From observations to nonlinear model in geoinformatics От

From observations to nonlinear model in geoinformatics От наблюдений к нелинейным моделям в геоинформатике A. Gvishiani, Geophysical Center RAS (GC RAS), Moscow, Russian Federation gvi@wdcb. ru e. GY-IGY Conference, Suzdal, 17 -19 September 2007

International Geophysical Year 1957 - 1958 – allowed scientists to participate in global observations of geoscience phenomena using common instruments and data processing – gathered data from around the world – established the World Data Centre system – ……… e. GY-IGY Conference, Suzdal, 17 -19 September 2007

52 centers in 12 countries e. GY-IGY Conference, Suzdal, 17 -19 September 2007

e. GY embraces and extends IGY principles… International cooperation and data sharing Universal access to data and information Timely and convenient access to data Global, cross-disciplinary scope Data preservation Capacity building, especially in developing countries Education, public outreach, information for decision making e. GY-IGY Conference, Suzdal, 17 -19 September 2007

WSIS The Declaration of Principles and Plan of Action approved by the participants recognized that science has a central role in the development of the Information Society. e. GY-IGY Conference, Suzdal, 17 -19 September 2007

Science in the Declaration of Principles 1 • Para 7. We recognize that science has a central role in the development of the Information Society. Many of the building blocks of the Information Society are the result of scientific and technical advances made possible by the sharing of research results. [. . . ] • Para 25. The sharing and strengthening of global knowledge for development can be enhanced by removing barriers to equitable access to information for economic, social, political, health, cultural, educational, and scientific activities and by facilitating access to public domain information, including by universal design and the use of assistive technologies. [. . . ] e. GY-IGY Conference, Suzdal, 17 -19 September 2007

DMA Algorithmic Scheme Fuzzy comparisons of positive numbers Proximity in finite metric space Limit in finite metric space Multidimensional Discrete Spaces MDS Finite Time Series FTS Density as limit measure Smooth FTS: Equilibrium Monotonous FTS Recognition of dense subsets: Crystal, Monolith. Prediction of FTS: Forecast Extremums on FTS Clusterization: Rodin Anomalies on FTS: DRAS, FLARS, FCARS Convex FTS Recognition of linear structures: Tracing Fuzzy logic and geometry on FTS: Geometric measures e. GY-IGY Conference, Suzdal, 17 -19 September 2007

RODIN overview • • • X – finite metric space with distance d(x, y) – measure of nearness of x to y Density of A X in x X: – given level of density, – given level of clusterness, K 0 – initial version of cluster, Kn – current version of cluster e. GY-IGY Conference, Suzdal, 17 -19 September 2007

Gulf Saint Malo This figure presents results of clustering of Euler solutions for the Saint Malo region. Initial set of Euler solutions included 34, 500 points. The clustering algorithm rejected almost 7000 points, finding dense clusters that outline isometric and linear structures of the region. A distinctive feature of the Euler solutions obtained is the linear clusters lineated N-S in the southern part of the map and NW-SE in its northern part. Inland, they coincide with the strike of doleritic dikes. Clustering solutions in the southern part of the map shows that besides N-S trending dikes, there is probably another dike swarm striking NNW-SSE or NW-SE. Using Euler solutions, the later dikes can be followed to the north to their intersection with the Cadomian belt and further they can be correlated with offshore dike swarm of NW-SE direction. e. GY-IGY Conference, Suzdal, 17 -19 September 2007

Gulf Saint Malo First figure shows the total magnetic anomaly for a small part of the region by shaded relief and isolines, as well as Euler solutions obtained with window size 9 x 9 points (2 x 2 km). About 40% of Euler solutions were rejected because of their extremely small singular values. Obviously it is too difficult to use this results for analysis of geological structure of the region. On second figure Euler solutions was selected using different “standard” criteria. Solutions with low tolerance, situated at depth larger than 2 km and located at a distance five times larger than window size from the center of window were rejected. Results are shown on the figure. In comparison with previous figure now Euler solutions outline isometric and linear bodies. However, position of possible causative sources remains unclear. Third figure shows the result of clusterization. RODIN algorithm was applied to the original set of Euler solutions using parameters α=0. 8 and r=0. 3. The result obtained is stable in the sense of possible changes of these parameters; i. e. close pictures were obtained for α ranged between 0. 78 and 0. 82 and for r ranged between 0. 25 and 0. 35. The algorithm found dense clusters, which more clearly outline possible causative sources. e. GY-IGY Conference, Suzdal, 17 -19 September 2007

