Krumbein, William Christian

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(b. Beaver Falls, Pennsylvania, 28 January 1902; d. Los Angeles, California, 18 August 1979),

geology, quantitative methodology, sediment analysis, stratigraphic analysis.

Krumbein is the pioneer in the application of quantitative methods in stratigraphy and sediment analysis. His many innovations include the logarithmic transform of particle size-distributions or phi scale, now almost universally used to describe the grain size of clastic sediments, the lithology tetrahedron to describe the complex interrelations between tectonic-sedimentary facies, the application of Markov chains to describe the stratigraphic memory of lithologic formations through geologic time, and the introduction of the general linear model in the statistical analysis of multidimensional geologic variables. Krumbein was a true trailblazer in the study of sediments and sedimentary rocks—years ahead of his time—and as a consequence, he was not recognized in his lifetime for his innovations by his more conventional colleagues.

Early Life . William Christian Krumbein, known as the father of mathematical geology, was born of German immigrants. An early marriage ended in failure and in 1946 he married Marjorie Kamm. Because Krumbein was a reserved person, little is known about his family, early childhood, and upbringing, other than that his parents, Carl and Hattie, moved to a largely German neighborhood in northwest Chicago when he and his older brother Henry J. were very young.

Academic Formation . Krumbein received his PhB in business administration from the University of Chicago in 1926 and worked for several years in the field of insurance adjustments with a finance company in downtown Chicago. Legend among his students had it that during the stock market crash of 1929, having witnessed a suicide fall from his skyscraper window, he returned to the University of Chicago and began graduate work in geology. His master’s thesis was titled “A Key for the Determination of Minerals by Means of Structure, Form, and Texture,” and he received an MSc in geology in 1930. His doctoral dissertation, written under J. Harlan Bretz and titled “The Mechanical Analysis of Related Samples of Glacial Till,” earned him a PhD in geology from the University of Chicago in 1932.

Krumbein’s interest in geology was stimulated when he was an undergraduate business student at the University of Chicago by Paul MacClintock, from whom he took an introductory geology course. Later, as a graduate student in geology taking Francis Pettijohn’s first course in sedimentation, Krumbein recognized the potentialities of statistical analysis, already familiar to him from business school, to the description of sediments. While in graduate school, Krumbein shared an office with M. King Hub-bert, who became famous for his petroleum resource prediction models; in hindsight this association proved to be very fruitful for petroleum geology.

Between 1933 and 1945, Krumbein rose from instructor to associate professor at the University of Chicago. During World War II he served with the Beach Erosion Board of the U.S. Army Corps of Engineers in Washington, D.C. (1942–1945), in the area of beach-landing intelligence, and then briefly worked for the Gulf Research and Development Corporation (1945–1946). In 1946 Krumbein became professor of geology at Northwestern University, and from 1960 until his mandatory retirement from Northwestern in 1970, he was William Deering Professor of Geological Sciences. In 1955 he was on leave for research at the National Bureau of Standards in Washington, D.C.

Scientific Production . Krumbein wrote more than 140 papers, including one published posthumously, and four books during his forty-seven years of productivity. His interests shifted appreciably from sediment analysis (39 publications) to stratigraphic analysis (27 publications) and finally to statistical methodology in geology (62 publications). His books followed the same trend: Manual of Sedimentary Petrography (1938), with Pettijohn, was followed by Stratigraphy and Sedimentation (1st ed., 1951; 2nd ed, 1963), with Laurence Sloss, and subsequently by An Introduction to Statistical Models in Geology (1965), with Franklin Graybill. Only six papers cannot be classified into these three categories; they are concerned mainly with educational matters and the use of computers in geology, as well as futuristic trends in geology.

He intended his first book, Down to Earth, coauthored with Carey Croneis in 1936, as an introductory text for a short course in geology. It attempted to develop an approach that was analytical instead of being solely a descriptive treatment of geological sciences. The purpose was to bring out the true relationship of geology to other sciences, following the “back to nature” trend observed by previous authors. In their 1936 preface Krumbein and Croneis explain the “back to nature trend” as the growing public awareness to understand the earth sciences—which they can see and feel—as opposed to the “layman’s relatively decent understanding of atoms electrons and genes which they cannot see.” This book is a factual depiction and philosophical interpretation of the history of Earth and its inhabitants through geologic time. It ends with a prognosis of humankind’s prospects on Earth that claims that “if he learns to control nature even reasonably completely, his mind and his studies will, indeed, have brought him immortality” (p. 484).

