Mathematical Models of Geological Structure of Outcrops and Stratigraphy Based on Graph Theory

― Representation of Turned Layers ―

Abstract

We have shown in the previous studies that the outcrop structure and the stratigraphy at an outcrop composed of unturned layers are mathematically expressed in the form of a structure graph G=(V, R^#, φ_V, φ_A) and a stratigraphic graph S= (V/E^*, U^*, φ_V, φ_A), respectively. The present study extends the idea to describe the outcrop composed of turned layers. The contact relation between geological bodies x and y is expressed in terms of binary relation R^# on a set of geological bodies V, where R^# =R∨I ∨F ∨L∨D∨T, R is the relation of a comformity or an uncomformity, I is the relation of a intrution, F is the relation of a fault, L is the relation of a inclusion, D is the relation of a interfingers, and T is the relation of a reverse. φ_A is a function which assigns the contact relation such as conformity, unconformity, intrusion, fault, intrusion, interfingers, conformity (reverse) and unconformity (reverse) to each arc. In the case of turned layers x and y, if x is under y : xTy, y is older than x: yU^*x, if φ_A (x, y) = conformity (reverse), φ_A (y, x) =conformity, if φ_A (x, y) =unconformity (reverse), φ_A (y, x) =unconformity. In the case that geologic bodies [g_i] are turned and geologic bodies [g_j] cover [g_i] with unconformity, let H₁ be [g_i] which exist in the underside spatially of the surface of unconformity, and let H₂ be [g_i] which exist in the spatially upper side of the surface of unconformity. The stratigraphic total order is expressed as P’= (H₁, unconformity, H₂). When the total order P’₁ and P’₂ of strata of H₁ and H₂ are obtained respectively, the total order P’ is infered as, P’= (P’₁, unconformity, P’₂).