Spatial-Temporal AnalysisA Cellular Automata Model of Mineral-Related Activity from 1998 to 2010 for

Idaho and Western Montana

Raines, G.L.1, Zientek, M.L.2, Causey, J.D.2, and Boleneus, D.E2.

1) U.S. Geological Survey, MS 176 c/o Mackay School of Mines, UNR, Reno, Nevada 89557; Email:graines@usgs.gov

2) U.S. Geological Survey, 904 W. Riverside Ave. Room 202, Spokane, WA 99201

 

Spatial Modeling Extension

• Kemp, L.D., Bonham-Carter, G.F. and Raines, G.L., 1999, Arc-WofE: Arcview extension for weights of evidence mapping.

– http://gis.nrcan.gc.ca/software/arcview/wofe

• Kemp, L.D., Bonham-Carter, G.F. and Raines, G.L., 2001, Arc-SDM: Arcview extension for spatial data modelling using weights of evidence, logistic regression, fuzzy logic and neural network analysis.

– http://gis.nrcan.gc.ca/software/arcview/sdm

What is a cellular automata?

• Cellular automata (CA) are defined by an array of cells.

• The state of each cell evolves by a simple transition rule, the automaton.

• Implementation of a CA in a GIS involves a summation filter with an if-then or logic rule.

Conway’s Game of LifeProbability of Life in Next Generation

Number of Live Neighbors

Status of Center CellAlive Dead

5 0 0

4 or 3 1 1

2 0 0Assumes using a 3x3 neighborhood or kernel, that is 8 neighbors plus the center cell.

How have CAs been used?

• Modeling evolution of cities– Project Gigalopolis

http://www.ncgia.ucsb.edu/projects/gis/project_gig.htm

• Flow of lava• Evolution of forest fires• Physics – diffusion, Brownian motion,

defraction• Biology – life processes

Statement of Problem

• Project mineral-related activity on public land to 2010– Objective is to make a probabilistic prediction

of total activity– Not necessary to know annual activity

• Data on mineral activity from 1989 to 1998 are available to calibrate the CA– Permitting rules changed in 1992.

Public Lands

USFSBLMPrivate Land

Neighbors Adjustment at Edges

SE Idaho

Butte Montana

2

987

065

43

1

Neighbors

Fuzzification of Resource Themes

Input Fuzzification Steepness

Midpoint

(km) Hedge

Permits Small 4 3 None

Significant Deposits Small 4 10 None

Patents Small 4 6 None

Exploration Activity Small 4 3 None

DMEA Small 4 3 None

Phosphate Proximity Small 5 5 More or less

Assessment Tract

Class Membership

Non-permissive Permissive Favorable

Porphyry Cu 0 0.75 0.95

Revette Cu 0 0.75 0.95

Small Fuzzification Algorithm

1

21

1f

fx

x

Where f1 is the steepness of the transition from a membership value

of 1 to 0 and f2 is the mid point where the membership value is 0.5

(Tsoukalas and Uhrig, 1997).

00.10.20.30.40.50.60.70.80.9

1

0 5 10 15 20

Crisp Values

Fu

zzy

Me

mb

ers

hip

Va

lue

Small

Somewhat Small

Very Small

Large

Resource Thresholds

> .90.75-.90<.750

Membership

SE Idaho

Butte Montana

• Echo Status (State of a cell, 1.6 x 1.6 km)– Stayed alive– Stayed died– Just born– Just died

• Trace (Sum of times a cell is alive)

States - Echo and Trace

Permit Activity 1988-1998

Stayed Alive

Stayed Dead

Just Born

Just Died

State

Expansive Period

Expansive Period

Stayed Alive

Stayed Dead

Just Born

Just Died

State

Expansive Period

Stayed Alive

Stayed Dead

Just Born

Just Died

State

Expansive Period

Stayed Alive

Stayed Dead

Just Born

Just Died

State

Expansive Period

Stayed Alive

Stayed Dead

Just Born

Just Died

State

Rule Change

Stayed Alive

Stayed Dead

Just Born

Just Died

State

Contractive Period

Stayed Alive

Stayed Dead

Just Born

Just Died

State

Contractive Period

Stayed Alive

Stayed Dead

Just Born

Just Died

State

Contractive Period

Stayed Alive

Stayed Dead

Just Born

Just Died

State

Contractive Period

Stayed Alive

Stayed Dead

Just Born

Just Died

State

Contractive Period

Stayed Alive

Stayed Dead

Just Born

Just Died

State

Trace 1988-1998

0

Years Active

11109876

543

2

1

SE Idaho

Butte Montana

Calibration – Echo StatesA:Stayed Dead

69000

69100

69200

69300

69400

69500

69600

69700

69800

69900

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998

B: Just Died

50

100

150

200

250

300

350

400

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998

C: Just Born

100

200

300

400

500

600

700

800

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998

D: Stayed Alive

200

400

600

800

1000

1200

1400

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998

Actual CA Calculated

Expansive Contractive

Voting Rule Probability of Life in Next Generation

Number of Live Neighbors

Status of Center Cell

Alive Dead

5 1 1

4 0 0

Does not differentiate between live and dead center cells!

