Geophysics Department, Stanford University, 281–289.
CLAERBOUT, J.F. (1992). Earth soundings analysis: Processing versus inversion. London, Blackwell Scientific Publications.
CLARK, D.A. (1981). A system for regional lithofacies mapping. Bulletin of Canadian Petroleum Geology, 20, 197–208.
COHEN, T.J. (1970). Source-depth determinations using spectral, pseudo-autocorrelation and cepstral analysis. Geophysical Journal International, 20, 223–231.
COLLINS, F. and LEE, C.C. (1956). Seismic wave attenuation characteristics from pulse experiments. Geophysics, 21, 16–40.
CONNOLLY, J.A.D. (2005). Computation of phase equilibria by linear programming: A tool for geodynamic modelling and its application to subduction zone decarbonation. Earth and Planetary Science Letters, 236, 524–541.
CONNOR, C.B., SPARKS, R.S.J., MASON, R.M., BONADONNA, C. and YOUNG, S.R.A. (2003). Exploring links between physical and probabilistic models of volcanic eruptions: The Soufrière Hills, Montserrat. Geophysical Research Letters, 30, 1701–1708 [http://dx.doi.org/10.1029/2003GL017384].
COOK, R.D. and WEISBERG, S. (1982). Residuals and influence in regression. London, Chapman and Hall.
CORTINI, M. and BARTON, C.C. (1994). Chaos in geomagnetic reversal records: A comparison between Earth’s magnetic field data and model disk dynamo data. Journal of Geophysical research: Solid Earth, 99 (B9), 18021–18033.
CRANDALL, I.B. (1926). Theory of vibrating systems and sound. New York, NY, Van Nostrand.
CURRIE, L.A. (1995). Nomenclature in evaluation of analytical methods including detection and quantification capabilities. Pure and Applied Chemistry, 67, 1699–1723.
CURRIE, L.A. (2004). Uncertainty in measurements close to detection limits: Detection and quantification capabilities. In: Quantifying uncertainty in nuclear analytical measurements. IAEA-TECDOC-1401. Vienna, International Atomic Energy Agency, 9–33.
CURRIE, R.G. (1973). Geomagnetic line spectra – 2 to 70 years. 21,. Astrophysics and Space Science, 21, 425–438.
DAHL, P.S. (1990). A PC- and LOTUS-based data acquisition/reduction system for an ICP spectrometer. Computers & Geosciences, 16, 881–896.
DANTZIG, G.B. (1949). Programming in a linear structure [abstract]. Econometrica, 17, 73–74.
DAVID, M. (1977). Geostatistical ore reserve estimation. Developments in geomathematics 2. Amsterdam, Elsevier Scientific.
DAVIS, J.C. (1986). Statistics and data analysis in geology. 2nd edn., New York, NY, John Wiley & Sons.
DAVIS, M. (1987a). Production of conditional simulations via the LU triangular decomposition of the covariance matrix. Mathematical Geology, 19, 91–98.
DAVIS, M.W.D. and DAVID, M. (1978). The numerical calculation of integrals of an isotropic variogram function on hypercubes. Journal of the International Association for Mathematical Geology, 10, 311–314.
DEAKIN, M.A.B. (1981). The development of the Laplace transform, 1737–1937. I. Euler to Spitzer, 1737–1880. Archive for History of the Exact Sciences, 25, 343–390.
DEAKIN, M.A.B. (1982). The development of the Laplace transform, 1737–1937. II. Poincaré to Doetsch, 1880–1937. Archive for History of the Exact Sciences, 26, 351–381.
DeMERS, M.N. (1990). SEDRULE: a rule-based system for interpreting some major sedimentary environments. Computers & Geosciences, 16, 833–846.
DIELMAN, T.E. (1984). Least absolute value estimation in regression models: an annotated bibliography. Communications in Statistics – Theory and Methods, 13, 513–541.
DODGE, Y. (1987). Statistical data analysis based on the L 1 norm and related methods. Amsterdam, North-Holland.
DODSON, J. (1742). The anti-Logarithmic canon. Being a table of numbers, consisting of eleven places of figures, corresponding to all Logarithms under 100000 London, James Dodson.
DOWD, P.A. (1991). A review of recent developments in geostatistics. Computers & Geosciences, 17, 1481–1500.
DURBIN, J. (1960). The fitting of time-series models. Review of the International Statistical Institute, 28, 233–243.
DUTKA, J. (1996). On Gauss’ priority in the discovery of the method of least squares. Archive for the History of Exact Sciences, 49, 355–370.
EGOZCUE, J.J., PAWLOWSKY-GLAHN, V., MATEU-FIGUERAS, G. and BARCELÓ-VIDAL, C. (2003). Isometric logratio transformations for compositional data analysis. Mathematical Geology, 35, 279–300.
EISENHART, C. (1935). A test for the significance of lithological variations. Journal of Sedimentary Petrology, 5, 137–145.
EMICH, F. and FEIGEL, F. (1932). Microchemical laboratory manual. London, Chapman and Hall.
ENCKE, J.F. (1841). On the method of least squares. In: TAYLOR, R. (ed.). Scientific memoirs, selected from the transactions of Foreign Academies of Science and Learned Societies, and from foreign journals. Vol. 2. London, Richard and John E. Taylor, 317–369.
EYNATTEN, H. von, BARCELO-VIDAL, C. and PAWLOWSKY-GLAN, V. (2003). Modelling compositional change: The example of chemical weathering of granitoid rocks. Mathematical Geology, 35, 231–252.
EZEKIEL, M. and FOX, K.A. (1930). Methods of correlation and regression analysis, linear and curvilinear. New York, NY, John Wiley & Sons.
FERBER, R.-G. (1984). Stabilization of normal-incidence seismogram inversion removing the noise-induced bias. Geophysical Prospecting, 33, 212–233.
FISHER, R.A. (1921). On the ‘probable error’ of a coefficient of correlation deduced from a small sample. Metron, 1, 3–32.
FISHER, A. (1923). Mathematical theory of probabilities. 2nd edn., New York, NY, Macmillan.
FISHER, R.A. (1925a). Statistical methods for research workers. Edinburgh, Oliver and Boyd.
FISHER, R.A. (1925b). Theory of statistical estimation. Proceedings of the Cambridge Philosophical Society, 22, 700–725.
FISHER, R.A. (1934). Two new properties of mathematical likelihood. Proceedings of the Royal Society, London. ser. A, 144, 285–307.
FITCH, T.J., McCOWAN, D.W. and SHIELDS, M.W. (1980). Estimation of the seismic moment tensor from teleseismic body wave data with applications to intraplate and mantle earthquakes. Journal of Geophysical Research – Solid Earth, ser. B, 85, 3817–3828.
FLINN, D. (1962). On folding during three-dimensional progressive deformation. Quarterly Journal of the Geological Society, London, 118, 385–433.
FOUGERE, P.F. (1985). A review of the problem of spontaneous line splitting in maximum entropy power spectral analysis. In: SMITH, C.R. and GRANDY, W.T. Jr. (eds.). Minimum-entropy and Bayesian methods in inverse problems. Fundamental Theories of Physics volume 14. Dordrecht, D. Reidel, 303–315.
