Journal of Biogeography
SUPPORTING INFORMATI ON
Geographical variability in the controls of giant kelp biomass dynamics
Tom W. Bell, Kyle C. Cavanaugh, Dan C. Reed and David A. Siegel
APPENDIX S1 Supplementary methods.
Modelled significant wave height methods
Predictions of significant wave height, Hs, for the entire time-series were made at each
coastline site using a generalized additive model (GAM) with the log-transformed
hourly CDIP-modelled Hs as the response variable with a Gaussian distribution, and the
hourly Harvest buoy Hs, period and direction as the predictors. The site-specific model
was trained with two-thirds of the available wave data from 1998–2012 and validated
using the remaining third. Hourly Hs was hindcasted from 1987–2011 by inputting
hourly Hs and period from the Harvest platform, and using a randomly selected
direction from a probability function for daily direction generated from direction data
from the Harvest buoy from 1998–2012 for dates (1987–1997) and known directional
data for 1998–2011. An estimate of hourly Hs was generated from the model 100 times
using a new randomly selected direction for each iteration. The mean Hs from these 100
model estimates was used as the Hs for that hour at the specific site.
Sea urchin observations
Sea urchin densities were measured by the Santa Barbara Coastal Long Term Ecological
Research Project (SBC LTER), the Partnership for Interdisciplinary Studies of Coastal
Oceans (PISCO), the US Geological Survey and the National Park Service. Sea urchin
density was only included at sites where urchin surveys were conducted inside a 500-m
coastline segment. GAMs were run for these coastline segments only for the period for
which sea urchin data were available. All surveys were completed on an annual basis,
with seasonal densities estimated from the annual survey closest in time, and density
was obtained from either 1-m2 quadrats along permanent 50-m transects or 30 m × 2 m
swath surveys. Annual surveys are a valid proxy for season densities because sea
urchins are long-lived (Ebert & Southon, 2003). The SBC LTER collected sea urchin
density data for sites on the mainland coastline of southern California; PISCO sites were
used for sites on the mainland coast of central and southern California, the northern
Channel Islands, and Santa Barbara Island, the US Geological Survey for sites on San
Nicolas Island and the National Park Service for sites in the northern Channel Islands.
Generalized additive models and prediction methods
The general concept of GAMs is that a response variable can be modelled as the sum of
non-linear functions of different predictor variables (Hastie & Tibshirani, 1990). The
underlying relationship between each predictor variable and kelp biomass was
determined using thin-plate penalized regression splines, which add penalties to wiggly
functions to avoid overfitting (Wood & Augustin, 2002). The weight of these penalties
was optimized using generalized cross-validation, which minimizes the root mean
square error between the fit and data points. Shrinkage smoothers were added to the
GAM allowing a predictor to be penalized over its entire range and be practically
removed from the model if no significant relationship was found. The basis dimension
(k) for each predictor function was kept at 7, and additional wiggleness penalty
constants (m) were added to all predictors at a value of 2. The function g(E(Y)) is the
link function along with a specified distribution for the response variable data; in this
case, the g is a log link. Each fi(Xi) represents a non-parametric smoothing function of
each additive predictor variable.
𝑔(E(proportional biomass))
= 𝛽0 + 𝑓1 (𝐻smax ) + 𝑓2 (mean NO3 ) + 𝑓3 (NPGO) + 𝑓4 (kelp occupancy)
+ 𝑓5 (harvest effort) + 𝑓6 (urchin density)
To investigate potential drivers not included in the model, biomass predictions were
generated for coastline segments where there was an extended absence of kelp canopy.
Proportional kelp biomass was predicted without the time periods of extended kelp
absence (more than two quarters from nearest season with measureable biomass). The
environmental predictors identified by the EOF analysis were found to affect kelp
biomass over short time-scales (lagged by one season), so it was assumed that periods
of extended kelp absence were most likely to be due to a driver that was not identified
or measured for that period. The model biomass predictions were validated using an
eight-fold cross-validation, where the model is trained with seven-eighths of the data to
predict the final eighth, and repeated eight times. Once validated, these initial
predictions of proportional kelp biomass were lagged by one quarter and used as the
kelp occupancy term for a second run of GAM predictions. This final prediction
represents a hindcast of kelp biomass at a site if it were only a function of the
environmental predictors evaluated, and can be compared to actual kelp biomass
measurements. Periods of mismatch from these comparisons were related to sitespecific data on grazer abundance.
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