Characterizing the Spatial Variability: examples from the Earth and - - PowerPoint PPT Presentation

Characterizing the Spatial Variability: examples from the Earth and Mars

Luis Samaniego and Andr´ as B´ ardossy Presented at GI Days 2008 June 16, M¨ unster

Outline

• 1. Motivation
• 2. Methods to describe the spatial variability
• 3. Landscape simulations
• 4. Conclusions

1

Motivation

• 1. How to analyze the spatial variability of an observable? e.g.:

■ Precipitation ■ Terrain elevation ■ Soil moisture

2

Motivation

• 1. How to analyze the spatial variability of an observable? e.g.:

■ Precipitation ■ Terrain elevation ■ Soil moisture

• 2. How to relate it with explanatory variables (i.e. dependence)?

2

Motivation

• 1. How to analyze the spatial variability of an observable? e.g.:

■ Precipitation ■ Terrain elevation ■ Soil moisture

• 2. How to relate it with explanatory variables (i.e. dependence)?
• 3. How to use this knowledge in practical applications? e.g.:

■ Weather generators ■ Simulation / interpolation in Geostatistics

2

Part II

Methods to describe the spatial variability

Observable in geophysics

z(x) = is a realization of a underlining random function Z(x)

4

Observable in geophysics

z(x) = is a realization of a underlining random function Z(x) Known:

■ Its variability is a consequence of interactions between natural process ⇒

deterministic

■ Generating processes are sensitive to boundary conditions ■ Exhaustive process description is not possible with known physical/chemical

laws

4

Observable in geophysics

z(x) = is a realization of a underlining random function Z(x) Known:

■ Its variability is a consequence of interactions between natural process ⇒

deterministic

■ Generating processes are sensitive to boundary conditions ■ Exhaustive process description is not possible with known physical/chemical

laws Unknown:

■ Conditions under which these process took place

4

Observable in geophysics

z(x) = is a realization of a underlining random function Z(x) Known:

■ Its variability is a consequence of interactions between natural process ⇒

deterministic

■ Generating processes are sensitive to boundary conditions ■ Exhaustive process description is not possible with known physical/chemical

laws Unknown:

■ Conditions under which these process took place

⇒ Stochastic methods commonly used to extract the spatial dependence

4

Working hypotheses

■ Second order stationarity Expected value of Z(x)

E[Z(x)] = m

Covariance of two random variables h apart

C(h) = E

• Z(x + h) − m
• Z(x) − m
• ■ Intrinsic

Variance of two random variables h apart

γ(h) = 1 2E

• Z(x + h) − Z(x)

2

5

An example

10000 20000 30000 h [m] 100 200 300 var(h) x103 [m²] 0.2 0.4 0.6 0.8 1 γ(h) [m²] n = 100 000

Variogram cloud and experimental variogram from a DEM of the Neckar region (50 × 50) m

6

Shortcomings of the covariance function / variogram

■ Describe the spatial dependence as an integral over the whole range of values

7

Shortcomings of the covariance function / variogram

■ Describe the spatial dependence as an integral over the whole range of values ■ Does not take into account that extremes can have a different spatial

dependence structure from the central values (Journel and Alabert, 1989)

7

Shortcomings of the covariance function / variogram

■ Describe the spatial dependence as an integral over the whole range of values ■ Does not take into account that extremes can have a different spatial

dependence structure from the central values (Journel and Alabert, 1989)

■ Strongly influenced by the marginal distribution (B´

ardossy, WRR, 2006)

7

Shortcomings of the covariance function / variogram

■ Describe the spatial dependence as an integral over the whole range of values ■ Does not take into account that extremes can have a different spatial

dependence structure from the central values (Journel and Alabert, 1989)

■ Strongly influenced by the marginal distribution (B´

ardossy, WRR, 2006)

■ ... What to do?

7

Spatial copula

■ Bivariate distribution function on the unit square with uniform marginals ■ Denotes the pure effect of dependence (Skar 1959, Nelsen 1999)

Cs(h, u, v) = P

• Fz
• Z(x)
• < u; Fz
• Z(x+h)
• < v
• = Cs
• Fz
• Z(x)
• , Fz
• Z(x+h)
• 8

Spatial copula

■ Bivariate distribution function on the unit square with uniform marginals ■ Denotes the pure effect of dependence (Skar 1959, Nelsen 1999)

Cs(h, u, v) = P

• Fz
• Z(x)
• < u; Fz
• Z(x+h)
• < v
• = Cs
• Fz
• Z(x)
• , Fz
• Z(x+h)
• Examples:

0.00 50.00 100.00 150.00 200.00 250.00 0.00 50.00 100.00 150.00 200.00 250.00 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00

Random field

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.85 0.90 0.95 1.00 1.05 1.10 1.15

Copula density, h=2

8

Spatial copula

■ Bivariate distribution function on the unit square with uniform marginals ■ Denotes the pure effect of dependence (Skar 1959, Nelsen 1999)

Cs(h, u, v) = P

• Fz
• Z(x)
• < u; Fz
• Z(x+h)
• < v
• = Cs
• Fz
• Z(x)
• , Fz
• Z(x+h)
• Examples:

50 100 150 200 250 50 100 150 200 250

0.00 0.01 0.02 0.03 0.04

Normal field

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

Copula density, h=4

8

Estimation of an empirical copula

Step 1: Select a spatial domain, e.g. a DEM.

