Two-Phase Sampling Approach for Augmenting Fixed Grid Designs to Improve Local Estimation for Mapping

Aquatic Resources  

Kerry J. Ritter

Molly Leecaster

N. Scott Urquhart

Ken Schiff 

Project Funding

• The work reported here was developed under the STAR Research Assistance Agreement CR-829095 awarded by the U.S. Environmental Protection Agency (EPA) to Colorado State University. This presentation has not been formally reviewed by EPA.  The views expressed here are solely those of the presenter and STARMAP, the Program they represent. EPA does not endorse any products or commercial services mentioned in this presentation.

• Southern Californian Coastal Water Research Project (SSCWRP)

Background• Maps of sediment condition are important for

making decisions regarding pollutant discharge• Maps in marine systems are rare• Special study by San Diego Municipal Wastewater

Treatment Plant• Objective : To build statistically defensible maps

of chemical constituents and biological indices around two sewage outfalls– Point Loma

– South Bay

Point Loma and South Bay Outfalls

TYPICAL DESIGN SITUATION

• Many features of the real situation are unknown.– Here: The nature of the semivariogram

• Multiple Responses What is a good solution for one response

may not be a good design for another!

• Time constraint– Answer was required by this past Monday

Two-Phase Approach• Phase I: Model spatial variability at various

spatial scales (eg. Variogram) – This summer

• Phase II: Use information from Phase I to design survey that meets accuracy requirements – next summer = 2005

How Should We Add Sites to Existing Grid in Order to

Estimate Variogram?

• What is best design configuration?

• More sites with less intensity or fewer sites with more intensity?

• Shorter sample spacing or larger sample spacing?

Variogram

distance

ga

mm

a

0 10 20 30 40 50

0.0

0.5

1.0

1.5

2.0

2.5

VARIOGRAM

}NUGGET=>

SILL=>

RANGE

Empirical Variograms(Point Loma 2000 Regional Survey)

distance

gam

ma

0 2 4 6 8

010

2030

4050

60

CHROMIUM

R=5.09 S=36.27 N =0.00distance

gam

ma

0 2 4 6 8

0.0

0.05

0.10

0.15

TOC

R=8.8 S=.077 N =0.0242distance

gam

ma

0 2 4 6 8

05

1015

2025

30

COPPER

R=2.75 S=22.53 N =0.00

distance

gam

ma

0 2 4 6 8

050

100

150

200

250

300

ZINC

R=6.14 S=218.55 N =0.00

Lag Distribution Variogram

lag distance (km)

No.

of p

airs

2 4 6 8

1020

3040

50

Design Considerations for Modeling the Variogram

• Sufficient replication at various spatial scales– Variogram model

– Parameter estimates

• Adequate spatial coverage– Stationarity

– Isotropy vs. Anisotropy

– Strata

• Allow for multiple responses

Choosing the Best DesignCase Study: Point Loma

• Three design configurations– S, STAR, and S with satellites

• Two sets of lag classes– Shorter vs. larger sample spacing

• Compare lag distributions• Simulation study

– Simulate response– Consider different models of spatial variability

• Compare relative performance of designs for estimating parameters

“STAR” and “S” Cluster Designs

S DESIGN

Xkm

Ykm

0 20 40 60 80 100

020

4060

8010

0

STAR DESIGN

Xkm

Ykm

0 20 40 60 80 100

020

4060

8010

0

“S” and “S with Satellites” Design

S DESIGN

Xkm

Ykm

0 20 40 60 80 100

020

4060

8010

0

S with SATELLITES DESIGN

Xkm

Ykm

0 20 40 60 80 100

020

4060

8010

0

Sample AllocationStar S S with Satellites

Grid Stations =12 Grid Stations =12 Grid Stations =12

5 “STAR” Clusters of Size 17

   3 grid station

2 sites of interest

1 “S” Cluster of Size 9

11 “S” Clusters of Size 9

      5 grid stations

6 sites of interest

8 “S” Clusters of Size 9

8 Satellites added to 3 S”

4 grid stations

4 sites of interest

Field duplicates=9 Field duplicates=6 Field duplicates=8

Total Samples =

12+3*(17-1) +2*(17)+9+9=112

Total Samples =

12+5*(9-1)+6*(9)+6=112

Total Samples =

12+4*(9-1) +6*(9)+6=112

“Star” Cluster Design

Point Loma 5 Star + 1 S Cluster

Xkm

Ykm

466 468 470 472 474

3610

3615

3620

3625

Point Loma 5 Star + 1 S Cluster

Xkm

Ykm

466 468 470 472 474

3610

3615

3620

3625

“S” Cluster Design

S DESIGN

Xkm

Ykm

466 468 470 472 474

3610

3615

3620

3625

S DESIGN

Xkm

Ykm

466 468 470 472

3610

3615

3620

3625

Lag = 0.05, 0.10, 0.20, 0.50 Lag = 0.05, 0.25, 1.00, 3.00

“S” Cluster with SatellitesS with SATELLITES DESIGN

Xkm

Ykm

466 468 470 472 474

3610

3615

3620

3625

S with SATELLITES DESIGN

Xkm

Ykm

466 468 470 472

3610

3615

3620

3625

Omnidirectional Lag Dist.

