Summarizing the state of the terrestrial biosphere in few dimensions

2.1 Data

Table 1 gives an overview of the data streams used in this analysis (for a more detailed description see Appendix A). For an effective joint analysis of more than a single variable, the variables have to be harmonized and brought to a single grid in space and time. The Earth System Data Lab (ESDL; https://www.earthsystemdatalab.net, last access: 23 April 2020; Mahecha et al., 2020) curates a comprehensive set of data streams to describe multiple facets of the terrestrial biosphere and associated climate system. The data streams are harmonized as analysis-ready data on a common spatiotemporal grid (equirectangular grid 0.25^∘ in space and 8 d in time, 2001–2011), forming a 4D hypercube, which we call a “data cube”. The ESDL not only curates Earth system data, but also comes with a toolbox to analyze these data efficiently. For this study, we chose all available variables in the ESDL v1.0 (the most recent version available at the time of analysis), divided the available variables into meteorological and biospheric variables and discarded the atmospheric variables. We also discarded variables with distributions that are badly suited for a linear PCA (e.g., burned area contains mostly zeros) and variables with too many missing values. The only dataset that was added post hoc was fAPAR, which represents an important aspect of vegetation which was not available in the data cube at the time of analysis (it is part of the most recent version of the data cube).

The datasets taken from Tramontana et al. (2016) and Jung et al. (2019) are derived from flux tower measurements (Baldocchi, 2020). The flux towers are not equally distributed in climate space; i.e., there are many flux towers in temperate areas but much fewer in tropic and arctic regions, which may lead to less accurate data in these regions. These datasets also exclude large arid areas such as the Sahara and Gobi deserts and parts of the Arabian Peninsula which may affect the resulting loadings of the PCA slightly.

In this study, each variable was normalized globally to zero mean and unit variance to account for the different units of the variables, i.e., transform the variables to have standard deviations from the mean as the common unit. Because the area of the pixel changes with latitude in the equirectangular coordinate system used by the ESDL, the pixels were weighted according to the represented surface area. Only spatiotemporal pixels without any missing values were considered in the calculation of the covariance matrix.

2.2 Dimensionality reduction with PCA

As a method for dimensionality reduction, we used a modified principal component analysis to summarize the information contained in the observed variables. PCA transforms the set of d centered and, in this case, standardized variables into a subset of p, $\mathrm{1}\le p\le d$, principal components (PCs). Each component is uncorrelated with the other components, while the first PCs explain the largest fraction of variance in the data.

The data streams consist of d=12 observed variables at the same time and location. Each observation is defined in a d-dimensional space, x_i∈ℝ^d, and we define the dataset by collecting all samples in the matrix $\mathbf{X}=\left[{\mathit{x}}_{\mathrm{1}}\right|\mathrm{\cdots }\left|{\mathit{x}}_n\right]\in {\mathbb{R}}^{d\times n}$. The observations are repeated in space and time and lie on a grid of $\mathrm{lat}\times \mathrm{long}\times \mathrm{time}$. In our case, we have $n=\left|\mathrm{lat}\right|\times \left|\mathrm{long}\right|\times \left|\mathrm{time}\right|=\mathrm{720}\times \mathrm{1440}\times \mathrm{506}=\mathrm{524},\mathrm{620},\mathrm{800}$ observations, where $|\cdot |$ denotes the cardinality of the dimension. Note that the actual number of observations was lower, $n=\mathrm{106},\mathrm{360},\mathrm{156}$, because we considered land points only and removed missing values.

The fundamental idea of PCA is to project the data to a space of lower dimensionality that preserves the covariance structure of the data. Hence, the fundament of a PCA is the computation of a covariance matrix, Q. When all variables are centered to global zero mean and normalized to unit variance, the covariance matrix can in principle be estimated as

However, in our case the data cube lies on a regular 0.25^∘ grid and estimating Q as above would lead to overestimating the influence of dynamics in relatively small pixels of high latitudes compared to lower latitudes where each data point represents a larger area. Hence, one needs a weighted approach to calculate the covariance matrix,

where w_i=cos (lat_i) and lat_i is the latitude of observation i, $w={\sum }_{i=\mathrm{1}}^n{w}_i$ is the total weight, and n is the total number of observations. Equation (2) has the additional property that it can be computed sequentially on very big datasets, such as our Earth System Data Cube, by a consecutively adding observations to an initial estimate.

Note that the actual calculation of the covariance matrix is even more complicated, because summing up many floating-point numbers one by one can lead to large inaccuracies due to precision issues of floating-point numbers and instabilities of the naive algorithm (Higham, 1993; the same holds for the implementations of the sum function in most software used for numerical computing). Here, we used the Julia package WeightedOnlineStats.jl (https://doi.org/10.5281/zenodo.3360311, repository: https://github.com/gdkrmr/WeightedOnlineStats.jl/, last access: 23 April 2020) (implemented by the first author of this paper), which uses numerically stable algorithms for summation, higher-precision numbers, and a map-reduce scheme that further minimizes floating-point errors.

Based on this weighted and numerically stable covariance matrix, the PCA can be computed using an eigendecomposition of the covariance matrix,

In this case, the covariance matrix Q is equal to the correlation matrix because we standardized the variables to unit variance. Λ is a diagonal matrix with the eigenvalues, ${\mathit{\lambda }}_{\mathrm{1}},\mathrm{\dots },{\mathit{\lambda }}_d$, in the diagonal in decreasing order and $\mathbf{V}\in {\mathbb{R}}^{d\times d}$, the matrix with the corresponding eigenvectors in columns. V can project the new incoming input data x_i (centered and standardized) onto the retained PCs,

where y_i is the projection of the observation x_i onto the d PCs.

