• CO\(_2\) flux measures the CO\(_2\) consumption and production of plants.

• Flux towers provide ground-based, direct and local observations of carbon exchange.

The moderate-resolution imaging spectroradiometer (MODIS) is a scientific instrument that was launched into Earth orbit by NASA on board the Terra satellite (1999) and the Aqua satellite (2002).

The latitude is 53.6289 and the longitude is -106.1978.

The CO\(_2\) flux unit used in this study is \(umol/m^2/s\).

Functional data smoothing (Ramsay, 2006) was applied. We represented each year as a functional object using cubic splines: \(f(t_{ij})=\sum_{j}\beta_j\phi_j(t_{ij}).\)

Landmarks are a key quantity of data registration.

We constructed a time transformation \(h_i(t)\) for each curve such that the registered curves with values

\[ x^{*}(t) = x_i[h_i(t)] \]

have identical argument values for any given landmark.

This warping function, \(h_i(t)\), is the bridge that connects the real time and the registered time.

In the regisgtered time scale, each year of data are represented by \(\sum\beta^{*}\phi(t)\). To obtain the simulations, we used multivariate normal distribution to simulate the \(\beta^*\).

The Dirichlet distribution of order \(K\ge2\) with parameters \(a_1,\dots,a_k>0\) has a probability density function

\[ f(x_1,\dots,x_k;\alpha_1,\dots,\alpha_k) = \frac{1}{B(\alpha)}\prod_{i=1}^Kx_i^{\alpha_i-1}, \]

where \(x_i>0\) and \(\sum_{i=1}^kx_i=1\).

\[ E(X_i) = \frac{\alpha_i}{\sum_k \alpha_k} \]

• FLUXNET (2014). Distribution of the global flux networks. Retrieved from: https://fluxnet.ornl.gov/introduction. Accessed: 2016-09-16.

• NASA (2016). NASA’s earth observing system: Project science office. Retrieved from: http://eospso.nasa.gov/. Accessed: 2016-09-16.

• Wood, S. (2006). Generalized additive models: an introduction with R. CRC press.

• Ramsay, J. O. (2006). Functional data analysis. John Wiley & Sons, Inc..