The notion of a . . . Main ideas behind . . . The practical . . . Traditional approach . . . Interval uncertainty Fuzzy uncertainty Exact Bounds for Interval Towards the precise . . . Other practical . . . and Fuzzy Functions Under Additional complexity First Problem: . . . Monotonicity Constraints, Second Problem: . . . Algorithms for . . . Possibility of . . . with Potential Applications Algorithms for . . . From Monotonicity to . . . to Biostratigraphy Future Work Acknowledgments Title Page Emil Platon ◭◭ ◮◮ Energy & Geoscience Institute, University of Utah ◭ ◮ Kavitha Tupelly, Vladik Kreinovich, Scott A. Starks Pan-American Center for Earth & Environ. Stud. Page 1 of 19 University of Texas, El Paso, TX 79968, USA Go Back Karen Villaverde Full Screen New Mexico State U., Las Cruces, NM, 88003, USA Close
The notion of a . . . Main ideas behind . . . 1. Biostratigraphy is important The practical . . . Traditional approach . . . • Biostratigraphy is concerned with the stratigraphic analysis of rocks based on Interval uncertainty their paleontologic content. Fuzzy uncertainty Towards the precise . . . • Generally speaking, stratigraphy analyses the rock strata and is concerned Other practical . . . with their succession and age relationship. Additional complexity • All aspects of rocks as strata are, however, of concern for stratigraphy. First Problem: . . . Second Problem: . . . • The analysis of fossil can also provide useful information regarding the envi- Algorithms for . . . ronment in which rocks have accumulated. Possibility of . . . • Example: a coral is an unambiguous indication of a warm ocean. Algorithms for . . . From Monotonicity to . . . Future Work Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 19 Go Back Full Screen Close
The notion of a . . . Main ideas behind . . . 2. The notion of a stratigraphic map The practical . . . Traditional approach . . . • Problem: how to determine the age of the fossil? Interval uncertainty Fuzzy uncertainty • Fact: in a normal sequence, the age increases with the depth in the well that Towards the precise . . . penetrates that sequence. Other practical . . . • Solution: if the rock accumulation rate is known, the depth x at which the Additional complexity fossil species was found can be used to determine its age y . First Problem: . . . Second Problem: . . . • Stratigraphic map: the dependence between the depth x and the age y . Algorithms for . . . • Once we know the depth x and the stratigraphic map y = f ( x ), we can Possibility of . . . determine the age y of the fossil. Algorithms for . . . From Monotonicity to . . . • Complication: a stratigraphic map is different for different locations, because Future Work it depends on the geological history (of accumulation rates) at this location. Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 19 Go Back Full Screen Close
The notion of a . . . Main ideas behind . . . 3. Main ideas behind constructing a stratigraphic map The practical . . . Traditional approach . . . • In every area, we have several fossils whose age y has been determined. Interval uncertainty Fuzzy uncertainty • For the selected fossil, we know the depth x i at which it was found, and we Towards the precise . . . know the estimated age y i . Other practical . . . • Based on the points ( x i , y i ), we must find the desired dependence y = f ( x ). Additional complexity First Problem: . . . • Since deeper layers are older, we should have a monotonic (increasing) de- Second Problem: . . . pendence y = f ( x ) for which y i = f ( x i ). Algorithms for . . . • So, ideally, we should have a monotonic function that passes through all the Possibility of . . . points. Algorithms for . . . From Monotonicity to . . . Future Work Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 19 Go Back Full Screen Close
The notion of a . . . Main ideas behind . . . 4. The practical construction of a stratigraphic map The practical . . . Traditional approach . . . is not that easy Interval uncertainty Fuzzy uncertainty • The conclusion about monotonicity is based on the idealized assumption : Towards the precise . . . • y i is the age of the oldest (for wells, youngest) of many fossils of this type. Other practical . . . Additional complexity • For some types, we do have many fossils, so the oldest of these fossils repre- First Problem: . . . sents a reasonable size sample. Second Problem: . . . • Corresponing values x i and y i are highly reliable. Algorithms for . . . Possibility of . . . • For other types of fossils, however, we may have only a few sample fossils of Algorithms for . . . this type in a given area. From Monotonicity to . . . • So, x i and y i are not very accurate. Future Work Acknowledgments • As a result of this inaccuracy, in practice, it is usually impossible to have a Title Page monotonic dependence that passes exactly through all the points ( x i , y i ). ◭◭ ◮◮ ◭ ◮ Page 5 of 19 Go Back Full Screen Close
