What is a density surface model?

Why model abundance spatially?

  • Use non-designed surveys
  • Use environmental information
  • Maps

Back to Horvitz-Thompson estimation

Horvitz-Thompson-like estimators

  • Rescale the (flat) density and extrapolate

\[ \hat{N} = \frac{\text{study area}}{\text{covered area}}\sum_{i=1}^n \frac{s_i}{\hat{p}_i} \]

  • \( s_i \) are group/cluster sizes
  • \( \hat{p}_i \) is the detection probability (from detection function)

Hidden in this formula is a simple assumption

  • Probability of sampling every point in the study area is equal
  • Is this true? Sometimes.
  • If (and only if) the design is randomised

Many faces of randomisation

plot of chunk randomisation

Randomisation & coverage probability

  • H-T equation above assumes even coverage
    • (or you can estimate)

Extra information

plot of chunk plottracks

Extra information - depth

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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

  • (only segments where counts > 0)

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

  • Include detectability
  • Account for effort
  • Flexible/interpretable effects
  • Predictions over an arbitrary area

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”: 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

Effective effort

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

Estimated abundance

  • 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 segment
    • “Segment” – change between segments
  • “Count model” only lets us use segment-level covariates
  • “Estimated abundance” lets us use either

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 bias and more

  • \( \hat{p} \) is not always simple!
  • Availability & perception bias somehow enter
  • We can make explicit models for this
  • More later in the course

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

Example of spatial data in QGIS

Recap

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