Simulation setup and model calibration

The VS model was used to explore the extent to which shifts in limiting factors and non-linear growth responses can explain the empirical evidence for changing relationships between climate and white spruce growth over recent decades in three locations in Northwestern North America (D’Arrigo et al., 2008): (1) Arrigetch (1586–1975, 716 m asl), a treeline site in the Brooks Range of Alaska studied by Jacoby & D’Arrigo(1995), (2) Twisted Tree-Heartrot Hill (TTHH;

1459–1999, 915 m asl), an elevational tree line site in the Yukon Territory, Canada studied by D’Arrigo et al.(2004), and (3) Thelon River Sanctuary (1288–2004, 160 m asl), a site located in the latitudinal treeline in Nunavut in the northern interior of Canada studied byD’Arrigo et al.

(2009). These three tree-ring sites were chosen because their change in climate response has been well documented in the literature using traditional correlation analysis but no modelling work

6.2. Data and methods 125

Table 6.1: Parameters of the VS and VS-Lite models. Parameter values are the defaults used byVaganov et al.

(2006) andEvans et al.(2006) for the VS model and byTolwinski-Ward et al.(2011) for VS-Lite. Where appro-priate, ranges for parameter values are indicated in brackets.

Parameter Description Value

Full VS model

Temperature response function

Tmin Minimum temperature for growth (^◦C) 5

Topt1 Lower optimal temperature for growth (^◦C) 18

Top2 Upper optimal temperature for growth (^◦C) 24

Tmax Maximum temperature for growth (^◦C) 31

Moisture response function

Wmin Minimum soil moisture for growth (wilting moisture;v/v) 0.04

Wopt1 Lower optimal soil moisture for growth (v/v) 0.2

Wopt2 Upper optimal soil moisture for growth (v/v) 0.8

Wmax Maximum soil moisture for growth (v/v) 0.9

Water balance

k¹ Fraction of precipitation reaching the soil (dimensionless) 0.72

k2 First coefficient for transpiration (mm/day) 0.12

k3 Second coefficient for transpiration (1/degrees) 0.175

Pmax Maximum daily precipitation infiltrating in the soil (mm) 20

Sr Rooting depth (mm) 1000

Tmelt Sum of temperature for starting soil thawing (^oC) 60

tmelt Period for summing temperatures for soil thawing (days) 10

a¹ First coefficient for soil thaw (mm/^oC) 10

a2 Second coefficient for soil thaw (1/day) 0.006

SNr Rate of snow melting (mm/^oC/day) 1

SNmt Minimum temperature for snow melting (^oC) 0

Λ Rate of soil water drainage (dimensionless) 0.001

Cambial module

Tbeg Temperature sum to starting growth (^oC) 60

tbeg Period for temperature sum (days) 10

Vcr Critical rate of cell transition to resting (µm/days) 0.04

b¹ Coefficients of growth rate function from position j (µ m/days) 0.42

b2 Vo(j) = b1+ b2j(µ m/days) 0.25

b3 Coefficients of minimum growth rate fucntion (µ m/days) -1.62

b4 Vmin= b3+ b4j(µ m/days) 0.54

dt Tangential tracheid size (µ m) 40

VS-Lite model

Temperature response function

T1 Minimum temperature for growth (◦C) [0 8.5]

T2 Optimal temperature for growth (^◦C) [9 20]

Moisture response function

M1 Minimum moisture for growth (v/v) [0.01 0.03]

M² Optimal moisture for growth (v/v) [0.1 0.5]

Water balance

α First coefficient for runoff (1/month) 0.093

µ Second coefficient for runoff (dimensionless) 5.8

m Third coefficient for runoff (dimensionless) 4.886

Wmax Maximum moisture held by soil (field capacity; v/v) 0.76

Wmin Minimum moisture held by soil (wilting point; v/v) 0.01

dr Rooting depth (mm) 1000

Integration period for growth

I⁰ Relative month for integration start -4

If Relative month for integration end 12

exploring the mechanisms has been published. Moreover, these sites are central to the issue of changing climate sensitivity of tree growth in recent decades (D’Arrigo et al., 2008). The

126

Chapter 6. Re-assessing moisture limitation on tree-ring growth in the boreal forest:

a case study in northwestern North America

data were obtained from the ITRDB. Note that these chronologies were not used in correlation analysis and are not included included in the network of 115 sites described in Supplementary Table E1 because not all sites spanned the complete common period 1950–2000.