Ominidirectional Lag Dist

Pairwise Lag distances

No. o

f pair

s

0 2 4 6 8

010

020

030

040

0

SD3StarD5SSATD3

Ominidirectional Lag Dist

Pairwise Lag distances

No. o

f pair

s

0 2 4 6 8

010

020

030

040

0

SStarSSAT

Lag = 0.05, 0.10, 0.20, 0.50 Lag = 0.05, 0.25, 1.00, 3.00

Directional Lag DistLag = 0.05, 0.10, 0.20, 0.50

{ Lag = 0.05, 0.25, 1.00, 3.00 is similar}

Direction = 0

Pairwise Lag distances

No

. o

f p

air

s

0 2 4 6 8

02

04

06

08

01

00

12

0

S0STAR0SSAT0

Direction = 45

Pairwise Lag distances

No

. o

f p

air

s

0 2 4 6 8

02

04

06

08

01

00

12

0

S45STAR45SSAT45

Direction = 90

Pairwise Lag distances

No

. o

f p

air

s

0 2 4 6 8

02

04

06

08

01

00

12

0

S90STAR90SSAT90

Direction = 135

Pairwise Lag distances

No

. o

f p

air

s

0 2 4 6 8

02

04

06

08

01

00

12

0

S135STAR135SSAT135

Simulation Study• 3 Grid Enhancements: S, STAR, S with Satellites• Two sets of lag classes of size 4

– 0.05, 0.10, 0.20, 0.50 (km)– 0.05, 0.25, 1, 3 (km)

• Spherical variogram– Range = 1, 2, 4, 6– Nugget = 0.00, 0.10– Sill = 1

• 1000 sims• Fit using automated procedure in Splus

– This may have introduced artifacts

Percent Difference from Target Range(Median Range) S=1, N= 0.10

True Range

Perc

ent o

f Tar

get

1 2 3 4 5 6

-10

010

2030

40

SStarSSAT

Lag = 0.05, 0.25, 1.00, 3.00

True Range

Perc

ent o

f Tar

get

1 2 3 4 5 6

-10

010

2030

40

SStarSSAT

Lag = 0.05, 0.10, 0.20, 0.50

Percent Difference from Target Sill(Median Sill) S=1, N= 0.10

True Range

Perc

ent o

f Tar

get

1 2 3 4 5 6

-10

-50

510

1520

SStarSSAT

Lag = 0.05, 0.25, 1.00, 3.00

True Range

Per

cent

of T

arge

t

1 2 3 4 5 6

-10

-50

510

1520

SStarSSAT

Lag = 0.05, 0.10, 0.20, 0.50

Percent Difference from Target Nugget(Median Nugget)

S=1, N= 0.10

True Range

Med

ian

1 2 3 4 5 6

-100

-50

050

100

SSTARSSAT

Lag = 0.05, 0.25, 1.00, 3.00

True Range

Med

ian

1 2 3 4 5 6

-100

-50

050

100

SSTARSSAT

Lag = 0.05, 0.10, 0.20, 0.50

Summary

STAR- performed better than S and S with Satellites for estimating variogram parameters- robust to different lag classes

S – lacks sufficient information at short distances for estimating nugget

S with Satellites- better than S design for estimating nugget, not as good as STAR

Larger lag classes generally did better than shorter lag classes

Further Research

• Choose another variogram model– Exponential

• Choose another variogram fitting algorithm– REML

• Simulate anisotropy• Investigate robustness to model misspecification• Explore other designs

END OF PLANNED PRESENTATIONS

• Questions and suggestions are welcome

Note

• Note that rest of slides show simulation results for N=0, S=1. They will not be included in presentation

Percent Difference from Target (Median Range)

S=1, N= 0

True Range

Perc

ent o

f Tar

get

1 2 3 4 5 6

-10

010

2030

40

SStarSSAT

Lag = 0.05, 0.25, 1.00, 3.00

True Range

Perc

ent o

f Tar

get

1 2 3 4 5 6

-10

010

2030

40

SStarSSAT

Lag = 0.05, 0.10, 0.20, 0.50 Lag = 0.05, 0.25, 1.00, 3.00

Percent Difference From Target(Median Sill)

S=1, N=0

True Range

Perc

ent o

f Tar

get

1 2 3 4 5 6

-50

510

SStarSSAT

Lag = 0.05, 0.25, 1.00, 3.00

True Range

Perc

ent o

f Tar

get

1 2 3 4 5 6

-50

510

SStarSSAT

Lag = 0.05, 0.10, 0.20, 0.50

Difference from Target (Median Nugget)

S=1, N= 0

True Range

Med

ian

1 2 3 4 5 6

0.0

0.00

20.

004

0.00

60.

008

SSTARSSAT

True Range

Med

ian

1 2 3 4 5 6

0.0

0.00

20.

004

0.00

60.

008

SSTARSSAT

Lag = 0.05, 0.25, 1.00, 3.00Lag = 0.05, 0.10, 0.20, 0.50

“S” Cluster Design

• 12 grid stations 12

• 11 “S” Clusters of Size 9 99-5 = 94– 5 grid stations– 6 sites of interest (some old stations, some Bight

stations, some new)

• 6 field duplicates 6

• Total samples = 112 112

“STAR” Cluster Design

• 12 grid stations 12• 5 “STAR” Clusters of Size 16 (17) 80

– 3 grid stations

– 2 site of interest (one Bight station, one old station) 2• 1 “S” Cluster of Size 8 (9) 9

– new station

• 9 field duplicates 9• Total samples = 112 112

“S” Cluster with Satellites

• 12 grid stations 12• 8 “S” Clusters of Size 8 (9)

– 4 grid stations (8) 32– 4 sites of interest (some old stations, some Bight

stations, some new) (9) 36• 8 Satellites added to 3 Clusters 24• 8 field duplicates 8• Total samples = 112 112