The notion of a . . . Main ideas behind . . . 5. Traditional approach and its drawbacks The practical . . . Traditional approach . . . • Problem: few-sample data points do not fit to a monotonic curve. Interval uncertainty Fuzzy uncertainty • Idea: we select a threshold n 0 and only consider points ( x i , y i ) which came Towards the precise . . . from samples of size ≥ n 0 . Other practical . . . • Remaining problems: we Additional complexity First Problem: . . . – ignore all the points ( x i , y i ) with lower accuracy, and Second Problem: . . . – consider all the points with higher accuracy as exact, ignoring the fact Algorithms for . . . that these points are not absolutely accurate. Possibility of . . . Algorithms for . . . • Objective: it is desirable to use the ignored information, to get a more accu- From Monotonicity to . . . rate stratigraphic map. Future Work Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 19 Go Back Full Screen Close
The notion of a . . . Main ideas behind . . . 6. Interval uncertainty The practical . . . Traditional approach . . . • For few-sample fossil types, the actual oldest age y i is different from the Interval uncertainty estimated oldest age � y i . Fuzzy uncertainty Towards the precise . . . • Due to chaotic rock movements, the ideal depth x i differs from the depth � x i Other practical . . . at which the fossil was found. Additional complexity • Problem: we have too few fossils to determine the probability of different First Problem: . . . def def values ∆ x i = � x i − x i and ∆ y i = � y i − y i . Second Problem: . . . Algorithms for . . . • What we do have: expert estimates for the upper bound ∆ i on ∆ x i . Possibility of . . . Algorithms for . . . • Interval uncertainty: for each fossil type i , we know the intervals x i = [ x i , x i ] = [ � x i − ∆ i , � x i + ∆ i ] and, similarly, y i = [ y i , y i ] that contain the From Monotonicity to . . . actual (unknown) values of x i and y i . Future Work Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 19 Go Back Full Screen Close
The notion of a . . . Main ideas behind . . . 7. Fuzzy uncertainty The practical . . . Traditional approach . . . • Interval information comes from the guaranteed bound on ∆ x i and ∆ y i . Interval uncertainty Fuzzy uncertainty • Additional information: often, an expert can also provide bounds that contain Towards the precise . . . ∆ y i with a certain degree of confidence. Other practical . . . • Usually, we know several such bounding intervals corresponding to different Additional complexity degrees of confidence. First Problem: . . . Second Problem: . . . • Such a nested family of intervals is also called a fuzzy set , because it turns Algorithms for . . . out to be equivalent to a more traditional definition of fuzzy set: Possibility of . . . • If a traditional fuzzy set is given, then: Algorithms for . . . From Monotonicity to . . . – different intervals from the nested family Future Work – can be viewed as α -cuts corresponding to different levels of uncertainty Acknowledgments α . Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 19 Go Back Full Screen Close
The notion of a . . . Main ideas behind . . . 8. Towards the precise formulation of the problem The practical . . . Traditional approach . . . • Interval uncertainty: Interval uncertainty Fuzzy uncertainty – We know the n boxes x i × y i corresponding to different types of fossils. Towards the precise . . . – We know that the monotonic dependence y = f ( x ) is such that y i = Other practical . . . f ( x i ) for some ( x i , y i ) ∈ x i × y i . Additional complexity – Objective: to find, for every depth x , the bounds of the possible values First Problem: . . . of age y = f ( x ) for all the dependencies that are consistent with the Second Problem: . . . given data. Algorithms for . . . Possibility of . . . • Fuzzy uncertainty: Algorithms for . . . – For each degree of confidence α , we must solve the problem correspond- From Monotonicity to . . . ing to the α -cut intervals. Future Work – Thus, for each x , we want to have a fuzzy set of possible values of f ( x ). Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 19 Go Back Full Screen Close
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