FRÉCHET, M. (1906). Sur quelques points du calcul fonctionnel [On several points of functional calculus; Doctoral dissertation, Paris, 1905]. Rendiconti del Circolo Matematico di Palermo, 22, 1–74.
FRIBERG, L.M. (1989). Garnet stoichiometry program using a LOTUS 1-2-3 spreadsheet. Computers & Geosciences, 15, 1169–1172.
GADDUM, J.H. (1945). Lognormal distributions. Nature,, 156, 463–466.
GAUSS, C.F. (1809a). Theoria motus corporum coelestium in sectionibus conicis solem ambientium [Theory of the motion of the heavenly bodies moving about the Sun in conic sections]. Hamburg, F. Perthes and I.H. Besser.
GAUSS, C.F. (1809b [1857]). Determination of an orbit satisfying as nearly as possible any number of observations whatever. In: Theory of the motion of the heavenly bodies moving about the Sun in conic sections [translated from Latin by C.H. DAVIS]. Boston, MS, Little, Brown & Co, 249–273.
GELLIBRAND, H. (1635). A discourse mathematical on the variation of the magneticall needle. Together with its admirable diminution discovered. London, William Jones.
GIBRAT, R. (1931). Lés inégalités economiques [Economic inequalities]. Paris, Recueil Sirey.
GOGORZA, C.S.G., SINITO, A.M., DI TOMMASO, I., VILAS, J.F., CREER, K.M. and NUÑEZ, H. (1999). Holocene geomagnetic secular variations recorded by sediments from Escondido Lake (south Argentina). Earth, Planets and Space, 51, 93–106.
GOOGLE RESEARCH (2012). Google Books Ngram Viewer (v. 2.0) [online: https://books.google.com/ ngrams/info].
GREENHALGH, S.A., ZHOU, B. and GREEN, A. (2006). Solutions, algorithms and inter-relations for local minimization search geophysical inversion. Journal of Geophysics and Engineering, 3, 101–113.
GREENWOOD, H.J. (1967). The N-dimensional tie-line problem. Geochimica et Cosmochimica Acta, 31, 465–490.
GRIFFITHS, L.J. (1975). Rapid measurement of digital instantaneous frequency. IEEE Transactions on Acoustics, Speech and Signal Processing, ASSP-23, 207–222.
GRIFFITHS, L.J. and PRIETO-DIAZ, R. (1977). Spectral analysis of natural seismic events using autoregressive techniques. IEEE Transactions on Geoscience Electronics, GE-15, 13–25.
GUBBINS, D. (2004). Time series analysis and inverse theory for geophysicists. Cambridge, Cambridge University Press.
HAMILTON, W.R. (1848). Researches concerning quaternions. First series. Transactions of the Royal Irish Academy, 21 (1), 199–296.
HAMON, B.V. and HANNAN, E.J. (1963). Estimating relations between time series. Journal of Geophysical Research, 68, 6033–6041.
HANNAN, E.J. (1966). Spectral analysis for geophysical data. Geophysical Journal International, 11, 225–236.
HARRIS, F.J. (1978). On the use of windows for harmonic analysis with the discrete Fourier transform. Proceedings of the IEEE, 66, 51–83.
HARRISON, P.W. (1957). New technique for three-dimensional fabric analysis of till and englacial debris containing particles from 3 to 40 mm in size. Journal of Geology, 65, 98–105.
HARVEY, G. (1822). On the method of minimum squares, employed in the reduction of experiments, being a translation of the appendix to an essay of Legendre’s entitled, “Nouvelles methodes pour la determination des orbites des cometes,” with remarks. The Edinburgh Philosophical Journal, 7, 292–301.
HEAVISIDE, O. ( 1892b). Electrical papers. v. 1. London, Macmillan.
HEAVISIDE, O. (1892c). On operators in physical mathematics. Part I. Proceedings of the Royal Society, London, 52, 504–529.
HEAVISIDE, O. (1893). On operators in physical mathematics. Part II. Proceedings of the Royal Society, London, 54, 105–142.
HEAVISIDE, O. (1899). Generalised differentiation and divergent series. In: Electromagnetic theory. London, The Electrician Printing and Publishing Co., 434–492.
HELSEL, D.R. (2005). Nondetects and data analysis. Hoboken, NJ, Wiley-Interscience.
HELSEL, D.R. and HIRSCH, R.M. (1992). Statistical methods in water resources. Amsterdam, Elsevier.
HERRING, R. (1980). The cause of line splitting in Burg maximum-entropy spectral analysis. IEEE Transactions on Acoustics, Speech and Signal Processing, 28, 692–701.
HOAGLIN, D.C. and WELSCH, R.E. (1978). The Hat matrix in regression and ANOVA. The American Statistician, 32, 17–22.
HOOKER, R.H. (1901). Correlations of the marriage rate with trade. Journal of the Royal Statistical Society, London, 64, 485–492.
HOSSACK, J.R. (1968). Pebble deformation and thrusting in the Bygdin area (S. Norway). Tectonophysics, 5, 315–339.
HOWARTH, R.J. (1973a). The pattern recognition problem in applied geochemistry. In: JONES, M.J. (ed.). Geochemical Exploration 1972. Institution of Mining and Metallurgy, London, p. 259–273.
HOWARTH, R.J. (1984). Statistical applications in geochemical prospecting: A survey of recent methods. Journal of Geochemical Exploration, 21, 41–61.
HOWARTH, R.J. (1996a). Sources for the history of the ternary diagram. British Journal for the History of Science, 29, 337–356.
HOWARTH, R.J. (1996b). History of the stereographic projection and its early use in geology. Terra Nova, 8, 499–513.
HOWARTH, R.J. (1999). Measurement, portrayal and analysis of orientation data in structural geology (1670–1967). Proceedings of the Geologists’ Association, 110, 273–309.
HOWARTH, R.J. (2001a). A history of regression and related model-fitting in the earth sciences (1636?–2000). Natural Resources Research, 10, 241–286.
HOWARTH, R.J. (2001b). Measurement, portrayal and analysis of orientation data in structural geology (1670–1967): Corrections and additions. Proceedings of the Geologists' Association, 112, 187–190.
HOWARTH, R.J. and MCARTHUR, J.M. (1997). Statistics for Strontium isotope stratigraphy: A robust LOWESS fit to the marine Sr-isotope curve for 0 to 206 Ma, with look-up table for derivation of numerical age. The Journal of Geology, 105, 441–456.
HUBAUX, A. and SMIRIGA-SNOECK, N. (1964). On the limit of sensitivity and the analytical error. Geochimica et Cosmochimica Acta, 28, 1199–1216.
HUBBERT, A.M.K. (1948). Line-integral method of computing the gravimetric effects of two-dimensional masses. Geophysics, 13, 215–225.
HUBBERT, M.K. (1959). Techniques of prediction with application to the petroleum industry. Preprint for presentation at the 44th Annual Meeting of the American Association of Petroleum Geologists, Dallas, Texas, Tuesday, March 17, 1959. Publication 204, Houston, TX, Shell Development Company, Exploration and Production Research Division.
IMBRIE, J. (1963). Factor and vector analysis programs for analyzing geologic data. United States Office of Naval Research, Geography Branch, Technical Report 6, ONR Task No. 389-135 [AD0420466], Evanston, IL, Northwestern University.
