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Geologische Rundschau

, Volume 74, Issue 1, pp 165–177 | Cite as

Application of Markov chain and entropy analysis to lithologic successions: An example from the Cretaceous of the Benue trough (Nigeria)

  • M. Hoque
  • C. S. Nwajide
Article

Keywords

Entropy Nigeria Facies Association Section Stratigraphiques Trial Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Zusammenfassung

Eine Markov'sche Kettenanalyse wird in der Untersuchung von zwei stratigraphischen Ausschnitten aus der Kreide des Benue-Troges in Nigeria eingesetzt. Vier Faziesabschnitte bestehend aus grobkörnigem, mittelkörnigem und feinem Sandstein sowie Tonstein, lassen sich trennen. Matrizen zu den Faziesübergängen werden nach einem eingeschlossenen und einem mehrstufigen Markovmodell gebildet. Daraus lassen sich unabhängige Probematrizen ableiten. Bei einem Vergleich dieser zwei Serien von Matrizen und der Anwendung von zwei statistischen Tests zu diesen Matrizen stellt man fest, daß die untersuchten Formationsausschnitte Markov'sche Eigenschaften aufweisen, die eine nach oben feiner werdende Zyklizität repräsentieren. Der Fazieszyklus läßt sich als eine Faziesabfolge von Flußrinnen, Gleithängen und Überflutungsebenen eines Flußsystems interpretieren. Mit der Methode einer Fazieszählung läßt sich die Zusammensetzung eines Modalzyklus ermitteln; eine übergeordnete Matrize zeigt wichtige Zusammenhänge auf, die in der eingeschlossenen Matrize nicht sichtbar sind. Im allgemeinen nehmen die Chi-Quadratzahlen bei der Abnahme der Zahl der Nullen in der Ubergangsmatrize zu. Deshalb hat die Wahl der Zählmethode Einfluß auf die Chi-Quadratzahlen. Die Zufälligkeit im Auftreten eines Faziestypes innerhalb eines Zyklus wird durch die Entropie ausgedrückt, die von den Markovmatrizen abgeleitet werden kann. Zwei Arten von Entropien werden für jede Fazies berechnet: die Entropie nach der Ablagerung und vor der Ablagerung bilden zusammen eine Entropieserie. Die Entropie des Gesamtsystems läßt sich ebenfalls berechnen. Diese Werte werden graphisch dargestellt und mit den idialisierten Diagrammen von Hattori verglichen, wobei sich zeigt, daß es sich hier im wesentlichen um einen abgeschnittenen, asymmetrischen Zyklus (Typ A-4 von Hattory) handelt, der in einem Flußsystem abgelagert wurde. Die Untersuchung zeigt weiterhin, daß die Anwendung der Entropiefunktion zusammen mit der Markov'schen Kettenanalyse charakteristische Züge einer Sedimentserie erkennen läßt, die als diagnostischer Hinweis bei der Bewertung von Ablagerungsbedingungen in einer stratigraphischen Serie gewertet werden können.

Abstract

Markov chain analysis is used in the study of two composite stratigraphic sections from the Cretaceous of the Benue trough of Nigeria. Four facies states, consisting of coarse sandstone, medium sandstone, fine sandstone and shale, are recognized in the sections. Faciès transition matrices are structured on embedded Markov and multistory Markov model, from which independent trials matrices are derived. By comparing these two sets of matrices, and applying two statistical tests to the matrices, it is established that the stratigraphic sections examined exhibit a Markovian property with a fining-upward cyclicity. The cycle is interpreted in terms of in-channel, point-bar and overbank facies association in a fluvial system. Methods of faciès count determine the composition of the modal cycle in a succession; multistory matrix reveals important relationships between facies states that are not apparent in the embedded matrix. Generally, Chi-square values increase with decrease in the number of zeros in the transition matrix; therefore, choice of count method has an effect on Chi-square value. The randomness in the occurrence of facies within a cycle is evaluated in terms of entropy which can be calculated from the Markov matrices. Two types of entropies are calculated for every facies state: entropy after deposition and entropy before deposition, which together form an entropy set; the entropy for the whole system is also calculated. These values are plotted and compared withHattori's idealized plots which indicate that the sequence is essentially a truncated asymmetric cycle (type A-4 ofHattori), and was deposited in a fluvial environment. The study further suggests that use of the entropy function together with Markov chain analysis reveals some characteristic features in a succession which can be used as diagnostic evidence in evaluating depositional environment of a stratigraphic sequence.

