This research is funded by
Review of Geostatistics in Aquatic Systems
U.S.EPA – Science To Achieve
Results (STAR) Program
Cooperative
# CR - 829095
Agreement
Joshua French and Scott Urquhart
Department of Statistics, Colorado State University
Fort Collins, Colorado
Exploring Spatial Correlation in Rivers
Developments Related to Aquatic Systems
Aquatic Applications
Choice of Distance Metric
Oceans, Seas, and Bays
Author: Joshua French
Advisor: Scott Urquhart
Development
Benefits
Pittsburgh, PA
Abstract
Inconclusive
40
Use of in-water distance instead of Euclidean
Data Set
Addresses problem of using Euclidean
distance in a river system
Great improvements in stream temperature
predictions
Computationally efficient method of
estimating in-water distance.
Yields valid autocovariance model
Coordinate transformation
Latitude (NAD27)
Cincinnati, OH
Use of in-water distance and stream order as distance
metric
39
Semivariograms are used to explore and quantify
spatial correlation of several particle size and
biological variables for the longitudinal profile of
the Ohio River
Multi-dimensional scaling
38
Louisville, KY
The data consisted of between 190 and 235
unique sampling sites (depending on the
variable) collected by the Ohio River Valley
Sanitation Commission (ORSANCO). The data
consisted of both particle size and biological
measurements.
37
Euclidean distance reasonable when variable of interest
depends on underlying geology
Simplifies analysis
-88
-86
-84
-82
Conan (1989)
Shelikof Strait walleye Pollock a bundance
estimated
Variograms and ordinary kriging kriging
Sullivan (1991)
Barabas et al. (2001)
Harvestable biomass of Nephrops norvegicus
Variograms and ordinary kriging
Conan et al. (1992)
Variance of Alaskan walleye pollock
Transitive method in one dimension
Williamson and Traynor (1996)
Nephrops norvegicus biomass
Variograms and point kriging
Maynou et al. (1998)
Euphausiid population size
Block averaging and kriging
Romaine et al. (2002)
Nitrogen “hotspot” locations
Variograms and Bayesian Transformed Gaussian
random field model
De Oliveira and Ecker (2002)
Conservation criteria
Indicator kriging
Stelzenmuller et al. (2004)
Distributional pattern of Atlantic cod
Variograms and ordinary kriging
Mello (2005)
Gardner et al. (2003)
Loland and Host
(2003)
Yuan (2004)
Estuaries
Invalid Covariance Structures
-80
Longitude (NAD27)
Invalid Covariance Structure
Exploratory analysis was conducted for each of the variables. When reasonable, the method of moments empirical semivariogram
was calculated for each variable. Maximum likelihood was then used to model the empirical semivariogram using the exponential,
Matern, or Gaussian semivariogram models. The variogram analysis fell into three categories: good results, poor results, and no
results. Good results were when the variogram model fit the sample variogram reasonably well, poor results were when the
variogram model fit the empirical variogram poorly, and no results were variables for which variogram analysis did not seem
reasonable.
Good results:
•Percent Gravel, Number of Individuals, Number of Species
Number of Fish
Number of Fish Species
Response
12
10
8
9
0.20
250
150
200
250
0
50
Lag Distance (Mi)
100
150
200
250
0
50
Lag Distance (Mi)
100
150
200
250
Lag Distance (Mi)
•Percent Sand, Percent Detritivore, Pecent Simple Lithophilic Individuals, Percent Invertivore
2.0
2.5
Variance
3.0
3.5
3.5
3.0
2.0
2.5
Variance
1.4
1.2
Reference
Method
0
50
100 150 200 250
Lag Distance (Mi)
0
50
100 150 200 250
Lag Distance (Mi)
0
50
100 150 200 250
Lag Distance (Mi)
0
50
Variograms and ordinary kriging
Simard et al. (1992)
Biomass of pelagic fish
Variograms and ordinary kriging
Simard et al. (1993)
Distribution of possible outcomes at unsampled
locations
Indicator kriging
Barabas (2001)
Spatial structure of species richness
Variograms
Rueda and Defeo (2003)
Spatial structure of fish abundance, size and
biovalue
Variograms
Rueda and Defeo (2003b)
Probability that biovalue exceeds desirable
thresholds
Indicator kriging
Rueda and Defeo (2003b)
Regions of sediment structure
Indicator kriging and block kriging
Caeiro et al. (2003)
Produces valid covariance models Kruvoruchko and Gribov (2004)
Valid covariance models
Ver Hoef et al. (2005)
Valid covariance models
Prediction/Estimation Development
Fitting of variogram models to highly skewed acoustic survey data
Variogram analysis is used with generalized least squares
A spatio-temporal model which allows the prediction in space and time with known
confidence
A spatial statistical model based on kriging is developed to analyze oceanographic
spatial data
A comprehensive introduction to the use of geostatistics to estimate the abundance and
distribution of fish
A Bayesian Transformed Gaussian random field model is combined with variograms
Multivariate methods such as principal component analysis, cluster analysis, and
discriminant analysis are combined with different types of kriging
New three-dimensional visualization and animation techniques are developed