IMBRIE, J. and PURDY, E.G. (1962). Classification of modern Bahamian carbonate sediments. In: HAM, W.E. (ed.). Classification of carbonate rocks: A symposium arranged by the Research Committee of the American Association of Petroleum Geologists. Including papers presented orally at Denver, Colorado, April 27, 1961. AAPG Memoir 1. Tulsa, OK, The American Association of Petroleum Geologists, 253–272.
IMBRIE, J. and VAN ANDEL, T.H. (1964). Vector analysis of heavy-mineral data. Bulletin of the Geological Society of America, 75, 1131–1156.
ISAAKS, E.H. and SRIVASTAVA, R.M. (1989). Applied geostatistics. Oxford, Oxford University Press.
JENSEN, J.L. (1988). Maximum-likelihood estimation of the hyperbolic parameters from grouped observations. Computers & Geosciences, 14, 389–408.
JOHNSON, D.S. and NURMINEN, J. (2007). The history of seafaring: navigating the World oceans. London, Conway.
JONES, H.E. (1937). Some geometrical considerations in the general theory of fitting lines and planes. Metron, 13, 21–30.
JÖRESKOG, K.G., KLOVAN, J.E. and REYMENT, R. (1976). Geological factor analysis. Amsterdam, Elsevier.
KAILATH, T. (1974). A view of three decades of linear filtering theory. IEEE Transactions on Information Theory, IT-20, 146–181.
KAISER, H. (1947). Die Berechnung der Nachweisempfindlichkeit [The calculation of detection sensitivity]. Spectrochimica Acta, 3, 40–67.
KANAL, L. and CHANDRASEKARAN, B. (1968). On dimensionality and sample size in statistical pattern classification. In: Proceedings of the 24th National Electronics Conference, December 9–11, 1968, Chicago, Illinois, National Electronics Conference Inc., Oak Brook, IL, 2–7.
KANTOROVICH, L. (1939). Mathematicheskie metody organizatsii i planirovania proizvodstva [Mathematical methods of organization and planning production]. Leningrad, Leningrad State University Publishers.
KANTOROVICH, L.V. (1960). Mathematical methods of organizing and planning production. Management Science, 6, 363–422.
KAY, S.M. and MARPLE, S.L., Jr. (1979). Sources of and remedies for spectral line splitting in autoregressive spectrum analysis. In: Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal processing, Washington, DC, Institute of Electrical and Electronic Engineers, New York, NY, [IEEE/IEE Electronic Library. 4, 151–154].
KERMACK, K.A. (1954). A biometrical study of Micraster coranginum and M. (isomicraster) senonensis. Philosophical Transactions of the Royal Society, London, ser. B, 237, 375–428.
KERMACK, K.A. and HALDANE, J.B.S. (1950). Organic correlation and allometry. Biometrika, 37, 30–41.
KIJKO, A. (1994). Seismological outliers: L1 or adaptive Lp norm application., 84. Bulletin of the Seismological Society of America, 84, 473–477.
KIRCHHOFF, G and BUNSEN, R. (1860). Chemische Analyse durch Spectralbeobachtungen [Chemical analysis by observation of spectra]. Annalen der Physik und Chemie, 110 (6), 161–189.
KLEINECKE, D. (1971). Use of linear programming for estimating geohydrologic parameters of groundwater basins. Water Resources Research, 7, 367–374.
KOCH, G.S. and LINK, R.F. (1970–71). Statistical analysis of geological data. v. 2. New York, NY, John Wiley & Sons.
KOCH, G.S., Jr. (ed.) (1990). Geological problem solving with Lotus 1-2-3 for exploration and mining geology. Computer methods in the earth sciences volume 8. Oxford, Pergamon Press.
KRIGE, D.G. (1960). On the departure of ore value distributions from the lognormal model in South African gold mines. Journal of the South African Institute of Mining and Metallurgy, 61, 231–244.
KRUHL, J.H. (ed.) (1994). Fractals and dynamical systems in earth science. Berlin, Springer-Verlag.
KRUMBEIN, W.C. (1934a). Size frequency distributions of sediments. Journal of Sedimentary Petrology, 4, 65–77.
KRUMBEIN, W.C. (1936a). Application of logarithmic moments to size frequency distributions of sediments. Journal of Sedimentary Petrology, 6, 35–47.
KRUMBEIN, W.C. (1936b). The use of quartile measures in describing and comparing sediments. American Journal of Science, ser. 5, 32, 98–111.
KRUMBEIN, W.C. (1939). Preferred orientation of pebbles in sedimentary deposits. Journal of Geology, 47, 673–706.
KRUMBEIN, W.C. (1948). Lithofacies maps and regional sedimentary-stratigraphic analysis. Bulletin of the American Association of Petroleum Geologists, 32, 1909–1923.
KRUMBEIN, W.C. (1952). Principles of facies map interpretation. Journal of Sedimentary Petrology, 22, 200–211.
KRUMBEIN, W.C. (1955a). Composite end members in facies mapping. Journal of Sedimentary Petrology, 25, 115–122.
KRUMBEIN, W.C. and GRAYBILL, F.A. (1965). An introduction to statistical models in geology. New York, NY, McGraw-Hill.
KRUMBEIN, W.C. and PETTIJOHN, F.J. (1938). Manual of sedimentary petrography.. New York, NY, NY, Appleton-Century.
KRUMBEIN, W.C. and SLOSS, L.L. (1951). Stratigraphy and sedimentation. San Francisco, CA, W.H. Freeman.
KRUSKAL, W.H. (1953). On the uniqueness of the line of organic correlation. Biometrics, 9, 47–58.
KYRIAKIDIS, P.C. (2005). Sequential spatial simulation using Latin hypercube sampling. v. 1. In: LEUANGTHONG, O. and DEUTCH, C.V. (eds.). Geostatistics Banff 2004: Seventh International Geostatistics Congress, Quantitative Geology and Geostatistics. Dordrecht, Kluwer, 65–74.
LAGRANGE, J.-L. (1788). Méchanique analytique [Analytical mechanics]. Paris, Desaint.
LAGRANGE, J.-L. (1795 [1877]). Leçons élémentaires sur les mathématiques données a l’École Normale [Elementary mathematics lessons taught at the Normal School], v. 7. In: SERRET, J.-A. (ed.). Œuvres de Lagrange. Paris, Gauthier-Villars, 183–287.
LAGRANGE, J.-L. (1797). Théorie de fonctions analytiques contenant les principes du calcul différentiel [Theory of analytic functions, containing the principles of differential calculus]. Paris, L’Imprimerie de la République.
LAMBERT, J.H. (1772). Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten [Notes and comments on the composition of Terrestrial and Celestial maps]. In: Beiträge zur Gebrauch der Mathematik und deren Anwendung [Contribution to the use of mathematics and its application]. v. III. Berlin, Verlage des Buchladens der Königliche Realschule, 105–199.
LANCASTER-JONES, E. (1929). The computation of gravitational effects due to irregular mass distributions. Geophysical Journal International, 2 (Supplement s3), 121–140.
LANCZOS, C. (1938). Trigonometric interpolation of empirical and analytic functions. Journal of Mathematics and Physics, 17, 123–199.