Krumbein’s scientific productivity is legendary. His clear analytical ability, which helped him essentially to draft entire papers in his mind, and an iron discipline at the desk and typewriter, supported this productivity. He revised, edited, and corrected his own writing at least twice before giving it to a colleague for review, usually Laurence Sloss or Michael Dacey; he then produced the final version himself, seldom using the service of his part-time secretary. His papers are characterized by clear, careful prose, and he explained everything, but only once; there was no repetition in his papers.

As Krumbein often said in class, “it would be fun” to examine his scientific productivity with his own statistical methods. Arthur Howland published a complete bibliography of Krumbein in “William C. Krumbein: The Making of a Methodologist” (1975). This bibliography enables one to calculate the number of works and pages he published per year as well as the average length of his papers, and also to classify them according to the primary main topics: sediment analysis, stratigraphic analysis, and statistical methodology. Additionally, he published papers on Earth science teaching, reviews of the state of the art in computers used in geology, and analysis of geological bibliography. The bibliography data matrix shows that to 1975 he published 132 papers and four books on geology, for totals of 2,805 and 2,645 pages respectively. His papers average twenty-one pages in length, but this figure is somewhat misleading because it mixes soft-copy reports and printed papers.

As noted briefly above, Krumbein’s research interests shifted through time. He concentrated from 1932 to 1944 on the analysis of sediments and on the statistical methodology it required. He published only two papers on the stratigraphy of clastic units in this period. During this time he worked at the University of Chicago and spent his sabbatical year at the hydraulics lab at the University of Iowa. In Iowa he worked with wave tanks and at the University of California–Berkeley on beach studies. This research focused on all aspects of grain-size parameters, including some interesting conclusions about permeability as a function of size parameters of unconsolidated sands, a pioneering work that remains relevant in the early twenty-first century. This period produced the previously mentioned books, Down to Earth and the classical Manual of Sedimentary Petrography. Few books have had so major an impact on a discipline as the Manual, which shaped the way sediments were analyzed from then on.

A major shift toward stratigraphic analysis occurred, apparently from mid-1945 to mid-1946, when Krumbein was working at the Gulf Research and Development Company labs. His interests probably changed because of the difference in expectations between the petroleum industry and academia and to the interchange of ideas and contacts with fellow researchers such as Charles Ryniker, Roy Hazzard, and Sigmund Hammer, who directed Krumbein’s attention respectively to limestone, subsurface, and geophysical studies. He devoted the next eight years mainly to the production of stratigraphic and sedimentological papers, the most important being the development of the facies tetrahedron, Lithofacies Maps: An Atlas of the United States and Southern Canada (1960) with Laurence Sloss and Ed Dapples. He studied the concepts of recent sedimentation, diagenesis and oil exploration, and shales and their environmental significance, as well as the importance of following a single stratigraphic horizon over a considerable area showing the regional stratigraphic and diagenetic patterns. This activity led to the classic book, Stratigraphy and Sedimentation (1951), coauthored with Sloss.

From 1953 on, Krumbein published a markedly greater number of papers on statistical methodology. The topics of sampling, populations, Latin squares, regression, discriminant analysis, trend surfaces (both polynomial and Fourier), time series, and Markov processes, culminating in the unifying theory of the general linear model were explored and written about in this period. Another classic book, An Introduction to Statistical Models in Geology (1965), or the “yellow bible,” as it was affectionately called by the students, was written with Graybill.

The Teacher . Krumbein was a passionate teacher and a relentless researcher. His twenty-five graduate students include many prominent scientists. He taught his students—of which this author was one—the value of questioning established theories and paradigms, and he also encouraged them to have fun doing so, because for him “statistics was fun.” He showed them many ways to gather information, process it, and most important of all, to interpret the meaning of their findings. During one of my daily tutoring sessions with him for my master’s thesis, I once commented, regarding some Venezuelan cyclothems, that “this does not mean anything.” He corrected me with words that still, after almost forty years, ring in my ears: “There does not fall a leaf from a tree that does not have a meaning.”

