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

Slide 1

Assessing the model skill of salinity stratification is like walking a dog …..

---anonymous

Slide 2

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

Slide 3

Typical Applications of Fréchet Distance

• Signature matching for online bank service

• Handwriting based web search

• GIS route recognition

Slide 4

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

FD is the minimum leash length required for the man and dog to finish their respective continuous route from start to end without backtracking

Slide 5

Fréchet Distance (FD)

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

Slide 6

Discrete Fréchet Distance (DFD)

DFD is the minimum leash length required if L1 (dog) and L2 (man) are discretized as piecewise linear curves

Slide 7

Average Discrete Fréchet Distance (ADFD)Slide 8

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.

How the dog and man walk through with only DFD leash length available

Slide 9

Step 1

Step 2

Step 3

Step 4

…….

Step 9

dfd happens to be d2

Example of CBay CB5.1 Salt ProfileSlide 10

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

DFD and ADFD (top down walk)Slide 12

Start

End

DFD=0.51355

ADFD=0.2138

Minimum Path has 31 steps

d26

dh

dv

(d26 )^2=dh^2+dv^2

α

d26 is contributed by dh and dv

Analyze Error (ADFD) ContributionADFD=( 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

d26

dh

dv

(d26 )^2=dh^2+dv^2

α

Slide 13

DFD and ADFD (bottom up walk)Slide 14

End

Start

DFD=0.51355

ADFD=0.25069

Minimum Path has 41 steps

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

DFD and ADFD of CH3D for CB5.1 (top down walk)

Slide 15

Start

End

DFD=0.21312

ADFD=0.104

Minimum Path has 29 steps

DFD and ADFD of CH3D for CB5.1 (bottom up walk)

Slide 16

End

Start

DFD=0.21312

ADFD=0.10206

Minimum Path has 29 steps

Modified Target DiagramSlide 17

Y axis: normalized model-data bias (B*)

X axis: ADFD

Sign of X axis:

sign of (model std – obs std)

standard deviation

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.

Slide 18