LANCZOS, C. (1956). Applied analysis. New York, Prentice-Hall.
LANCZOS, C. (1961). Linear differential operators. New York, NY, Van Nostrand.
LANGELIER, W.F. and LUDWIG, H.F. (1942). Graphical methods for indicating the mineral character of natural waters. Journal of the American Water Works Association, 34, 335–352.
LAPLACE, P.-S. (1782 [1785]). Théorie des attractions des sphéroides et de la figure des planètes [Theory of the attraction of spheroids and the figure of the planets]. Histoire de l’Académie Royale des Sciences année 1782 avec les Mémoires de Mathématique & de Physique, pour la même année 1782: tirés des registres de cette Académie, for 1782, 113–196.
LAPLACE, P.-S. (1798a). Traité Méchanique Céleste [Treatise on celestial mechanics]. v. II. Paris, Crapelet.
LAPLACE, P.-S. (1798b). Exposition du système du monde [Explanation of the system of the World]. 2nd edn., Paris, Crapelet.
LAPLACE, P.-S. (1809). The system of the World [translated from French by J. POND]. London, Richard Phillips.
LAPLACE, P.-S. (1812). Théorie analytique des probabilités [Analytical probability theory.]. Paris, Mme. Ve. Courcier.
LAU, Y.-S., HUSSAIN, Z.M. and HARRIS, R. (2004). A time-dependent LMS algorithm for adaptive filtering. WSEAS Transactions on Circuits and Systems, 3, 35–42.
LAURENT, P.-A. (1843). Extension du théorème de M. Cauchy relatif à la convergence du développement d'une fonction suivant les puissances ascendantes de la variable x [Extension of Cauchy's theorem on the convergent expansion of a function according to ascending powers of x]. Comptes rendus hebdomadaires des Séances de l’Académie des Sciences, Paris, 17, 348–349.
LAW, J. (1944). A statistical approach to the interstitial heterogeneity of sand reservoirs. Transactions of the American Institute of Mining and Metallurgical Engineers, 155, 202–222.
Le MAITRE, R.W. (1982). Numerical petrology. Statistical interpretation of geochemical data. Developments in petrology 8. Amsterdam, Elsevier Scientific Publishing.
Le ROUX, J.P. and RUST, I.C. (1989). Composite facies maps: a new aid to palaeo-environmental reconstruction. South African Journal of Geology, 92, 436–443.
LEATHEM, J.G. (1905). Volume and surface integrals used in physics. Cambridge, Cambridge University Press.
LEBAILLY, J., MARTIN-CLOUAIRE, R. and PRADE, A. (1987). Use of fuzzy logic in a rule-based system in petroleum geology. In: SANCHEZ, E. and ZADEH, L. (eds.). Approximate reasoning in intelligent systems, decision and control. Oxford, Pergamon Press, 125–144.
LEBESGUE, H. (1902). Intégrale, longueur, aire [Integral, length, area]. Doctoral dissertation, University of Paris. Milan, Bernandon de C. Rebeschini]. Annali di Mathematica Pura ed Applicata, 7, 231–359.
LEBESGUE, H. (1904). Leçons sur l’intégration et la recherche des fonctions primitives [Lessons on integration and investigating primitive functions]. Paris, Gauthier-Villars.
LEGENDRE, A.-M. (1785). Recherches sur l’attraction des sphéroïdes homogènes [Researches on the attraction of homogeneous spheroids]. Mémoires de Mathématiques et de Physique, présentés à l’Académie Royale des Sciences, par divers savans, et lus dans ses Assemblées, Paris, 10, 411–435.
LEGENDRE, A.-M. (1805). Appendice sur la méthode des moindres quarrés [Appendix on the method of minimum squares]. In: Nouvelles méthodes pour la détermination des orbites des comètes [New methods for the determination of the orbits of comets]. Paris, Courcier, 72–80.
LEGGE, J.A. and RUPNIK, J.J. (1943). Least squares determination of the velocity function V = V 0 + kz for any set of time depth data. Geophysics, 8, 356–361.
LEIBNIZ, G.W. (1684). Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus [A new method for maxima and minima as well as tangents, which is neither impeded by fractional nor irrational quantities, and a remarkable type of calculus for them]. Acta Eruditorum, 3, 467–473 [partial English translation in STRUIK (1986), 271–280; see also PARMENTIER (1995), 96–117].
LEIBNIZ, G.W. (1710). Symbolismus memorabilis calculi algebraici et infinitesimalis, in comparatione potentiarum et differentiarum; et de lege homogeneorum transcendentali [The remarkable symbolic algebraic calculus, infinitesimals; the transcendental law of homogeneity]. Miscellanea Berolinensia ad incrementum scientiarum, 1, 160–165.
LEVENBERG, K. (1944). A method for the solution of certain non-linear problems in least squares. Quarterly of Applied Mathematics, 2, 164–168.
LEVINSON, N. (1947). The Wiener RMS (root mean square) error criterion in filter design and prediction. Journal of Mathematics and Physics, 25, 261–278.
LINDSEY, J.P. (1960). Elimination of seismic ghost reflections by means of a linear filter. Geophysics, 25, 130–140.
LINK, R.F. and KOCH, G.S. (1975). Some consequences of applying lognormal theory to pseudolognormal distributions. Journal of the International Association for Mathematical Geology, 7, 117–128.
LODE, W. ( 1925). Versuche über den Einfluß der mittleren Hauptspannung auf die Fließgrenze [Experiments on the influence of the average major strain on the yield point]. Zeitschrift für angewandte Mathematik und Mechanik, 5, 142–144.
LODE, W. (1926). Versuche über den Einfluß der mittleren Hauptspannung auf das Fließen des Metalle, Eisen, Kupfer, und Nickel [Experiments on the influence of the average major strain on the flow of the metals, iron, copper, and nickel]. Zeitschrift für Physik, 36, 913–939.
LOMB, N.R. (1976). Least squares frequency analysis of unequally-spaced data. Astrophysics and Space Science, 39, 447–462.
LORENZ, E.N. (1963). Deterministic non-periodic flow. Journal of Atmospheric Science, 20, 130–141.
LOUIS, H. (1907). On a deficiency in the nomenclature of mineral deposits. Transactions of the Institution of Mining Engineers, 34, 286–287.
MAGIDIN, A. (2010). Lebesque integral basics [online: http://math.stackexchange.com/questions/7436/lebesgue-integral-basics].
MAIRE, C. and BOSCOVICH, R.J. (1755). De litteraria expeditione per pontificam ditionem ad dimetiendos duos meridiani gradus et corrigendam mappam geographicam. Jussu, et auspiciis Benedicti XIV, Pont. Max [An account of an expedition to measure two degrees of the meridian]. Rome, Nicolaus & Marcus Palearini.
MAIRE, C. and BOSCOVICH, R.J. (1770). Voyage astronomique et geographique, dans l’état de l’eglise, entrepris par l’ordre et sous l’auspices du Pape Beniot XIV, pour mesurer deux dégrés du méridien, and corriger la Carte de l’Etat ecclésiastique [Astronomical and geographical travel in the state of the Church, undertaken by the order and under the auspices of Pope Benedict XIV, to measure two degrees of the meridian, and to correct the map of the ecclesiastical state]. Paris, N.M. Tillard.
