Sampling Design, Spatial Allocation, and Proposed Analyses Don Stevens Department of Statistics Oregon State University

Sampling Environmental Populations Environmental populations exist in a spatial matrix Population elements close to one another tend to be more similar than widely separated elements Good sampling designs tend to spread out the sample points more or less regularly Simple random sampling (SRS) tends to result in point patterns with voids and clusters of points

Sampling Environmental Populations Systematic sample has substantial disadvantages –Well known problems with periodic response –Less well recognized problem: patch-like response –Inflexible point density doesnt accommodate Adjustment for frame errors Sampling through time

Random-tessellation Stratified (RTS) Design Compromise between systematic & SRS that resolves periodic/patchy response Cover the population domain with a randomly placed grid Select one sample point at random from each grid cell

RTS Design Does not resolve systematic sample difficulties with –variable probability (point density) –unreliable frame material –Sampling through time

Generalized Random-tessellation Stratified (GRTS) Design Design is based on a random function that maps the unit square into the unit interval. The random function is constructed so that it is 1- 1 and preserves some 2-dimensional proximity relationships in the 1-dimensional image. Accommodates variable sample point density, sample augmentation, and spatially-structured temporal samples.

Spatial Properties Of Reverse Hierarchical Ordered GRTS Sample The complete sample is nearly regular, capturing much of the potential efficiency of a systematic sample without the potential flaws. Any subsample consisting of a consecutive subsequence is almost as regular as the full sample; in particular, the subsequence., is a spatially well-balanced sample. Any consecutive sequence subsample, restricted to the accessible domain, is a spatially well-balanced sample of the accessible domain (critical for sediment sample).

Spatially Balanced Sample Assess spatial balance by variance of size of Voronoi polygons, compared to SRS sample of the same size. Voronoi polygons for a set of points: The i th polygon is the collection of points in the domain that are closer to s i than to any other s j in the set.

Sampling Through Time Detection of a signal that is small relative to noise magnitude requires replication Spatial replication (more samples per year) addresses spatial variation Need temporal replication (more years) to address temporal variation Detection of trend in slowly changing status requires many years

Sampling Through Time Repeat sampling of same site eliminates a variance component if the site retains its identity through time. Design based on assumption that sediment does retain identity, but water does not. Both water and sediment samples have spatial balance through time, but sediment sample includes revisits at 1, 5, and 10 year intervals.

Proposed Analyses Annual descriptive summaries –Mean values, proportions, distributions, precision estimates based on annual data Mean concentration confidence limits Percent area in non-compliance confidence limits Histograms Distribution function plots confidence limits Subpopulation comparisons

Proposed Analyses Composite estimation: Annual status estimates that incorporate prior data –Model that predicts current value at site s based on prior observation: –Composite estimator is weighted combination of mean of current observation and mean of predicted values based on prior observations –Results in increased precision for annual estimates –Can also be used to borrow strength from spatially proximate data

Proposed Analyses Trend Analyses. –Need to describe trend at the segment or Bay level. –Usual approach: trend in mean value. –Also consider: trend in spatial pattern, trend in population distribution, distribution of trend, and mean value of trend. –Trend analyses will exploit repeat visit pattern for sediment samples.

Proposed Analyses Space-Time Models –Use random field approach to account for correlation through space and time –Panel structure (repeat visits) in sediment sample is a good structure to estimate space- time correlation –Long-term: need 10+ years to get sufficient data to estimate model parameters

Proposed Analyses Bayesian Hierarchical Models –Good way to incorporate ancillary information into status estimates E.g., loading estimates, flow data, metrological data –Distribution of response is modeled as a function of parameters whose distribution in turn depends on ancillary data, hence, hierarchical

Proposed Analyses Spatial displays –Contour plots –Perspective plots –Hexagon mosaic plots –Multivariate displays