What is a density surface model?

David L Miller

Why model abundance spatially?

  • Use more information
  • Greater explanatory power
  • Spatially explicit estimates (of abundance and uncertainty)
  • Variance reduction

Extra information

plot of chunk plottracks

Extra information - depth

plot of chunk plotdepth

Extra information - depth

plot of chunk plotdepth-notspat

  • NB this only shows segments where counts > 0

Extra information - SST

plot of chunk plotsst

Extra information - SST

plot of chunk plotsst-notspat

  • NB this only shows segments where counts > 0
What is going on here?

"You should model that"

Modelling outputs

  • Abundance and uncertainty
    • Arbitrary areas
    • Numeric values
    • Maps
    • Extrapolation (with caution!)
  • Covariate effects
    • count/sample as function of covars

Modelling requirements

  • Account for effort
  • Flexible
  • Explicit spatial terms
  • Interpretable effects
  • Predictions over an arbitrary area
  • Theoretical basis for model validation
  • Include our detectability information

Accounting for effort

Effort

plot of chunk tracks2

  • Have transects
  • Variation in counts and covars along them
  • Want a sample unit w/ minimal variation
  • “Segments” – approx. square chunks of effort

Chopping up transects

Physeter catodon by Noah Schlottman

Physeter catodon by Noah Schlottman

Flexible, interpretable effects

Smooth response

plot of chunk plotsmooths

Explicit spatial effects

plot of chunk plot-spat-smooths

Predictions

Predictions over an arbitrary area

plot of chunk predplot

  • Don't want to be restricted to predict on segments
  • Predict within survey area
  • Extrapolate outside (with caution)
  • Working on a grid of cells

Detection information

Including detection information

  • Two options:
    • adjust areas to account for effective effort
    • use Horvitz-Thompson estimates as response

Adjusting areas

  • Area of each segment \( A_j \) and use \( A_j\hat{p}_j \)
  • (2-D) Equivalent to effective strip width
    • \( \hat{\mu} = w\hat{p} \)
  • Response is counts per segment
  • “Adjusting for effort”
  • “Count model”

Horvitz-Thompson estimates

  • Estimate H-T abundance per segment
  • Effort is area of each segment
  • “Estimated abundance” per segment

\[ \hat{n}_j = \sum_{i \text{ in segment } j } \frac{s_i}{\hat{p}_i} \]

Detectability and covariates

  • 2 covariate “levels” in detection function
    • “Observer”/“observation” – change within transect
    • “Segment” – change between segments
  • “Estimated abundance” lets us use observer-level covariates in detection function
  • “Count model” only lets us use segment-level covariates

When to use each approach?

  • Generally “nicer” to adjust effort
  • Keep response (counts) close to what was observed
  • Unless you want observation-level covariates
    • These can make a big difference!

Availability/perception/etc

  • Availability & perception bias via \( \hat{p} \)
  • \( \hat{p} = \hat{p}_\text{availability}\hat{p}_\text{perception}\hat{p}_\text{detection} \)
  • Not going to cover this much here
  • See bibliography for more info

DSM flow diagram

DSM process flow diagram

Spatial models

Abundance as a function of covariates

  • Two approaches to model abundance
  • Explicit spatial models
    • When: Good coverage, fixed area
  • “Habitat” models (no explicit spatial terms)
    • When: Poorer coverage, extrapolation
  • We'll cover both approaches here

Data requirements

What do we need?

  • Need to “link” data
  • Distance data/detection function
  • Segment data
  • Observation data to link segments to detections

Jason demo of segmenting etc

  • Show each table
  • Their relations
  • Spatial representation

Recap

  • Model counts or estimated abundace
  • The effort is accounted for differently
  • Flexible models are good
  • Incorporate detectability
  • 2 tables + detection function needed