MANDELBROT, B. (1975a). Les objects fractales: Forme, hasard, et dimension [Fractals: Form, chance and dimension]. Paris, Flammarion.
MANDELBROT, B.B. (1982). The fractal geometry of nature. San Francisco, CA, W.H. Freeman.
MANDELBROT, B.B. (1995). Measures of fractal lacunarity: Minkowski content and alternatives. In: BANDT, C., GRAF, S. and ZÄHLE, M. (eds.). Fractal geometry and stochastics. Basel, Birkhäuser Verlag, 15–42.
MANN, C.J. (1987). Misuses of linear regression in the earth sciences. In: SIZE, W.B. (ed.). Use and abuse of statistical methods in the earth sciences. Oxford, Oxford University Press, 74–108.
MARK, D.M. and CHURCH, M. (1977). On the misuse of regression in earth science. Journal of the International Association for Mathematical Geology, 9, 63–77.
MARQUARDT, D.W. (1963). An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics, 11, 431–441.
MARTÍN-FERNÁNDEZ, J.A. and THIO-HENESTROSA, S. (2006). Rounded zeros: some practical aspects for compositional data. In: BUCCIANTI, A., MATEU-FIGUERAS, G. and PAWLOWSKY-GLAHN, V. (eds.). Compositional data analysis in the geosciences: From theory to practice. London, The Geological Society, 191–202.
MARTÍN-FERNÁNDEZ, J.A., BARCELÓ-VIDAL, C. and PAWLOWSKY-GLAHN, V. (2003). Dealing with zeros and missing values in compositional data sets using nonparametric imputation. Mathematical Geology, 35, 253–278.
MATEU-FIGUERAS, G. (2003). Models de distribució sobre el símplex [Distribution models on the simplex]. Doctoral dissertation, Catalunya, Universitat Politècnica de Catalunya.
MATHESON, I.B.C. (1990). A critical comparison of least absolute deviation fitting (robust) and least squares fitting: the importance of error distributions. Computers and Chemistry, 14, 49–57.
MAXWELL, J.C. (1873). A treatise on electricity and magnetism. 2 vols. Oxford, Clarendon Press.
MAY, R.M. (1972). Limit-cycles in predator-prey communities. Science, 177, 900–902.
McALISTER, D. (1879). The law of the geometric mean. Proceedings of the Royal Society, London, 29, 367–376.
McARTHUR, J.M., HOWARTH, R.J. and SHIELDS, G.A. (2012). Strontium isotope stratigraphy. In: GRADSTEIN, F., OGG, J., SCHMITZ, M. and OGG, G. (eds.). The geologic timescale 2012. Vol. 1. Oxford, Elsevier, 127–144.
McCARN, D.W. and CARR, J.R. (1992). Influence of numerical precision and equation solution algorithm on computation of kriging weights. Computers & Geosciences, 18, 1127–1167.
McCARTHY, J. (1960). Recursive functions of symbolic expressions and their computation by machine, Part I. Communications of the ACM, 3, 184–195.
McKAY, M.D., BECKMAN, R.J. and CONOVER, W.J. (1979). A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 21, 239–245.
McWILLIAMS, T.P. (1986). Sensitivity analysis of geologic computer models: A formal procedure based on Latin hypercube sampling. Mathematical Geology, 19, 81–90.
MEIJERING, E. (2002). A chronology of interpolation. From ancient astronomy to modern signal and image processing. Proceedings of the IEEE, 90, 319–342.
MENDEZ, L.A. (2016). A practical introduction to fuzzy logic using LISP. Studies in fuzziness and soft computing 327. Cham, Springer International Publishing.
MIESCH, A.T. (1976b). Q-mode factor analysis of geochemical and petrologic data matrices with constant row-sums. United States Geological Survey Professional Paper 574-G, Washington, DC, United States Government Printing Office.
MIESCH, A.T. and CHAPMAN, R.P. (1977). Logtransformations in geochemistry. Journal of the International Association for Mathematical Geology, 9, 191–198.
MILLER, J. (ed.) (2015a). Earliest known uses of some of the words of mathematics [online: http://jeff560.tripod.com/mathword.html].
MILLER, J. (2015c). Legendre functions of the first and second kind [online: http://www.solitaryroad.com/c679.html].
MILLER, R.L. (1949). An application of the analysis of variance to paleontology. Journal of Paleontology, 23, 635–640.
MILLER, R.L. and KAHN, J.S. (1962). Statistical analysis in the geological sciences. New York, John Wiley & Sons.
MINASAY, B. and MCBRATNEY, A.B. (2002). Uncertainty analysis for pedotransfer functions. European Journal of Soil Science, 53, 417–430.
MINASAY, B. and MCBRATNEY, A.B. (2006). A conditioned Latin hypercube method for sampling in the presence of ancillary information. Computers & Geosciences, 32, 1378–1388.
MITKIN, V.N., ZAYAKINA, S.B. and ANOSHIN, G.N. (2003). New technique for the determination of trace noble metal content in geological and process materials. Spectrochimica Acta Part B: Atomic Spectroscopy, 58, 311–328.
MITRA, S. (1976). A quantitative study of deformation mechanisms and finite strain in quartzites. Contributions to Mineralogy and Petrology, 59, 203–226.
MONTA, P. (2015). Analysis of Briggs’ first logarithm table of 1617, Logarithmorum Chilias Prima [online: http://www.pmonta.com/tables/logarithmorum-chilias-prima/index.html].
MOOKERJEE, M. and PEEK, S. (2014). Evaluating the effectiveness of Flinn’s k-value versus Lode’s ratio. Journal of Structural Geology, 68 (Part A), 33–43.
MOORE, R.C. (1949). Meaning of facies. In: MOORE, R.C. (ed.). Sedimentary facies in geological history. Memoir 39. Washington, DC, Geological Society of America, 1–34.
MORTON, N.E. (1995). LODs [logarithms of odds] past and present. Genetics, 140 (May), 7–12.
MULLER, H.-G. (1987). Weighted local regression and kernel methods for nonparametric curve-fitting. Journal of the American Statistical Association, 82, 231–238.
NÁDAI, A. (1927). Der bildsame Zustand der Werkstoffe [The plastic state of materials]. Berlin, Springer-Verlag.
NÁDAI, A. (1931). Plasticity. A mechanics of the plastic state of matter [translated by A.M. WAHL]. Engineering Societies Monograph. New York, NY, McGraw-Hill.
NAPIER, J. (1614). Mirifici logarithmorum canonis descriptio [A description of the wonderful canon of logarithms]. Edinburgh, Andrew Hart.
NAPIER, J. and BRIGGS, H. (1618). A description of the admirable table of logarithmes: with a declaration of the most plentifull, easie, and speedy use thereof in both kinds of trigonometry, as also in all mathematicall calculations [translated from Latin by E. WRIGHT]. London,. Simon Waterson.
NAPIER, J. and MACDONALD, W.R. (1889). The construction of the wonderful canon of logarithms by John Napier [translated from Latin into English with notes and a catalogue of the various editions of Napier’s works by W.R. MACDONALD]. Edinburgh, William Blackwood.