The Geology Department of Northwestern University at the end of this golden era had immensely influential professors: Krumbein (geostatistics), Robert Garrels (geochemistry), Laurence Sloss (stratigraphy), E. H. Timothy Whitten (tectonics), Arthur Howland (petrography), and others. In the department’s Common Room, students and professors gathered informally over coffee and donuts for open discussion of everyone’s scientific problems. Many ideas were tossed around, polished, and examined for consistency and truth. One left these informal gatherings fortified not only with coffee—or occasionally beer— but also with a clearer understanding of one’s research problems. During a particularly severe winter storm over Lake Michigan, Krumbein showed us the importance of the seventh wave, a wave that is stronger and higher than the preceding ones, a theory that I later successfully applied to Caribbean beaches.

As Krumbein’s research assistant while a master’s candidate in the late 1960s, I was encouraged by him to work constantly on my thesis. The work was interrupted only for several daily trips to the computing center to hand in boxes of computer cards and retrieve sheets of printout, which he immediately inspected. There followed a half hour of tutoring, which did not feel like pedagogy at all, because he was thinking aloud, examining the meaning of the transition probability matrices and stabilization vectors of Markov simulations. After that he would invariably say, “wouldn’t it be nice to …” sending me away with new ideas and work instructions to keep me busy for the rest of the evening. Early the next morning he would knock at the door of my little office and inquire about the results of the night’s computer runs. This work regime kept me busy and “off the streets,” as he liked to say. In hindsight, it was also an invaluable training for my professional life.

One of the most remarkable qualities of this extraordinary teacher was that he followed the same discipline in his revisions of his students’ master’s theses and PhD dissertations. Chapters handed in during the afternoon (written over a couple of days) were returned early the next morning with corrections. He always had a word of encouragement, often along with new ideas to be developed during the week. Even when working on Saturday, he never closed his office door. Of the many ideas that he tossed out during these sessions was one that would shape my future research—the observation by M. L. Banks, a colleague during his short stay at Gulf Research and Development Labs in Pittsburgh, who had observed and published the intimate relationship of coal beds in the Oficina Formation of Eastern Venezuela to the amount of petroleum that occurred in adjacent sandstone reservoirs.

The annual Geology Department field trip in the fall quarter revealed other facets of Krumbein’s personality. During long, tedious travel along highways of the North American Midwest, he kept us awake by counting makes and colors of passing vehicles. He elaborated transition probability matrices and discussed the value of the equilibrium vector as the end product of a customer survey.

We applied this methodology in the evenings to makes of beers and colors of ladies’ skirts in local bars. He introduced many graduate students to time series and Markov processes this way. More serious applications to batholith crystallization patterns and thin-section analysis followed, producing lively discussions at outcrops. Krumbein liked to observe geologists on the outcrops, never getting too involved in the nitty-gritty of geology or paleontology, but always attentive to any theory that might arise and suggesting ways to prove or disprove the paradigm.

Krumbein considered that in geology, “most statistical methods are powerful enough to handle data distributed with a peak more or less in the middle.” He was adamant about correctly identifying populations and sampling strategies: “[Sampling] is inherent to the model— you must have a conceptual picture in your mind—and the question you want to answer.” He pointed out that “in many geological studies there is a large amount of biased observations—contacts, etc. and virtually nothing in between,” that “statistical inferences are made from samples to sample populations,” something frequently forgotten because “we take samples and literally beat them to death,” without knowing “how much is my own variation—which I contribute to the study.” In those days of large computer centers, when nobody envisioned the personal computer, he used to say that “the day will come when there is a computer at every drugstore,” but if you put “garbage in—you get garbage out—no matter how much you beat it.”

Krumbein philosophized in lectures that “a well trained geologist can get a pretty good mean value, but has larger variance than randomized samples,” something that traditionally educated geologists absorbed with difficulty. He asked students to look “at your leisure” at the important papers on statistics in geology, and in the next class would begin with the statement that “the important attribute of the variance is that it be additive, that makes it much simpler to pay with.” This latter statement stayed in my mind long after his death and was instrumental to produce Krumbein’s posthumous paper “CORSURF—A Covariance-Matrix Trend-Analysis FORTRAN IV Computer Program” written with Daniel Merriam in 1995.