20 50 80 100 10 30 60 90 80 50 25 40 20 10 5

DEM

9

Estimation of an empirical copula

Step 2: Estimate its empirical distribution function Fn(z).

20 50 80 100 10 30 60 90 80 50 25 40 20 10 5

DEM

25 50 75 100

z

0.2 0.4 0.6 0.8 1

F(z) Empirical Distribution Function of z

10

Estimation of an empirical copula

Step 3: Built a grid having the corresponding Fn(z) values.

20 50 80 100 10 30 60 90 80 50 25 40 20 10 5 0.38 0.69 0.88 1.00 0.06 0.25 0.50 0.75 0.94 0.88 0.69 0.44 0.56 0.38 0.25 0.13

DEM Marginal distribution

• f the DEM

25 50 75 100

z

0.2 0.4 0.6 0.8 1

F(z) Empirical Distribution Function of z

11

Estimation of an empirical copula

Step 4: Sample N pairs from it: (u = Fn(z), v = Fn(z + h))

0.38 0.69 0.88 1.00 0.06 0.25 0.50 0.75 0.94 0.88 0.69 0.44 0.56 0.38 0.25 0.13

Marginal distribution

• f the DEM

12

Estimation of an empirical copula

Step 5: Store all pairs (u, v) and (v, u). Tally the frequencies

0.38 0.69 0.88 1.00 0.06 0.25 0.50 0.75 0.94 0.88 0.69 0.44 0.56 0.38 0.25 0.13

Marginal distribution

• f the DEM

0.2 0.4 0.6 0.8 1 u 0.2 0.4 0.6 0.8 1 v

13

Estimation of an empirical copula

Step 6: Normalize both marginals and calculate the copula density cs(u, v).

0.38 0.69 0.88 1.00 0.06 0.25 0.50 0.75 0.94 0.88 0.69 0.44 0.56 0.38 0.25 0.13

Marginal distribution

• f the DEM

0.2 0.4 0.6 0.8 1 u 0.2 0.4 0.6 0.8 1 v 0.2 0.4 0.6 0.8 1 u 0.2 0.4 0.6 0.8 1 v

• E. copula density

h=1

Σ = 1 Σ = 1

14

Part II

Examples

Variation of the spatial dependency with h

0.2 0.4 0.6 0.8 1 u

0.2 0.4 0.6 0.8 1

v

h = ~ 1.0 km r = ~ 0.97

~ 5 km 1 2 3 4 5 7 9 10.5

Andean landscape

0.2 0.4 0.6 0.8 1

u

1 2 3

E

Order Disorder

16

Variation of the spatial dependency with h

0.2 0.4 0.6 0.8 1 u

0.2 0.4 0.6 0.8 1

v

h = ~ 2.5 km r = ~ 0.89

~ 5 km 1 2 3 4 5 7 9 10.5

Andean landscape

0.2 0.4 0.6 0.8 1

u

1 2 3

E

Order Disorder

17

Variation of the spatial dependency with h

0.2 0.4 0.6 0.8 1 u

0.2 0.4 0.6 0.8 1

v

h = ~ 5.0 km r = ~ 0.69

~ 5 km 1 2 3 4 5 7 9 10.5

Andean landscape

0.2 0.4 0.6 0.8 1

u

1 2 3

E

Order Disorder

18

Relation of the spatial copula and the geomorphologic genesis

0.2 0.4 0.6 0.8 1

u

0.2 0.4 0.6 0.8 1 v

h = ~ 20 km

~ 120 km

r = ~ 0.89

2 5 8 10.5

Xanthe Terra (Mars)

19

Relation of the spatial copula and the geomorphologic genesis

0.2 0.4 0.6 0.8 1

u

0.2 0.4 0.6 0.8 1 v

h = ~ 2.5 km

~ 5 km

r = ~ 0.88

1.5 3 4.5 6 7.5 9 10.5

Cotopaxi Volcano (Ecuador)

20

Relation of the spatial copula and the geomorphologic genesis

0.2 0.4 0.6 0.8 1

u

0.2 0.4 0.6 0.8 1 v

h = ~ 1.00 km

~ 2 km

r = ~ 0.86

1.5 3 4.5 6 7.5 9 10.5

Headwater Neckar (Germany)

21

Relation of the spatial copula and the geomorphologic genesis

0.2 0.4 0.6 0.8 1

u

0.2 0.4 0.6 0.8 1 v

h = ~ 0.36 km

~ 10 km

r = ~ 0.89

1.5 3 4.5 6 7.5 9 10.5

Marzuq Desert (Libya)

22

Shannon’s entropy of the spatial copulas

E(u) = −

• u2

cs(u, u2) log cs(u, u2)du2 (1)

0.2 0.4 0.6 0.8 1

u

1 2 3

E

Neckar Cotopaxi Sahara Xanthe Terra (Mars) Random field Gaussian field

Order Disorder

23

Part III

Landscape simulations

Research questions

z(x) = DEM

• 1. Which are the minimum necessary conditions to simulate (reconstruct) a

landscape?