NELDER, J. and MEAD, R. (1965). A simplex method for function minimisation. Computer Journal, 7, 308–313.
OMER, G.C.O. (1947). Differential-motion seismographs. Bulletin of the Seismological Society of America, 37, 197–215.
OPPENHEIM, A.V. and SCHAFER, R.W. (2004). From frequency to quefrency: a history of the cepstrum. IEEE Signal Processing Magazine, 21 (5), 95–106.
ORTIZ, E.L. (1969). The tau method. SIAM Journal on Numerical Analysis, 6, 480–492.
ORTIZ, E.L. (1994). The tau method and the numerical solution of differential equations: Past and present research. In: BROWN, J.D., CHU, M.T., ELLISON, D.C. and PLEMMONS, R.J. (eds.). Proceedings of the Cornelius Lanczos International Centenary Conference. SIAM Proceedings Series. Philadelphia, PA, Society for Industrial and Applied Mathematics, 77–82.
PARKS, J.M. (1966). Cluster analysis applied to multivariate geological problems. The Journal of Geology, 74, 703–715.
PEARSON, K. (1898). Mathematical contributions to the theory of evolution. V. On the reconstruction of the stature of prehistoric races. Philosophical Transactions of the Royal Society, London, Series A, 192, 169–244.
PEBESMA, E.J. and HEUVELINK, G.B.M. (1999). Latin hypercube sampling of Gaussian random fields. Technometrics, 41, 303–312.
PERUGINI, D., POLI, G. and VALENTINI, L. (2004). Strange attractors in plagioclase oscillatory zoning: petrological implications. Contributions to Mineralogy and Petrology, 149, 482–497.
PHILLIPS, F.C. (1954). The use of the stereographic projection in structural geology. London, Edward Arnold.
PITMAN, E.J.G. (1939). Tests of hypotheses concerning location and scale parameters. Biometrika, 31, 200–215.
PODKOVYROV, V.N., GRAUNOV, O.V. and CULLERS, R.L. (2003). A linear programming approach to determine the normative composition of sedimentary rocks. Mathematical Geology, 35, 459–476.
POINCARÉ, H. (1881). Mémoire sur les courbes définies par une équation différentielle [Memoir on curves defined by a differential equation]. I. Journal de Mathématiques Pures et Appliquées, ser. 3, 7, 375–442.
POINCARÉ, H. (1882). Mémoire sur les courbes définies par une équation différentielle [Memoir on curves defined by a differential equation]. II. Journal de Mathématiques Pures et Appliquées, ser. 3, 8, 251–296.
POISSON, S.-D. (1835). Recherches sur la probabilité des jugements, principalement en matière criminelle [Research on the probability of judgements, mainly in criminal matters]. Comptes rendus hebdomadaires des Séances de l’Académie des Sciences, Paris, 1, 473–494.
QUENOUILLE, M.H. (1949b). Problems in plane sampling. The Annals of Mathematical Statistics, 20, 355–375.
RADNER, R. (1959). The application of linear programming to team decision problems. Management Science, 5, 143–150.
RAMBERG, H. (1959). Evolution of ptygmatic folding. Norsk geologisk Tidsskrift, 39, 99–152.
RAMSAY, J.G. (1967). Folding and fracturing of rocks. New York, McGraw-Hill.
RAMSAY, J.G. and HUBER, M.I. (1983). The techniques of modern structural geology. Vol. 1: Strain analysis. London, Academic Press.
RAZUMOVSKY, N.K. (1941). On the role of the logarithmically normal law of frequency distribution in petrology and geochemistry. Comptes Rendus (Doklady) de l’Academie des Sciences de l’URSS, 33, 48–49.
REED, G.W. (1959). Activation analysis applied to geochemical problems. In: ABELSON, P.H. (ed.). Researches in Geochemistry. New York, NY, John Wiley & Sons, 458–475.
REHDER, S. and FRANKE, D. (2012). How to include ignorance into hydrocarbon-resource assessments? A case study applied to the presence of source rock at the Argentine Deep Water Margin. Natural Resources Research, 21, 301–309.
REICHE, P. (1938). An analysis of cross-lamination. The Coconino sandstone. Journal of Geology, 46, 905–932.
REIMANN, C. and FILZMOSER, P. (2000). Normal and lognormal data distribution in geochemistry: death of a myth. Consequences for the statistical treatment of geochemical and environmental data. Environmental Geology, 39, 1001–1014.
REIMANN, C., FILZMOSER, P., GARRETT, R.G. and DUTTER, R. (2008). Statistical data analysis explained. Applied environmental statistics with R. Chichester, John Wiley & Sons.
REITER, D. and STROUJKOVA, A. (2005). Improved depth-phase detection at regional distances. In: PATTERSON, E., (ed.). Proceedings of the 27th Seismic Research Review: Ground-based nuclear explosion monitoring techniques, Rancho Mirage, California, 20–22 September 2005, United States Department of Energy, Oak Ridge, TN, 403–412.
RENDU, J.-M.M. (1988). Applications in geology. In: CROW, E.L. and SHIMIZU, K. (eds.). Lognormal distributions. Theory and applications. Statistics textbooks and monographs.v. 88. New York, Marcel Dekker, 357–366.
REYMENT, R.A., BLACKITH, R.E and CAMPBELL, N.A. (1984). Multivariate morphometrics. 2nd edn., London, Academic Press.
REYMENT, R.A. and SAVAZZI, E. (1999). Aspects of multivariate statistical analysis in geology. Amsterdam, Elsevier.
RICE, J.R. and WHITE, J.S. (1964). Norms for smoothing and estimation. SIAM Review, 6, 243–256.
RICE, R.B. (1962). Inverse convolution filters. Geophysics, 27, 4–18.
ROBINSON, E.A. (1967b). Statistical communication and detection with special reference to digital signal processing of radar and seismic signals. London, Griffin.
RODRIGUEZ-ITURBE, I. and NORDIN, C.F. (1968). Time series analyses of water and sediment discharges. Hydrological Sciences Journal, 13, 69–84.
RUTHERFORD, E. (1937). The search for isotopes of hydrogen and helium of mass 3. Nature, 140, 303–305.
RUTHERFORD, E. and GEIGER, H. (1908). An electrical method of counting the number of α-particles from radioactive substances. Proceedings of the Royal Society, London, ser. A, A81, 141–161.
SABINE, E. (1841). Contributions to terrestrial magnetism. No. II. Philosophical Transactions of the Royal Society, London, 131, 11–35.
SANDER, B. (1930). Gefügekunde der Gesteine mit besonderer Berücksichtigung der Tektonite [Microstructure of rocks with special emphasis on tectonite]. Vienna, Springer.
SANDRI, M. (1996). Numerical calculation of Lyapunov exponents. The Mathematica Journal, 6 (3), 78–84.
SANTOS, E.T.F. and BASSREI, A. (2007). L- and θ-curve approaches for the selection of regularization parameter in geophysical diffraction tomography. Computers & Geosciences, 33, 618–629.
SAUVAGEAU, M. and KUMRAL, M. (2015). Analysis of mining engineering data using robust estimators in the presence of outliers. Natural Resources Research, 24, 305–316.