Résumé

On utilise une analyse par chaîne de Markov pour étudier deux sections stratigraphiques composites du Crétacé dans l'auge de Benue au Nigeria. Ces sections comportent quatre facies: grès grossier, grès moyen, grès fin et schiste. Les matrices de transition des facies sont structurées d'après un modèle de Markov emboîté et un modèle à étages multiples, à partier de quoi on a dérivé des matrices d'essais indépendantes. En comparant ces deux ensembles de matrices et en appliquant deux tests statistiques à ces matrices, on a établi que les sections stratigraphiques examinées ont une propriété Markovienne avec une cyclicité des matériaux par affinement vers le haut. Le cycle est interprété comme l'association des facies de chenal, de berge (»point-bar«) et de rive (»overbank«) dans un système fluvial. Les méthodes de comptage du facies déterminent la composition du cycle modal dans une séquence. La matrice à étages multiples révêle des relations importantes entre les facies, qui ne sont pas apparentes dans la matrice emboîtée. En général, les valeurs chi-carré augmentent lorsque le nombre de zéros décroît dans la matrice de transition. Par conséquent, le choix de la méthode de comptage a un effet sur la valeur chi-carré. Le caractère plus ou moins aléatoire de l'apparition des facies à l'intérieur d'un cycle s'évalue en termes d'entropie que l'on peut calculer à partir des matrices de Markov. On calcule deux sortes d'entropies pour chaque faciès: l'entropie après dépôt et l'entropie avant dépôt, qui forment ainsi un ensemble d'entropies. On calcule aussi l'entropie du système entier. Ces valeurs sont reportées et comparées avec les tracés idéaux d'Hattori, ce qui indique que la séquence est essentiellement un cycle assymétrique tronqué (type A4 d'Hattori), et qu'elle s'est déposée dans un milieu fluvial. De plus l'étude suggère que l'usage de la fonction d'entropie, conjointement à l'analyse par chaîne de Markov, révèle certains traits caractéristiques dans une succession. Ces traits peuvent être utilisés comme arguments diagnostiques dans la détermination du milieu de dépôt d'une séquence stratigraphique.

Краткое содержание

При исследовании дву х стратиграфических отделов мелового возраста в т роге Benue, Нигерия, применили процесс Ма ркова. Установили 4 фац ия, составленных грубоз ернистыми, среднезер нистыми и мелкозернистыми пе счаниками и сланцеватой глиной. М атрицы при фациальны х переходах образовал и по закрытой многост упенчатой модели Маркова. Удало сь вывести независим ые матрицы. При сравнени и этих двух серий матриц и применив к ни м два статистических метода установили, что иссле дованный отрезок формации действител ьно проявляет свойст ва, описываемые Марковы м, представляющие соб ой цикличность со все ум еньшающимся грануло метрическим составом. Фациальные циклы удается интерпретировать, ка к последовательност ь фаций русла рек, откосов ско льжения и заливаемых системой рек равнин. С помощью этих статист ических методов можно устано вить состав модально го цикла. В общем, числа кв адратов СЫ возрастает при умень шении числа нулей в пе реходной матрице. Поэтому выбо р метода подсчета влияет на число квадр атов СЫ. Случайность при появлении некого типа фация внутри цик ла выражена энтропией, к оторую можно вывести из матрицы Маркова. Подс читали два рода энтро пии для каждой фации: энтр опия после отложения и до отложения образу ют одну серию энтропи и. Также можно подсчита ть и энтропию всей системы. Полученные д анные, представленны е графически, могут быт ь сравнимы с идеально й диаграммой Hattory: причем о казывается, что здесь в основном имее т место отдельный асимметрический цикл — тип А — 4 Hattory —, отлаг ающийся в системе рек. Исследо вания показали также, что применение функц ии энтропии вместе с м атрицами Маркова разрешает ус тановить основные характерные черты ос адконакопления, кото рые можно использовать, к ак диагностические указание на условия о садконакопления в стратиграфической серии.

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Copyright information

© Ferdinand Enke Verlag Stuttgart 1985

Authors and Affiliations

  • M. Hoque
    • 1
  • C. S. Nwajide
    • 2
  1. 1.Department of GeologyUniversity of NigeriaNsukkaNigeria
  2. 2.Geological SurveyKadunaNigeria

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