CRYSTAL overview • CRYSTAL goal: to identify -dense subsets against a general background. • Definition. Subset A X is -dense against the background X if • CRYSTAL block-scheme: • Definition. x* – foundation in X, if (PX(X), PX(x*)) 0. 5, where – fuzzy comparison. (initial crystals: K(1) =x 1, K(2) =x 2, . . . ) • “Growth I” block: Kn– current version of current crystal K. • “Growth I” necessary condition: Growth Kn Kn+1=(Kn, xn+1), xn+1 is possible only if • “Growth II” block: If the “Growth I” condition is fulfilled then Kn+1={Kn, xn+1} will be dense only if the following condition is fulfilled: • “Growth II” sufficient condition: e. GY-IGY Conference, Suzdal, 17 -19 September 2007

Hoggar region, Algeria Magnetic anomaly field T Result of Crystal: 670 points in 38 clusters Result of Euler deconvolution: 2720 points e. GY-IGY Conference, Suzdal, 17 -19 September 2007

Discrete mathematical analysis (DMA) in nonlinear approach to anomaly recognition on time series How to identify anomalies? • Anomaly identification in large time series data sets needs automated technique to make it feasible and independent of expert, who processes the data. • The presented fuzzy logic based algorithms meet this objective in case of electric time series. e. GY-IGY Conference, Suzdal, 17 -19 September 2007

FTS anomaly recognition algorithms: DRAS, FLARS and FCARS • DRAS (Difference Recognition Algorithm for Signals ) - 2003 • FLARS (Fuzzy Logic Algorithm for Recognition of Signals) – 2005 • FCARS (Fuzzy Comparison Algorithm for Recognition of Signals) 2007 realize “smooth” modeling (in fuzzy mathematics sense introduced by L. Zade) of interpreter’s logic, that searches for anomalies on FTS. DRAS global level FTS DRAS, FLARS, FCARS local level Rectification of FTS FLARS global level FTS Anomalies FCARS global level Examples of FTS rectification functionals Length of the fragment, energy of the fragment, difference of the fragment from its regression of order n. e. GY-IGY Conference, Suzdal, 17 -19 September 2007

Piton La Fournaise volcano. La Réunion island. 2007, 2006, 2005, 2004, 2003, 2002, 2001, 2000, 1998, 1992, 1991, 1990, 1985 -88, 1983 -84, 1981, 1979, 1977, 1976, 1975 -76, 1973, 1972, 1966, 1964 -65, 1964, 1963, 1961, 1960, 1959, 1958, 1957, 1955 -57, 1954, 1953, 1952, 1951, 1950, 1949, 1948, 1947, 1946, 1945, 1944, 1943, 1942, 1941, 1938 -39, 1938, 1937, 1936, 1935, 1933 -34, 1932, 1931, 1930, 1929, 1926 -27, 1925 -26, 1924, 1921, 1920, 1917, 1915, 1913, 1910, 1909, 1908, 1907, 1905, 1904, 1903, 1902, 1901, 1900, 1899, 1898, 1897, 1894, 1890 -91, 1889, 1884, 1882, 1878, 1876, 1875, 1874, 1872, 1871, 1870, 1869, 1868, 1865, 1863 -64, 1861, 1860, 1859, 1858 -59, 1852, 1851, 1850, 1849, 1848, 1847, 1846, 1845, 1844, 1843, 1842, 1830, 1824, 1821, 1820, 1817, 1816, 1815, 1814, 1813, 1812, 1810, 1809, 1807, 1802, 1801 -02, 1800, 1797, 1795, 1794, 1792, 1791, 1789, 1787, 1786, 1784 -85, 1776, 1775, 1774, 1772, 1771, 1768, 1766, 1760, 1759, 1753, 1751, 1734, 1733, e. GY-IGY Conference, 1721, 1709, 1708, 1703, 1672, 1671, 1669, 1640 Suzdal, 17 -19 September 2007

DRAS: application to electric signals associated with the volcanic activity of La Fournaise volcano (Reunion Island). Anomalies observed one day before the eruption of 9 March 1998 e. GY-IGY Conference, Suzdal, 17 -19 September 2007

Local level of interpreter’s logic Local level is defined in the same way for all three algorithms DRAS, FLARS and FCARS: Interpreter estimates activity of sufficiently small fragments of the time series by assigning positive numbers to the fragments (or to their centers). In this way, interpreter proceeds from initial record to a nonnegative function. We call this function by rectification of the initial time series. Indeed, greater values of this function correspond to more anomalous fragments. e. GY-IGY Conference, Suzdal, 17 -19 September 2007

Local level. Formal interpretation. Discrete positive semi axe h+={kh; k=1, 2, 3, …} Record = time series (TS) y={yk=y(kh), k=1, 2, 3, …} Registration period T h+ Parameter of local observation Δ=lh, l=1, 2, … Fragment of local observation Δk y={yk-Δ/h , … , yk+Δ/h} Δ h+1 Definition. A non-negative mapping defined on the set of fragments {Δk y} 2Δ/h+1 we call by a rectifying functional of the given record “y”. A function y k Δky is called by rectification of the record “y”. e. GY-IGY Conference, Suzdal, 17 -19 September 2007