SCARGLE, J.D. (1982). Studies in astronomical time series analysis. II. Statistical aspects of spectral analysis of unevenly spaced data. The Astrophysical Journal, 263, 835–853.
SCARGLE, J.D. (1989). Studies in astronomical time series analysis. III. Fourier transforms, autocorrelation functions, and cross-correlation functions of unevenly spaced data. The Astrophysical Journal, 343, 874–887.
SCHMIDT, W. (1925). Gefugestatistik [Microstructural (petrofabric) statistics]. Tschermak’s Mineralogische und Petrographische Mitteilungen, 38, 392–423.
SCHUENEMEYER, J.H. and DREW, L.J. (1983). A procedure to estimate the parent populastion of the size of oil and gas fields as revealed by a study of economic truncation. Journal of the International Association for Mathematical Geology, 15, 145–162.
SCHULZ, M. and STATTEGGER, K. (1997). SPECTRUM: Spectral analysis of unevenly spaced paleoclimatic time series. Computers & Geosciences, 23, 929–945.
SCHWARZACHER, W. (1964). An application of statistical time-series analysis of a limestone-shale sequence. Journal of Geology, 72, 195–213.
SCHWERDTFEGER, H. (1950). Introduction to linear algebra and the theory of matrices. Groningen, P. Noordhoff.
SEYEDGHASEMIPOUR, S.J. and BHATTACHARYYA, B.B. (1990). The loghyperbolic: An alternative to the lognormal for modelling oil field size distribution. Mathematical Geology, 22, 557–571.
SHAH, B.K. and DAVE, P.H. (1963). A note on log-logistic distribution. Journal of the Maharaja Sayajirao University of Baroda, 12 (2–3), 15–20.
SHERIFF, R.E.(1984). Encyclopedic dictionary of exploration geophysics. 2nd edn., Tulsa, Society of Exploration Geophysicists.
SICHEL, H.S. (1947). An experimental and theoretical investigation of bias error in mine sampling with special reference to narrow gold reefs. Transactions of the Institution of mining and Metallurgy, London, 56, 403–474.
SIMPSON, S.M. (1954). Least squares polynomial fitting to gravitational data and density plotting by digital computer. Geophysics, 19, 255–269.
SIMPSON, S.M., Jr. (1955). Similarity of output traces as a seismic operator criterion. Geophysics, 20, 254–269.
SOBEL, D. (1995). Longitude: the true story of a lone genius who solved the greatest scientific problem of his time. New York, NY, Walker.
SORNETTE, D. and SAMMIS, G.C. (1995). Complex critical exponents from renormalisation group theory of earthquakes: Implications for earthquake predictions. Journal de Physique. ser. 1, 5, 607–619.
SPEARMAN, C.E. (1904b). ‘General intelligence’ objectively determined and measured. American Journal of Psychology, 15, 201–293.
SPEIDELL, J. (1619). New logarithmes. London, John Speidell.
SPROTT, J.C. (2009). Simplifications of the Lorenz Attractor. Nonlinear Dynamics, Psychology, and Life Sciences, 13, 271–278.
STARK, C.P. and HOVIUS, N. (2001). The characterisation of landslide size distributions. Geophysical Research Letters, 28, 1091–1094.
STOICA, P. (1993). List of references on spectral line analysis. Signal Processing, 31, 329–340.
STRINGHAM, I. (1893). Uniplanar algebra. Berkley, CA, The Berkley Press.
STRUIK, D.J. (ed.) (1986). A source book in mathematics, 1200–1800. Princeton, NJ, Princeton University Press.
SWAIN, J.J. (1990). Nonlinear regression. In: WADSWORTH, H.M. (ed.). Handbook of statistical methods for engineers and scientists. 2nd edn., New York, NY, McGraw-Hill, 18.1–18.31.
SYLVESTER, J.J. (1883a). Lectures on the principles of universal algebra. American Journal of Mathematics, 6, 270–286.
SYLVESTER, J.J. (1883b). On the equation to secular inequalities in the planetary theory. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, ser. 5, 16, 267–629.
TAYLOR, H.L. (1981). The L1 norm in seismic data processing. Developments in Geophysical Exploration, 2, 53–76.
TCHÉBYCHEF, P.-L. (1867). Des valeurs moyennes [translated from Russian by N. DE KHANIKOV]. Journal de mathématiques pures et appliquées. ser. 2, 12, 177–184.
TEISSIER, G. (1948). La relation d’allometrie sa signification statistique et biologique [The allometric relationship: its statistical and biological significance]. Biometrics, 4, 14–53.
ter BRAAK, C.J.F. and JUGGINS, S. (1993). Weighted averaging partial least squares regression (WA-PLS): an improved method for reconstructing environmental variables from species assemblages. Hydrobiologia, 269, 485–502.
THANASSOULAS, C., TSELENTIS, G.-A. and DIMITRIADIS, K. (1987). Gravity inversion of a fault by Marquardt’s method. Computers & Geosciences, 13, 399–404.
THOMPSON, G.T. (1992). The grand unified theory of least squares: f2(N) = f (2N). Computers & Geosciences, 18, 815–822.
THOMPSON, S.E. and KATUL, G.G. (2012). Multiple mechanisms generate Lorentzian and 1/f^α power spectra in daily stream-flow time series. Advances in Water Resources, 37, 94–103.
THOMSON, W. [Lord Kelvin] (1847). On a mechanical representation of electric, magnetic and galvanic forces. Cambridge and Dublin Mathematical Journal, new ser., 2, 61–64.
THOMSON, W. [Lord Kelvin] (1899). On the reflexion and refraction of solitary plane waves at a plane interface between two isotropic elastic mediums – fluid, solid or ether. London and Edinburgh Philosphical Magazine, ser. 5., 47, 179–191.
THURSTONE, L.L. (1931). Multiple factor analysis. Psychological Revue, 38, 406–427.
TODHUNTER, I. (1873). A history of the mathematical theories of attraction and the Figure of the Earth. From the time of Newton to that of Laplace. London, Macmillan.
TROUTMAN, B.M. and WILLIAMS, G.P. (1987). Fitting straight lines in the earth sciences. In: SIZE, W.B. (ed.). Use and abuse of statistical methods in the earth sciences. Oxford, Oxford University Press, 107–128.
TUKEY, J.W. (1958a). Bias and confidence in not-quite large samples [abstract]. The Annals of Mathematical Statistics, 29, 614.
TUKEY, J.W. (1959b). An introduction to the measurement of spectra. In: GRENANDER, U. (ed.). Probability and statistics. The Harald Cramér volume. New York, NY, John Wiley & Sons, 300–330.
TUKEY, J.W., (1965). The future of processes of data analysis. Proceedings of the Tenth Conference on the Design of Experiments in Army Research. Development and Testing. ARO-D Report 65-3. Durham, NC, United States Army Research Office, 691–729 [Reprinted in: BRILLINGER, D.R. (ed.) (1984): The Collected Works of John W. Tukey. Vol. IV. Philosophy: 1965–1986. Monterey, CA, Wadsworth, 517–549].
TUKEY, J.W. and HAMMING, R. W. (1949). Measuring noise color. I. Memorandum MM-49-110-119, 1 December 1949, Murray Hill, NJ, Bell Telephone Laboratory, 1–120 [Reprinted in: BRILLINGER, D.R. (ed.) (1984). The collected works of John W. Tukey. Vol. 1. Time series: 1949–1964. Wadsworth, Pacific Grove, CA, 1–127].