Use of constrained kriging or covariance-matching constrained kriging to make nonlinear predictions
Rivers and Streams
Cressie et al. (2005)
Reference
Maravelias et al. (1996)
Hobert et al. (1997)
Cressie and Majure (1997)
Aranuvachapun and Maskell (1997)
Rivoirard et al (2000)
De Oliveira and Ecker (2002)
Caerio et al (2003)
Torgersen et al. (2004)
Response
Method
Lag Distance (Mi)
•Percent Cobble, Percent Hardpan, Percent Fines, Percent Boulder, Percent Tolerant Individuals, Percent Nonnative
Individuals, Percent Piscivore
Spatial distribution of copper, lead, and zinc
Variograms
Zhang and Silinus (1997)
Spatio-temporal modeling of nitrate concentration
Variograms and ordinary kriging
Cressie and Majure (1997)
Patterns of spatial heterogeneity and experiment
design
Variograms
Cooper et al. (1997)
Movement and distribution of large woody debris in a
stream
Variograms
Wing et al. (1999)
Spatial heterogeneity in nutrient concentrations
Variograms
Dent and Grimm (1999)
Spatial distribution of DDT in a river
Variograms and three-dimensional kriging
Ouyang et al. (2003)
Distribution of sediment mercury in a river
Variograms and three-dimensional kriging
Ouyang et al. (2003b)
Spatial structure of a river channel and changes due to
channel change
Variograms and ordinary kriging
Chappell et al. (2003)
Fluvial response variables
Variograms and generalized least squares
Legleiter et al. (2003)
Acid Neutralizing Capacity
Variograms and multiple regression
Kellum (2003)
Stressor levels in unsampled Maryland streams
Variograms
Yuan (2004)
Spatial structure in the distribution of coastal cutthroat
trout
Variograms
Torgersen et al. (2004)
Daily change of dissolved oxygen throughout a river
network
Variograms and covariance-matching constrained
kriging
Cressie et al. (2005)
Characteristics and spatial distribution of pesticide
chlordane
Variograms and ordinary kriging
Ouyang et al. (2005)
Cressie et al. (2005)
Lakes
Sampling Design and Optimization
Development
Choose sample locations in order to enhance the reliability of the variogram
Summary of Results
Transformation
Trend Removed
38.1082+.0330x
Natural Log
Natural Log (outliers removed)
Square Root
Square Root
Square Root (outliers removed)
Square Root
17.7849-.0042x
15.5364-.0023x
6.5207-.0039x
Reference
100 150 200 250
No results:
Response
Percent Gravel
Percent Sand
Number of Fish
Number of Fish
Number of Native Species
Percent Lithophilic Fish
Percent Detritivore
Percent Detritivore
Percent Invertivore
Reference
Biomass of northern shrimp
1.5
1.5
1.0
Variance
Percent Invertivore
4.0
Percent Simple Lithophilic Individuals
1.8
Percent Detritivore
1.6
400 450 500 550 600 650 700
Variance
Percent Sand
Benefits
Prediction/Estimation Methods
11
Variance
0.30
0.25
Variance
350
300
Variance
Poor results
100
Development
Cost-weighted distance as a solution to traveling across
a barrier
Covariance models that incorporate flow and in-water
distance
Covariance models that incorporate both in-water and
Euclidean distance
Reference
Rathbun (1998), Ganio et al. (2005)
Ganio et al. (2005)
Ver Hoef et al. (2005)
13
14
400
Percent Gravel
Whittle
Gaussian and Matern
Spherical
Reference
Variograms and kriging
Cairo, IL
Results
50
Method
Harvestable resources of Pandalid shrimp
Use of in-water distance leads to invalid covariance models:
0
Response
Reference
Little et al. (1997)
Rathbun (1998)
Model
Nugget Sill
Range
Exponential 286.09 335.53 72.9 miles
Gaussian 520.88 658.32 71.67 miles
Gaussian
0.29
0.39 44.19 miles
Exponential 0.2
0.27 37.69 miles
Gaussian
10.1
11.87 39.93 miles
Matern
0.92
2.76 44.02 miles
Exponential 1.09
1.57 24.08 miles
Exponential 0.94
1.4 19.17 miles
Exponential 1.4
2.97 13.43 miles
Response
Reference
Warrick and Myers (1987)
Method
Reference
Effect changes in sulfate depositions on fish species
richness
Variograms
Hobert et al. (1997)
Benthic invertebrate counts
Variograms
Dolan et al. (2000)
Patterns of diatom distribution
Variograms
Kienel and Kumke (2002)
Lake Level
Triple diagram method based on kriging
Altunkaynak et al (2003)
Present a method using universal kriging with limited sampling stations
Posa and Rossi (1991)
Compare optimal sample designs for a classical approach and a geostatistical
approach in the context of shoreline recession and accretion
Dolan et al. (1992)
Enhance theory regarding design of sampling transects for characterizing water
quality in estuaries
Jassby et al. (1997)
Demonstrate that river systems can be stratified to improve kriging results
Cressie et al. (2005)
Optimal sample spacing of beach profile sample
intervals
Variograms
Phillips (1985)
Perform two simulation studies to assess the usefulness of multi-lag cluster designs
for estimating variogram parameters
Ritter and Leecaster (2007)
Alongshore pattern of shore erosion
Variograms
Phillips (1986)
Three-dimensional beach morphology
Variograms and ordinary kriging
Swales (2001)
Accuracy of beach volume estimates
Ordinary kriging
Swales (2001)
Coastal Systems
Response
Method
Reference