• 2. How can the spatial variability of a realization be evaluated?

25

Algorithm

Hypotheses ⇒ objective functions H01: Runoff generation ⇒ number of sinks ⇒ Φ1 =

i SNi

H02: Spatial structure ⇒ covariance function ⇒Φ2 =

h | ˆ

C(h) − C(h)| H03: Local continuity ⇒ nugget smoothing ⇒ Φ3 =

i | ˆ

C(0)i − C(0)i|

26

Algorithm

Hypotheses ⇒ objective functions H01: Runoff generation ⇒ number of sinks ⇒ Φ1 =

i SNi

H02: Spatial structure ⇒ covariance function ⇒Φ2 =

h | ˆ

C(h) − C(h)| H03: Local continuity ⇒ nugget smoothing ⇒ Φ3 =

i | ˆ

C(0)i − C(0)i| Method

• 1. Generate a permutation of z(x), e.g. n = 70 × 70 cells Pn = n! = 4900! ≈ 10104.2
• 2. Find a solution using simulated annealing:

ˆ z(x)r → min

i

wp

i Φp i

1

p

p > 6

• 3. Estimate ˆ

Csr and the goodness of fit statistic

• 4. Repeat 1. - 3. R times
• 5. Perform the test (e.g. Kolmogorow-Smirnow and Cramer von Mises distances)

26

Example

DEM: Neckar ≈ (21 × 21) km

0 m 5000 m

300 350 400 450 500 550 600 650 700 750 800

27

Effects of the hypotheses on the simulated DEM

1: Runoff generation

0 m 5000 m

300 350 400 450 500 550 600 650 700 750 800 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 u 0.0 0.2 0.4 0.6 0.8 1.0 v

10000 20000 30000

h [m]

• 0.04
• 0.02

0.02 0.04

C(h) (normalized)

C(h) Original DEM C(h) Simulated DEM

28

Effects of the hypotheses on the simulated DEM

2: Runoff generation + spatial structure

0 m 5000 m

300 350 400 450 500 550 600 650 700 750 800 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 u 0.0 0.2 0.4 0.6 0.8 1.0 v

10000 20000 30000

h [m]

• 0.04
• 0.02

0.02 0.04

C(h) (normalized)

C(h) Original DEM C(h) Simulated DEM

29

Effects of the hypotheses on the simulated DEM

3: Runoff generation + local continuity

0 m 5000 m

300 350 400 450 500 550 600 650 700 750 800 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 u 0.0 0.2 0.4 0.6 0.8 1.0 v

10000 20000 30000

h [m]

• 0.04
• 0.02

0.02 0.04

C(h) (normalized)

C(h) Original DEM C(h) Simulated DEM

30

Effects of the hypotheses on the simulated DEM

4: Runoff generation + spatial structure + local continuity

0 m 5000 m

300 350 400 450 500 550 600 650 700 750 800 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 u 0.0 0.2 0.4 0.6 0.8 1.0 v

10000 20000 30000

h [m]

• 0.04
• 0.02

0.02 0.04

C(h) (normalized)

C(h) Original DEM C(h) Simulated DEM

31

Simulated vs. real DEM

5: Sim. copula ↔ obs. copula: null hypotheses rejected p-value<10%

0 m 5000 m

300 350 400 450 500 550 600 650 700 750 800 1 2 3 4 5

10000 20000 30000

h [m]

• 0.04
• 0.02

0.02 0.04

C(h) (normalized)

C(h) Original DEM C(h) Simulated DEM

0.0 0.2 0.4 0.6 0.8 1.0 u 0.0 0.2 0.4 0.6 0.8 1.0 v

32

Conclusions

■ The strength of the stochastic dependency is not uniformly distributed over the

range of values of a DEM

33

Conclusions

■ The strength of the stochastic dependency is not uniformly distributed over the

range of values of a DEM

■ The spatial dependence can be clearly visualized and quantified by a spatial

copula

33

Conclusions

■ The strength of the stochastic dependency is not uniformly distributed over the

range of values of a DEM

■ The spatial dependence can be clearly visualized and quantified by a spatial

copula

■ Covariance functions alone may not be sufficient to model the spatial variability

(e.g. in water-eroded landscapes)

33

Conclusions

■ The strength of the stochastic dependency is not uniformly distributed over the

range of values of a DEM

■ The spatial dependence can be clearly visualized and quantified by a spatial

copula

■ Covariance functions alone may not be sufficient to model the spatial variability

(e.g. in water-eroded landscapes)

■ An empirical copula can be used for validation or for spatial interpolation

(B´ ardossy, WRR, 2006).

33

Thank you!

34

Appendix

35

Appropriate Measures for the Goodness of the Fit

Kolmogorow-Smirnow distance D1 = sup

• | ˆ

Cs(u, v) − Cs(u, v)|; (u, v) ∈ [0, 1]2 Cramer von Mises distance D2 = 1 1

• ˆ

Cs(u, v) − Cs(u, v) 2 dudv

36