Awards and Honors . Krumbein was a charter member of the International Association for Mathematical Geology and was its first president. The association named its premier award the William Christian Krumbein Medal. He was a Fellow of Geological Society of America, the American Association for the Advancement of Science, and the American Statistical Association. He belonged to the Society for Sedimentary Geology (and was its president in 1950), the Society of Economic Geologists, the American Association of Petroleum Geologists, the American Geophysical Union, and the Illinois Academy of Science. He received a Guggenheim Fellowship, was a Fulbright lecturer, and was a President’s Fellow at Northwestern University. In 1978 Krumbein received the William H. Twenhofel Medal from the Society of Sedimentary Geology.

In 1975 Krumbein’s colleagues honored him with a Festschrift, William C. Krumbein, The Making of a Methodologist, in recognition of “his stimulating teaching and guidance … and continuing leadership and research.” It was published by the Geological Society of America. Krumbein died of a heart attack in 1979. He was survived by his second wife, Marjorie.


For a complete bibliography of Krumbein’s works to 1975, see A.L. Howland, “William C. Krumbein: The Making of a Methodologist,” in Quantitative Studies in the Geological Sciences, a Memoir in Honor of William C. Krumbein, edited by E. H. Timothy Whitten (Boulder, CO: Geological Society of America, 1975).


“A History of the Principles and Methods of Mechanical Analysis.” Journal of Sedimentary Petrology 2 (1932): 89–124.

With Carey Croneis. Down to Earth: An Introduction to Geology. Chicago: University of Chicago Press, 1936.

With Francis J. Pettijohn. Manual of Sedimentary Petrography. NewYork: D. Appleton-Century, 1938.

With Laurence L. Sloss and Edward C. Dapples. Integrated Facies Analysis. Boulder, CO: Geological Society of America, 1949.

———. “Sedimentary Tectonics and Sedimentary Environments.” American Association of Petroleum Geologists Bulletin 33 (1949): 1859–1891.

With Robert M. Garrels. “Origin and Classification of Chemical Sediments in Terms of pH and Oxidation-Reduction Potentials.” Journal of Geology 60 (1952): 1–32.

With R. L. Miller. “Design of Experiments for Statistical Analysis of Geological Data.” Journal of Geology 61 (1953): 510–532.

“Regional and Local Components in Facies Maps.” American Association of Petroleum Geologists Bulletin 40 (1956): 2163–2194.

“Trend-Surface Analysis of Contour-Type Maps with Irregular Control-Point Spacing.” Journal of Geophysical Research 64 (1959): 823–834.

“The ‘Geological Population’ as a Framework for Analysing Numerical Data in Geology.” Liverpool and Manchester Geological Journal 2 (1960): 341–368.

With Edward C. Dapples and Laurence L. Sloss. Lithofacies Maps: An Atlas of the United States and Southern Canada. New York: John Wiley and Sons, 1960.

“Open and Closed Number Systems in Stratigraphic Mapping.” American Association of Petroleum Geologists Bulletin 46 (1962): 2229–2245.

With Laurence L. Sloss. Stratigraphy and Sedimentation. 2nd ed. San Francisco, CA: W.H. Freeman, 1963.

With Franklin A. Graybill. An Introduction to Statistical Models in Geology. New York: McGraw-Hill, 1965. “The Cyclothem as a Response to Sedimentary Environment and Tectonism.” In Symposium on Cyclic Sedimentation, edited by Daniel F. Merriam. Lawrence: Kansas Geological Survey Bulletin, 1966.

With Wolfgang Scherer. Structuring Observational Data for Markov and Semi-Markov Models in Geology. Washington, DC: U.S. Office of Naval Research, Technical Report 15, Contract N00014.67-A0356-0018 (formerly Nonr-1228(36); NR 389-150), 1968.

“Markov Models in the Earth Sciences.” In Concepts in Geostatistics, edited by Richard B. McCammon. New York: Springer-Verlag, 1975.

With Daniel F. Merriam and Wolfgang Scherer. “CORSURF: A Covariance-Matrix Trend-Analysis FORTRAN IV Computer Program.” Computers and Geosciences 21 (1995): 1065–1089.


Howland, Arthur L. “William C. Krumbein: The Making of a Methodologist.” In Quantitative Studies in the Geological Sciences, a Memoir in Honor of William C. Krumbein, edited by E. H. Timothy Whitten. Boulder, CO: Geological Society of America, 1975.

Merriam, Daniel F. “William C. Krumbein (1902–1979).” In Encyclopedia of Sediments and Sedimentary Rocks, edited by Gerard V. Middleton. Boston: Kluwer Academic, 2003.

Wolfgang Scherer