От наблюдений к нелинейным моделям в геоинформатике

From observations to nonlinear model in geoinformatics A.Gvishiani, Geophysical Center RAS (GC RAS), Moscow, Russian [email protected]

eGY-IGY Conference, Suzdal, 17-19 September 2007

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DMA Algorithmic SchemeMultidimensionalDiscrete SpacesMDSFuzzy comparisons of positive numbersProximity in finite metric spaceLimit in finite metric spaceDensity as limit measure Finite Time SeriesFTS Recognition of dense subsets:Crystal, Monolith.Clusterization:RodinRecognition of linear structures:TracingSmooth FTS:EquilibriumMonotonous FTSPrediction of FTS:ForecastExtremums onFTSAnomalies on FTS:DRAS, FLARS, FCARSConvexFTSFuzzy logic andgeometry on FTS:Geometric measures


RODIN overviewX finite metric space with distance d(x,y) measure of nearness of x to y Density of AX in xX: given level of density, given level of clusterness, K0 initial version of cluster, Kn current version of cluster


Gulf Saint MaloThis 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.


Gulf Saint MaloFirst 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 9x9 points (2x2 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 =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 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.


CRYSTAL overviewCRYSTAL goal: to identify -dense subsets against a general background.Definition. Subset AX is -dense against the background X ifCRYSTAL block-scheme:

Definition. x* foundation in X, if (PX(X), PX(x*))0.5, where fuzzy comparison. (initial crystals: K(1) =x1, K(2) =x2, ...)Growth I block: Kn current version of current crystal K.

Growth I necessary condition: Growth KnKn+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:


Hoggar region, AlgeriaMagnetic anomaly field TResult of Euler deconvolution: 2720 pointsResult of Crystal: 670 points in 38 clusters


Discrete mathematical analysis (DMA) in nonlinear approach to anomaly recognition on time seriesHow 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.


FTS anomaly recognition algorithms: DRAS, FLARS and FCARSDRAS (Difference Recognition Algorithm for Signals ) - 2003FLARS (Fuzzy Logic Algorithm for Recognition of Signals) 2005FCARS (Fuzzy Comparison Algorithm for Recognition of Signals) - 2007realize smooth modeling (in fuzzy mathematics sense introduced by L. Zade) of interpreters logic, that searches for anomalies on FTS.Examples of FTS rectification functionalsLength of the fragment, energy of the fragment, difference of the fragment from its regression of order n.


Piton La Fournaise volcano. La Runion island.2007, 2006, 2005, 2004, 2003, 2002, 2001, 2000, 1998, 1992, 1991, 1990, 1985-88, 1983-84, 1981, 1979, 1977, 1976, 1975-76, 1973, 1973, 1972, 1966, 1964-65, 1964, 1963, 1961, 1960, 1959, 1958, 1957, 1955-57, 1954, 1953, 1952, 1951, 1950, 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, 1924, 1921, 1920, 1917, 1915, 1913, 1910, 1909, 1908, 1907, 1905, 1904, 1903, 1902, 1901, 1901, 1900, 1899, 1898, 1898, 1897, 1894, 1890-91, 1889, 1884, 1882, 1878, 1876, 1875, 1874, 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, 1832, 1830, 1824, 1824, 1821, 1820, 1817, 1816, 1815, 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, 1734, 1733, 1721, 1709, 1708, 1703, 1672, 1671, 1669, 1649, 1640


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


Local level of interpreters logicLocal 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 non-negative function. We call this function by rectification of the initial time series. Indeed, greater values of this function correspond to more anomalous fragments.


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 =lh, l=1,2,Fragment of local observation k y={yk-/h , , yk , , yk+/h}h+1

Definition. A non-negative mapping defined on the set of fragments {k y}2/h+1we call by a rectifying functional of the given record y.A function ykky is called by rectification of the record y.


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 .


Local and global levels of interpreters logic. Record = TSLocal level TS rectification Global level establishing uplifts on rectification


DRAS: Difference Recognition Algorithm for Signals.Left and Right background measures Record rectification Record fragmentation Potential anomaly on the recordGenuine anomaly on the recordRecordParis, 3-5 November 2004


DRAS: global level. Recognition of potential anomalies. Left and right background measures of silence - horizontal level of backgroundPotential anomaly on the record y: - vertical level of backgroundPA={khY : min((Ly)(k), (Ry)(k)) < }Regular behavior of the record y:B={khY : min((Ly)(k), (Ry)(k)) }


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 D(k) = L(k) - R(k). The beginning of A(n) is the first positive maximum of D(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 D(k).


DRAS: recognition of potential anomaly.


DRAS: recognition of genuine anomaly. Genuine anomalies on the record y, A = {alternating-sign decreasing segments for (Dy)(k)}


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.


FLARS: global level. Anomaly fuzzy measure (k). - parameter of global observationkk - model of global observation record at the point k argument for minimality (regularity) of the point kh argument for maximality (anomaly) of the point khThe measure is a result of comparison of the arguments


FLARS: Global level. recognition of genuine anomaly. [0,1] vertical level of how extreme are the measure valuesRegular behavior/ potential anomaly NA = { kh Y : (k)

FLARS: global level. Recognition of potential anomaly. We introduce the function that possesses the following properties:One-sided background measures - the parameter of intermediate observation:

DRASFLARS DRAS and FLARS recognition comparison


FCARS FCARS anomaly recognition


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-Laborartory (@EMG) on electric signals monitoringIn 2000 e-group on electric monitoring of La Fournaise volcano was launched by OPGC, Clermont-Ferrand and 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 its activities onto seismic and electromagnetic studies in Kamchatka.


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От наблюдений к нелинейным моделям в геоинформатике

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