For each site, daily precipitation and temperature data from the nearest meteorological station having the longest and more continuous records were used to drive the model: (1) University Experimental Station for Arrigetch – 400 km south and ∼850 m lower than the sampling site, (2) Baker Lake for Thelon – 378 km west and ∼140 m lower than the sampling site, and (3) Dawson for TTHH – 152 km south and ∼906 m lower than the trees. Data were obtained from the Global Historical Climatology Network (GHCND, ftp://ftp.ncdc.noaa.

gov/pub/data/ghcn/daily/; Klein Tank et al., 2002) and the Adjusted Historical Canadian Climate Data (AHCCD,http://www.cccma.ec.gc.ca/hccd/) developed by the Meteorological Service of Canada (Vincent et al.,2002;Mekis & Vincent,2011). No seasonal soil freeze/thaw dynamics was used in the simulations because the simulated thaw dynamics appeared unrealistic and had a large impact on volumetric soil moisture. Therefore, a constant soil depth of 50 cm was used.

Difficulties in constraining model parameters have hampered modelling studies in these re-mote settings. In this study for the first time, a generic Bayesian approach as outlined by Van Oijen et al. (2005) was used to conduct a probabilistic calibration of the VS model and provide uncertainty estimates in simulated tree growth at each site. In this framework, the uncertainty in model parameterisation is represented as a probability distribution over the pa-rameters, which expresses our initial knowledge about these uncertain quantities. This initial probability distribution, and hence our knowledge about the parameters, is then updated con-ditional on new information from experimental or field observations (e.g., standardised tree-ring chronologies), expressed in terms of the likelihood of the model output being equal to the avail-able data. In a statistical sense, given a model parameter vector θ and data D, the Bayesian calibration of a dynamical model M involves computing the posterior distribution p(θ|D) ac-cording to Bayes’s Theorem:

p(θ|D) = cp(D|θ)p(θ) (6.6)

where c = p(D)⁻¹. c is a normalisation constant and does not need to be computed explicitly.

p(D|θ) is known as the likelihood function of θ, and p(θ) as the prior distribution for θ.

A subset of 10 critical model parameters was selected for calibration (Table 6.2).

This includes the 8 parameters of the response functions for moisture and temperature, the rate of drainage, and the critical temperature sum for starting growth. Therefore, θ = (T_min, T_opt1, T_opt2, T_max, W_min, W_opt1, W_opt2, W_max, Λ, T_beg). Each parameter was given a flat (uniform) prior distribution [θ^min_i , θ^max_i ] with physically plausible minimum and upper bounds based on the literature (Vaganov et al., 2006; Evans et al., 2006; Anchukaitis et al., 2006) and previous experimentation with the model (Table 6.2). If the strength of the cli-mate signal in the sampled tree-ring chronologies is sufficiently strong, the uncertainty (spread) of the posterior distribution of the parameters should decrease, otherwise the posteriors will not differ much from their flat priors.

Model calibration was then performed by maximizing the negative logarithm of a likelihood function (Van Oijen et al.,2005;Schoups & Vrugt,2010), which quantifies the probability that

6.2. Data and methods 127

the observed tree-ring chronology at a given site was generated by a particular parameter set, accounting for measurement error. For simplicity, the measurement error was set to 10% of the mean value and assumed to be uncorrelated and normally distributed with zero mean.

An efficient Markov-Chain Monte Carlo (MCMC) scheme based on the differential evolution adaptive Metropolis algorithm (DREAM) was used to sample the joint posterior parameter distribution (Vrugt et al., 2009; Laloy & Vrugt, 2012). DREAM runs multiple Markov chains (one for each parameter) in parallel and uses a discrete proposal distribution to efficiently evolve the sampler toward the posterior distribution. The first 10% of the iterations were discarded as unrepresentative ‘burn-in’ period where the chains are more related to their initial starting points (Gelman et al., 2013). Convergence to the stationary posterior distribution is reached when the variance between chains no longer exceeds the variance within each individual chain.

This was evaluated using the ˆR statistic (Gelman & Rubin,1992).

The last 20% of the iterations were considered as a representative sample of the posterior and then were thus used to compute statistics of expectation and spread of the distribution of each parameter and characterise model predictive uncertainty. The maximum a posteriori (MAP) estimation of θ, i.e., the iteration that maximises the likelihood, was used as the calibrated parameter set to run the model. The calibration was performed independently for the SPL10 and NEGEX chronologies at each site using the initial 20 years of the corresponding meteorological record. However, only the results for the SPL10 chronologies are shown here. The ability of the calibrated model to simulate tree growth was verified by comparing the site chronologies with the simulations over the remaining common period, which includes the most recent decades when the apparent changes in climate sensitivity of tree growth have occurred.

The VS–Lite model was used to characterise the general patterns of climate limitation on tree growth across the entire chronology network using the same gridded temperature and pre-cipitation data utilised for empirical correlation analysis. The model was calibrated over the pe-riod 1950–1970 using an inbuilt Bayesian approach described in detail byTolwinski-Ward et al.

(2013). The lack of snow dynamics on simulated soil moisture is the main caveat of the appli-cation of this model in the boreal region.

6.2.4 Relationship between tree growth and variations in growing season and