IASPEI General Assembly CAPE TOWN 10 16 JAN

IASPEI General Assembly CAPE TOWN 10 -16 JAN 2009 Memory Effects in Parameterizations of Mining-Induced Seismic Process Stanislaw Weglarczyk 1, Stanislaw Lasocki 2 1. Cracow University of Technology, Faculty of Environmental Engineering, Poland 2. AGH University of Science and Technology, Faculty of Geology, Geophysics and Environmental Protection, Poland, lasocki@agh. edu. pl

Mining-Induced Seismic Process From the Statistical Point of View Series of seismic events = n-element sample from a MIS stochastic process. Parameters of seismic events: Observables: tk, xk, yk, zk, Mk or M 0 k, Ek, … Derived parameters: interevent time: interevent distance: maximum magnitude: activity rate: ………………. Series of event parameter values = n-element sample from a seismic parameter process.

Stochastic Properties of Seismic Parameter Processes A seismic parameter process can be either fully random i. e. : • stationary: F • ( • , t; {pk}) = F • ( • ; {pk}) and pk(t)= pk • and memoryless: i, k, i k F ( i) = F ( k) and cov( i , k) = 0 or not i. e. non-stationary or having memory or both. A fully random process is unpredictable

Crucial Questions • Do parameter processes of MIS have memory? • If yes, which kind of memory, short, long, both kinds? Long memory test: Hurst rescaled range analysis (R/S analysis) with estimation significance of its results. Short memory test: Interval estimation of autocorrelation function.

Long memory: Hurst Rescaled Range Analysis (R/S Analysis) With Significance Test For every interval k, k=1, …, d: Process sample (X 1, …, Xn) Selecting interval length s No. of intervals d=n/s local range: Rk=max{Yi}-min{Yi} standard deviation: Sk=std{Yi} Hurst (1951): H > 0. 5: persistent process H = 0. 5: random walk (no memory) H < 0. 5: antipersistent process Rescaled range:

Example is only an uncertain estimate of the Hurst’s coefficient. Can we conclude that for this process the actual H > 0. 5? Remedy: permuting Empirical distribution and quantiles of H* when there is no memory Sample data X = (X 1, …, Xn) Reshuffling sample elements Permuted sample X *= (X 1*, …, Xn*) Casual links are cut m permuted samples {X *} m estimates of H* when there is no memory in the samples

Short memory: Interval Estimation of Autocorrelation Function Xt – uncorrelated process Xt – internally correlated process

Function r( ) is only an uncertain estimate of the true autocorrelation function. Can we conclude that for this process the actual ACF( )>0. 0 for some ? Remedy: resampling by blocks of blocks bootstrap technique

Blocks of Blocks Bootstrap Resampling B times random sampling with replacement from blocks of blocks B bootstrap estimates of ACF quantiles

Analyzed Data Eight time space seismic zones linked to mining in Rudna copper mine in Legnica-Glogow Copper District in Poland Zone ID Period of Activity No of Events No of Localized Events Largest Event Energy [J] (magnitude) Z_20. 1 04/04/85 -05/09/04 1242 857 Z_23. 1 12/04/80 -23/09/04 1592 1261 Z_26 20/11/84 -16/09/04 2678 1995 Z_27 19/04/86 -22/09/04 2207 1565 Z_28 28/03/88 -19/09/04 620 525 1. 8 109 (3. 9) 1. 5 109 (3. 8) 3. 6 108 (3. 5) 2. 1 108 (3. 4) 2. 8 108 (3. 4) Z_30. 1 27/04/90 -18/05/02 817 669 Z_31 01/01/80 -20/10/90 2664 929 1. 7 108 (3. 3) 1. 1 108 Z_35 26/11/91 -11/09/04 711 536 2. 1 108 (3. 2) (3. 4)

Extraction of stationary series based on cumulative number of events versus time graphs.

Studied Parameters • Interevent time, IET • Interevent distance, IED • log. Es, m. E (magnitude) Series No. Origin IET and m. E IED Period of Activity No of Events 1 Z_20. 1 09/05/90 -05/09/04 720 13/10/91 -05/09/04 504 2 Z_23. 1 15/02/97 -23/09/04 1300 23/04/96 -23/09/04 1088 3 Z_26 27/01/88 -16/09/04 2250 10/01/88 -15/12/03 1870 4 Z_27 29/03/91 -24/12/99 900 28/03/92 -22/09/04 1254 5 Z_28 21/04/93 -19/09/04 480 27/02/92 -19/09/04 432 6 Z_30. 1 21/03/91 -18/05/02 800 24/03/93 -18/05/02 600 7 Z_31 07/02/80 -30/12/84 1560 01/01/87 -22/08/90 522 8 Z_35 14/05/93 -11/09/04 660 28/04/98 -26/08/04 408

Long Memory Studies: R/S Analysis

Short Memory Studies: Internal Estimation of ACF

Results Number Series of No. Elements Inter-event Time H ACF Magnitude H ACF Number Inter-event Distance of H ACF Elements 1 720 0. 768 + 0. 698 - 504 0. 833 + 2 1300 0. 789 - 0. 648 - 1088 0. 806 ++ 3 2250 0. 845 ++ 0. 655 - 1870 0. 755 ++ 4 900 0. 755 + 0. 629 - 1254 0. 781 + 5 480 0. 825 - 0. 636 - 432 0. 625 + 6 800 0. 705 ++ 0. 654 - 600 0. 767 - 7 1560 0. 567 + 0. 630 - 522 0. 705 + 8 660 0. 745 - 0. 646 - 408 0. 771 +

Conclusions Interevent time and interevent distance processes have long and short memory. MIS occurrences and MIS locations are internally interrelated. Interevent distance process is a Markov chain(? ) Internal relations among MIS magnitudes are weaker and limited to long term interactions.

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