1. Discrete Mathematical Analysis: theory and geophysical applications

2. DMA definition Discrete mathematical analysis (DMA) is an approach to studying of multidimensional massifs and time series, based on modeling of limit in a finite situation, realized in a series of algorithms. The basis of the finite limit was formed on a more stable character, compared to a mathematic character, of human idea of discontinuity and stochasticity. Fuzzy mathematics and fuzzy logic are sufficient for modeling of human ideas and judgments. That was reason why they became technical foundation of DMA.

3. Construction scheme of DMA

4. La Fournaise volcano (Reunion Island)

5. DRAS: application to electric signals associated with the volcanic activity of La Fournaise volcano (Reunion Island) II

6. DRAS and FLARS Algorithms DRAS (Difference Recognition Algorithm for Signals ) and FLARS (Fuzzy Logic Algorithm for Recognition of Signals) are a results of “smooth” modeling (in fuzzy mathematics sense) of interpreter’s logic, which searches for anomalies at the record. DRAS and FLARS is really based on fuzzy mathematic principals.

7. Interpreter’s Logic Local level: Interpreter glances at the record and estimates activity of its sufficiently small fragments by positive numbers. At the same time, he puts some numeric marks to the centres of the fragments. In this way, from initial record interpreter necessarily proceeds to a non-negative function. It is naturally to call this function by rectification of the initial time series. Indeed, greater values of this function correspond to more anomalous points (centres of fragments). Global level: The anomalies on the record are the uplifts on its rectification.

8. Interpreter’s Logic. Illustration. Record Local level - rectification of the record Global level - searching the uplifts on rectification

9. DRAS and FLARS: local level - rectification Discrete positive semiaxes  h + ={kh; k=1,2,3,…} Record y={y k =y(kh), k=1,2,3,…} Registration period Y   h + Parameter of local observation Δ=lh, l=1,2,… Fragment of local observation Δ k y={y k-Δ/h,…, y k,…, y k+Δ/h }   Δ  h+1 Definition. A non-negative mapping  defined on the set of fragments  {Δ k y}  2Δ/h+1 we call by a rectifying functional of the given record “y”. We call any function  y  k  Δ k y  by rectification of the record “y”.

10. Examples of rectifications 1 Length of the fragment: 2 Energy of the fragment: 3 Difference of the fragment from its regression of order n: here as usual is an optimal mean squares approximation of order n of the fragment. If n=0 we get the previous functional “energy of the fragment”: 4 Oscillation of the fragment:

11. Illustration of rectification Record Rectification “Energy” 

12. DRAS: block-scheme of the algorithm Background measures Record rectification Record fragmentation Potential anomaly on the record Anomaly on the record Record

13. DRAS: recognition of potential anomaly on the record. Illustration.

14. DRAS: recognition of anomaly on the record. Genuine anomalis on the record y = A = {alternating-sign decreasing segments for (DαΦy)(k)}

15. DRAS: application to electric signals associated with the volcanic activity of La Fournaise volcano (Reunion Island) I Station – DON, direction - EW

16. Geometrical measures

17. Geometrical measure “background”

18. Geometrical measure “Beginning of mountain”

19. Geometrical measure “left slope”

20. Geometrical measure “peak”