TURCOTTE, D.L. (1997). Fractals and chaos in geology and geophysics. 2nd edn., Cambridge, Cambridge University Press.
ULRYCH, T.J. (1972). Maximum entropy power spectrum of truncated sinusoids. Journal of Geophysical Research, 77, 1396–1400.
ULRYCH, T.J. and OOE, M. (1979). Autoregressive and mixed autoregressive-moving average models and spectra. In: HAYKIN, S. (ed.). Nonlinear methods of spectral analysis. Berlin, Springer-Verlag, 73–125.
ULRYCH, T.J., SMYLIE, D.E., JENSEN, O.G. and CLARKE, G.K.C. (1973). Predictive filtering and smoothing of short records by using maximum entropy. Journal of Geophysical Research, 78, 4959–4964.
UNWIN, D.J. and WRIGLEY, N. (1987). Towards a general theory of control point distribution effects in trend-surface models. Computers & Geosciences, 13, 351–355.
van der POL, B. (1926). On relaxation-oscillations. Philosophical Magazine, ser. 7, 2, 978–992.
van der POL, B. and van der MARK, J. (1927). Frequency demultiplication. Nature, 120, 363–364.
VAUGHAN, S., BAILEY, R.J. and SMITH, D.G. 2011. Detecting cycles in stratigraphic data: Spectral analysis in the presence of red noise. Paleogeography, 26, PA4211 [online: http://dx.doi.org/10.1029/ 2011PA002195].
VERHULST, P.-F. (1845). Recherches mathématiques sur la loi d’accroissement de la population [Mathematical research on the law of population growth]. Nouveaux Mémoires de l’Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique, 18, 1–32.
VISTELIUS, A.B. (1961). Sedimentation time trend functions and their application for correlation of sedimentary deposits. Journal of Geology, 69, 703–728.
VISTELIUS, A.B. (ed.) (1967). Studies in mathematical geology. New York, NY, Consultants Bureau (Plenum Press).
VISTELIUS, A.B. (1980). Osnovy matematičeskoj geologii [Essential mathematical geology]. Leningrad, AN SSSR Izdatel’stvo nauk.
VISTELIUS, A.B. (1992). Principles of mathematical geology [translated by S.N. BANERGEE]. Dordrecht, Kluwer.
WALKER, M.R. and JACKSON, A. (2000). Robust modelling of the Earth’s magnetic field. Geophysical Journal International, 143, 1353–1368.
WALTER, É. (2014). Numerical methods and optimization. A consumer guide. Cham, Springer International.
WARING, E. (1779). Problems concerning interpolations. Philosophical Transactions of the Royal Society, London, 69, 59–67.
WATTIMENA, R.K. (2013). Predicting probability stability of rock slopes using logistic regression. International Journal of the Japanese Committee for Rock Mechanics, 9, 1–6.
WEBSTER, R. (1997). Regression and functional relations. European Journal of Soil Science, 48, 557–566.
WEEDON, G.P. (2003). Time series analysis and cyclostratigraphy. Cambridge, Cambridge University Press.
WEISSTEIN, E.W. (ed.) (2015). MathWorld – A Wolfram web resource [online: http://mathworld.wolfram.com].
WESSON, R.L. (1970). A time integration method for computation of the intensities of seismic rays. Bulletin of the Seismological Society of America, 60, 307–316.
WIDROW, B. and HOFF, M.E. (1960). Adaptive switching circuits. In: IRE WESCON Convention Record: at the Western Electronic Show and Convention, Los Angeles, California, August 23–26, 1960, Institute of Radio Engineers, 96–104.
WIDROW, B. and STEARNS, S.D. (1985). Adaptive signal processing. Englewood Cliffs, NJ, Prentice-Hall.
WIENER, N. (1942). The extrapolation, interpolation and smoothing of stationary time series with engineering applications. D.I.C. Contract 6037, A research pursued on behalf of the National Defence Research Council (Section D) February 1, 1942. Cambridge, MA, The Massachusetts Institute of Technology.
WIENER, N. (1949). Extrapolation, interpolation, and smoothing of stationary time series: with engineering applications. Cambridge, MA, Technology Press, Massachusetts Institute of Technology.
WILCZYNSKI, E.J. (1911). One-parameter families and nets of plane curves. Transactions of the American Mathematical Society, 12, 473–510.
WILSON, E.B. and LUYTEN, W.J. (1925). The frequency distribution of some measured parallaxes and of the parallaxes themselves. Proceedings of the National Academy of Sciences of the United States of America, 11, 270–274.
WILSON, E.B. and WORCESTER, J. (1945). The normal logarithmic transform. The Review of Economics and Statistics, 27, 17–22.
WINSTON, P.H. and HORN, B.K.P. (1981). LISP. Reading, MS., Addison-Wesley.
WRIGHT, C.J., MCCARTHY, T.S. and CAWTHORN, R.G. (1983). Numerical modelling of trace element fractionation during diffusion controlled crystallization. Computers & Geosciences, 9, 367–389.
YAGER, R.M. (1998). Detecting influential observations in nonlinear regression modelling of groundwater flow. Water Resources Research, 34, 1623–1633.
YANG, C.-S. and KOUWE, W.F.P. (1995). Wireline log-cyclicity analysis as a tool for dating and correlating barren strata: an example from the Upper Rotliegend of The Netherlands. In: DUNAY, R.E. and HAILWOOD, E.A. (eds.). Non-biostratigraphical methods of dating and correlation. Special Publication 89. London, The Geological Society, 237–259.
YUEN, D.A. (ed.) (1992). Chaotic processes in the geological sciences. The IMA Volumes in Mathematics and its Applications. 41. New York, NY, Springer-Verlag.
ZEEMAN, P. (1896). Over den invloed eener magnetisatie op den aard van het door een stof uitgezonden licht [On the influence of magnetism on the nature of the light emitted by a substance]. Amsterdam, Koninklijke Akademie van Wetenschappen te Amsterdam.
ZHDANOV, M.S. (2002). Geophysical inverse theory and regularization problems. Methods in geochemistry and geophysics 36. Amsterdam, Elsevier.
ZHOU, D., CHEN, H. and LOU, Y. (1991). The logratio approach to the classification of modern sediments and sedimentary environments in northern South China Sea. Mathematical Geology, 23, 157–165.
ZITKOVIC, G. (2013). The Lebesgue integral [online: https://www.ma.utexas.edu/users/gordanz/notes/lebesgue_integration.pdf].
ZOBEL, O.J. (1923a). Electrical wave filter. United States Patent Office, Patent number 1,615,212 [filed 1923, granted 1927].
ZOBEL, O.J. (1923b). Theory and design of uniform and composite electric wave filters. Bell Systems Technical Journal, 2, 1–46.
ZOBEL, O.J. (1923c). Transmission characteristics of electric wave filters. Bell System Technical Journal, 2, 567–620.
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Howarth, R.J. (2017). L. In: Dictionary of Mathematical Geosciences . Springer, Cham. https://doi.org/10.1007/978-3-319-57315-1_12
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