For mineral-related activity results in rapid death of all activity.

Annealed Voting Rule Probability of Life in Next Generation

Number of Live Neighbors

Status of Center Cell

Alive Dead

>5 1 1

4 or 5 = Annealing Window 0.5 0.5

<4 0 0

Softened (annealed) the boundary between 4 or 5 neighbors so mineral-related activity lasts a little longer.

Still does not differentiate the status of center cell!

Modified Annealed Voting Rule Probability of Life in Next Generation

Number of Live Neighbors

Status of Center Cell

Alive Dead

> Annealing Window Very Likely Likely

Annealing Window Likely Somewhat Likely

< Annealing Window Very Somewhat

Likely

Unlikely

Differentiating between live and dead center cells!

Complexity deals with small portion of mineral-related activity!

Definitions of ParametersParameters Definition

Very Likely Square root of Likely (Larger)

Likely A high probability of life.

Somewhat Likely An intermediate probability of life

Very Somewhat Likely Square root of Somewhat Likely (Larger)

Unlikely A low probability of life

Resource Threshold Minimum fuzzy membership defining where a reasonable explorationist would explore

Anneal Window Position and width control response of CA

Threshold Parameters Selected

CA Parameters

Time Periods

Expansive Contractive

Likely 0.87 0.75

Somewhat Likely 0.05 0.60

Unlikely 0.01 0.015

Resource Threshold 0.75 0.90

Anneal Window 5-1 7-4

Number of Live Neighbors

Status of Center Cell

Alive Dead

> Annealing Window Very Likely Likely

Annealing Window Likely Somewhat Likely

< Annealing Window Very Somewhat

Likely

Unlikely

Modified Annealed Voting Rule

Calibration – Echo StatesA:Stayed Dead

69000

69100

69200

69300

69400

69500

69600

69700

69800

69900

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998

B: Just Died

50

100

150

200

250

300

350

400

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998

C: Just Born

100

200

300

400

500

600

700

800

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998

D: Stayed Alive

200

400

600

800

1000

1200

1400

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998

Actual CA Calculated

Expansive Contractive

Cross-Tabulation 1989-1998Number of

CellsCA Trace

0 1 2 3 4 5 6 7 Sum

Actual

Trace

0 66364 1671 267 176 50 7 11 0 68446

1 354 136 76 70 24 4 2 0 666

2 154 42 57 49 18 4 1 0 325

3 129 69 102 133 47 29 20 3 532

4 33 32 52 78 34 20 16 1 266

5 8 11 15 25 42 31 11 5 148

6 8 4 22 34 24 22 14 3 131

7 17 4 17 34 81 125 70 25 373

Sum 66967 1969 608 599 320 242 145 37 70887

Selected Spatial StatisticsSpatial Statistic Trace CA Trace

Number of Patches 2,658 3,752

Mean Patch Size 6,829 4,836

Patch Size Coefficient of Variation 3,985 4,625

Total Edge 84,907,200 90,936,000

Edge Density 4.68 5.0

Mean Shape Index 1.19 1.17

Area Weighted Shape Index 26.43 27.32

Shannon’s Diversity Index 0.21 0.29

Shannon’s Evenness Index 0.10 0.14

Interspersion/Juxtaposition 70.21 58.71

Projected Probability of Activity 1998-2010

SE Idaho

Butte Montana

00-1616-3131-4747-6363-79

Probability

Avenue Code for Life• g = (g.FocalStats(#GRID_STATYPE_SUM, NbrHood.MakeRectangle(3,3,false), false) * 2 - g - 6).Abs <= 1OR• FiltGrid =

theGrid.FocalStats(#GRID_STATYPE_SUM, NbrHood.MakeRectangle(3,3,false), false)

• TimeGrid = ((FiltGrid = 3) or (FiltGrid = 4) ).con(1.AsGrid,0.AsGrid)

Modified Annealed Voting Rule• Ei+1,j = f (LO, Rj>RTp, Ppj(nij(x),Eij))• Where

– i = the year, p = the expansive or contractive period, and j = the cell number,– Eij = Status (alive or dead) of cell j in year i– LO = Private or public land ownership, no activity modeled on private land– Rj = Resource fuzzy-membership value of cell j– RTp = Period Resource threshold– Ppj = Random annual probability of activity in cell j for period p, which is a

function of nij(x), and Eij

– nij(x) = For year i, the number of active neighbors in the 3x3 neighborhood of cell j that is rescaled as a function of x, the maximum number of pubic-land neighbors of cell j. The value of x ranges from 1 to 9, and n(x) ranges from 0 to 9.