Stratification Skill Assessment Based on Average Discrete Fréchet Distance (ADFD)

1. Slide 1 Stratification Skill Assessment Based on Average Discrete Fréchet Distance (ADFD) Wen Long (UMCES) Raleigh Hood (UMCES) Kevin Sellner (CRC) SURA Testbed Estuarine Hypoxia Group Meeting Oct 1, 2010

2. Slide 2 Assessing the model skill of salinity stratification is like walking a dog ….. ---anonymous

3. Slide 3 Introduction to Fréchet Distance • Fréchet Distance (Fréchet, 1906) is the minimum leash length required for a dog-man’s problem: • Dog walks on curve L1 from start to end • Man walks on curve L2 from start to end • No backtracking is allowed in the walk • when FD=0, i.e. leash length =0, L1 and L2 must be the same

4. Slide 4 Typical Applications of Fréchet Distance • Signature matching for online bank service • Handwriting based web search • GIS route recognition References: [1] Eiter, T. and Mannila, H., Computing Discrete Frechet Distance, Apr 25, 1994, Report # CD-TR 94/64 [2] Mascret, A., Devogele, T., Le Berre, I., Henaff, A., Coastline Matching Process based on the discrete Frechet distance, 2009 [3] Fréchet, M., Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Mathematico di Palermo, 22:1-74, 1906

5. Slide 5 Fréchet Distance (FD) FD is the minimum leash length required for the man and dog to finish their respective continuous route from start to end without backtracking

6. Slide 6 Properties of FD • FD(L1, L2) = FD (L2,L1), i.e. exchanging routes between dog and man makes NO difference • When FD=0, L1, L2 must collapse

7. Slide 7 Discrete Fréchet Distance (DFD) DFD is the minimum leash length required if L1 (dog) and L2 (man) are discretized as piecewise linear curves

8. Slide 8 Average Discrete Fréchet Distance (ADFD) ADFD is the average distance of the “minimum path” ADFD=(d1+d2+…+d9)/9 “Minimum Path” is a series of point-pairs that the dog and the man have to follow to finish their routes while staying as close to each other as possible. (See next slide) ADFD is an optimal measure of the discrepancy of the two curves L1 and L2 !!! Hence ADFD can be used for skill assessment.

9. Step 1 Step 2 dfd happens to be d2 Step 3 Step 4 ……. Step 9 Slide 9 How the dog and man walk through with only DFD leash length available

10. Slide 10 Example of CBay CB5.1 Salt Profile

11. Slide 11 CB5.1 Salt Profile (with normalized axis)

12. Start DFD=0.51355 ADFD=0.2138 Minimum Path has 31 steps d26 α dv dh (d26 )^2=dh^2+dv^2 End d26 is contributed by dh and dv Slide 12 DFD and ADFD (top down walk)

13. d26 α dv dh (d26 )^2=dh^2+dv^2 Slide 13 Analyze Error (ADFD) Contribution ADFD=( d1+d2 + …+d26+…+ d31)/31  mean length of minimum path pairs • d26 is contributed by dh and dv • dh is associated with salinity error on the same depth level • dv is associated with mislocation of isopycnals • Angle α may be used to define ADFS (Average Discrete Frechet Slope) as a measure of model-obs stratification

14. End DFD=0.51355 ADFD=0.25069 Minimum Path has 41 steps Start Slide 14 DFD and ADFD (bottom up walk) When reversing the direction of walk, DFD is the same, ADFD can be DIFFERENT. In our skill assessment we should use mean value of the ADFD’s of the two opposite directions

15. Start DFD=0.21312 ADFD=0.104 Minimum Path has 29 steps End Slide 15 DFD and ADFD of CH3D for CB5.1 (top down walk)

16. End DFD=0.21312 ADFD=0.10206 Minimum Path has 29 steps Start Slide 16 DFD and ADFD of CH3D for CB5.1 (bottom up walk)

17. Slide 17 Modified Target Diagram Y axis: normalized model-data bias (B*) X axis: ADFD Sign of X axis: sign of (model std – obs std) standard deviation

18. Slide 18 Conclusion • ADFD is an optimal measure of curve mismatch and can be used for skill assessment of salinity stratification. • ADFS (Average Discrete Frechet Slope) could be introduced and be used for analyzing error due to same stratification strength but with isopycnal shift. • Target Diagram can be modified to incorporate ADFD.