Examples of rectifying functionals Length of the fragment: Energy of the fragment: Difference of the fragment from its regression of order n: here we use an optimal mean squares approximation of order n of the fragment. e. GY-IGY Conference, Suzdal, 17 -19 September 2007

Local and global levels of interpreter’s logic. Record = TS Local level – TS rectification Global level – establishing uplifts on rectification e. GY-IGY Conference, Suzdal, 17 -19 September 2007

DRAS: Difference Recognition Algorithm for Signals. Left and Right background measures Record rectification Potential anomaly on the record Genuine anomaly on the record Record fragmentation Record e. GY-IGY Conference, Suzdal, 17 -19 September 2007

DRAS: global level. Recognition of potential anomalies. - vertical level of background Left and right background measures of silence - β– horizontal level of background Potential anomaly on the record y: PA={kh Y : min((LαΦy)(k), (RαΦy)(k)) < β} Regular behavior of the record y: B={kh Y : min((LαΦy)(k), (RαΦy)(k)) β} e. GY-IGY Conference, Suzdal, 17 -19 September 2007

DRAS: global level. Recognition of genuine anomalies. Potential anomalies PA = UP(i), n=1, 2, N. is a union of coherent components DRAS recognizes genuine anomalies A(n) as parts of P(n) by analyzing operator DΦ(k) = LΦ(k) RΦ(k). The beginning of A(n) is the first positive maximum of DΦ(k) on P(n). Indeed , the difference between “calmness” from the left and anomaly behavior from the right is the biggest in this point. By the same reason, the end of A(n) is in the last negative minimum of DΦ(k). e. GY-IGY Conference, Suzdal, 17 -19 September 2007

DRAS: recognition of potential anomaly. e. GY-IGY Conference, Suzdal, 17 -19 September 2007

DRAS: recognition of genuine anomaly. Genuine anomalies on the record y, A = {alternating-sign decreasing segments for (DαΦy)(k)} e. GY-IGY Conference, Suzdal, 17 -19 September 2007

DRAS: application to electric signals associated with the volcanic activity of La Fournaise volcano (Reunion Island). Station – DON, direction - EW Anomalies registered 7 -8 June 1999 - a year later after the eruption of 9 March 1998. e. GY-IGY Conference, Suzdal, 17 -19 September 2007

FLARS: global level. Anomaly fuzzy measure μ(k). - parameter of global observation δk k - model of global observation record at the point k “argument” for minimality (regularity) of the point “kh” “argument” for maximality (anomaly) of the point “kh” The measure is a result of comparison of the “arguments” e. GY-IGY Conference, Suzdal, 17 -19 September 2007

FLARS: Global level. recognition of genuine anomaly. α [0, 1] – vertical level of how extreme are the measure values Genuine anomaly on the record y A = { kh Y : μ(k) α} Regular behavior/ potential anomaly NA = { kh Y : μ(k)<α} e. GY-IGY Conference, Suzdal, 17 -19 September 2007

FLARS: global level. Recognition of potential anomaly. We introduce the function that possesses the following properties: One-sided background measures - Θ – the parameter of intermediate observation: Δ<Θ≤Δ. β – horizontal level of background, (-1, 1) Potential anomaly on the record y PA={kh NA : min((LαΦy)(k), (RαΦy)(k)) < β} Regular behavior of the record y B={kh NA : min((LαΦy)(k), (RαΦy)(k)) β} e. GY-IGY Conference, Suzdal, 17 -19 September 2007

DRAS and FLARS recognition comparison DRAS FLARS e. GY-IGY Conference, Suzdal, 17 -19 September 2007

FCARS anomaly recognition FCARS e. GY-IGY Conference, Suzdal, 17 -19 September 2007

What algorithm to apply to FTS data sets? • DRAS. Calm and anomaly points are quite well distinguished, but genuine anomalies are not evident. DRAS is useful in searching big anomalies. • FLARS. High amplitude anomalies are quite obvious and small anomalies are not so evident on the background of noise. Useful to search very small isolated anomalies. • FCARS. Important in searching oscillating anomalies and identification of the beginning and ends of the signals. e. GY-IGY Conference, Suzdal, 17 -19 September 2007

E-Laborartory (@EMG) on electric signals monitoring • In 2000 e-group on electric monitoring of La Fournaise volcano was launched by OPGC, Clermont-Ferrand Schmidt IPE, Moscow. • In 2006 this e-group was reinforced by IPG, Paris and GC RAS, Moscow. Another focus of the e-group was added in seismic and electromagnetic studies of Corinth Gulf. • In 2008 the e-group plans to extend it’s activities onto seismic and electromagnetic studies in Kamchatka. e. GY-IGY Conference, Suzdal, 17 -19 September 2007