The canonical measure of the quality of a PCA is the fraction of explained variance by each component, ${\mathit{\sigma }}_i^{\mathrm{2}}$, calculated as

To get a more complete measure of the accuracy of the PCA, we used the “reconstruction error” in addition to the fraction of explained variance. PCA allows a simple projection of an observation onto the first p PCs and a consecutive reconstruction of the observations from this p-dimensional projection. This is achieved by

where Y_p is the projection onto the first p PCs, V_p the matrix with columns consisting of the eigenvectors belonging to the p largest eigenvalues, and X_p the observations reconstructed from the first p PCs.

The reconstruction error, e_i, was calculated for every point, x_i, in the space–time domain based on the reconstructions from the first p principal components:

As this error is explicit in space, time, and variable, it allows for disentangling the contribution of each of these domains to the total error. This can be achieved by estimating the (weighed) mean square error,

This approach can give a better insight into the compositions of the error than a single global error estimate based on the eigenvalues.

2.3 Pixel-wise analyses of time series

The principal components estimated as described above are ideally low-dimensional representations of the land surface dynamics that require further interpretation. These components have temporal dynamics that need to be understood in detail. One crucial question is how the dynamics of a system of interest deviate from its expected behavior at some point in time. A classical approach is inspecting the “anomalies” of a time series, i.e., the deviation from the mean seasonal cycle at a certain day of year.

Another key description of such system dynamics are trends. We estimated trends of the indicators as well as of their seasonal amplitude using the Theil–Sen estimator. The advantage of the Theil–Sen estimator is its robustness to up to 29.3 % of outliers (Theil, 1950a, b, c; Sen, 1968), while ordinary least-squares regression is highly sensitive to such values. The calculation of the estimator consists simply in computing the median of the slopes spanned by all possible pairs of points,

where z_i is the value of the response variable at time step i and t_i the time at time step i. In our experiments, we computed the slopes separately per pixel and principal component with time as the predictor and the value of the principal component as the response variable.

To test the slopes for significance, we used the Mann–Kendall statistics (Mann, 1945; Kendall, 1970) and adjusted the resulting p values with the Benjamini–Hochberg method to control for the false discovery rate (Benjamini and Hochberg, 1995). Slopes with an adjusted p<0.05 were deemed significant.

To identify disruptions in trajectories, breakpoint detection provides a good framework for analysis. For the estimation of breakpoints, the generalized fluctuation test framework (Kuan and Hornik, 1995) was used to test for the presence of breakpoints. The framework uses recursive residuals (Brown et al., 1975) such that a breakpoint is identified when the mean of the recursive residuals deviates from zero. We used the implementation in Zeileis et al. (2002). For practical reasons, here we only focus on the largest breakpoint.

Figure 1Example polygons and their areas, Eq. (10); the arrows indicate the directionality. (a) Clockwise polygon with a negative area. (b) Counterclockwise polygon with a positive area. (c) Chaotic polygon with a very low area. (d) Polygon with a single intersection and both a clockwise and counterclockwise portion. The clockwise portion is slightly larger than the counterclockwise portion; therefore the area is slightly negative.

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The analysis of a different type of dynamic considers bivariate relations. In the context of oscillating signals it is particularly instructive to quantify their degree of phase shift and direction – even if both signals are not linearily related. A “hysteresis” would be such a pattern describing how the pathways A→B and B→A between states A and B differ (Beisner et al., 2003). We estimated hysteresis by calculating the area inside the polygon formed by the mean seasonal cycle of the combinations of two components.

where n=46, the number of time steps in a year, and x_i and y_i are the mean seasonal cycle of two PCs at time step i. The polygon is circular; i.e., the indices wrap around the edges of the polygon so that x₀=x_n and $x_{n+\mathrm{1}}=x_{\mathrm{1}}$. This formula gives the actual area inside the polygon only if it is non-self-intersecting and the vertices run counterclockwise. If the vertices run clockwise, the area is negative. If the polygon is shaped like an 8, the clockwise and counterclockwise parts will cancel each other (partially) out. Trajectories that have larger amplitudes will also tend to have larger areas as illustrated in Fig. 1.

3.1 Quality of the PCA

Figure 2a shows the explained fraction of variance (Eq. 5) for the global PCA based on the entire data cube. The two leading components explain 73 % of the variance from the 12 variables; additional components contribute relatively little additional variance (PC₃ contributes 9 % and all subsequent PCs less than 7 %) each. This results in a “knee” at component 3, which suggests that two indicators are sufficient to capture the major global dynamics of the terrestrial land surface, but we will also consider the third components in the following analyses (Cattell, 1966).

Figure 3Reconstruction error of the data cube using varying numbers of principal components aggregated by the mean squared error. Reconstruction errors aggregated over all time steps and variables are shown in the left column: (a) using only the first component, (c) using the first two, (e) and using the first three. Corresponding right plots (b, d, f) show the mean reconstruction error aggregated by latitude.

We estimated the reconstruction error sequentially up to the first three principal components (Fig. 3). Regions that do not fit the model well show a higher reconstruction error. Considering one component only, the highest reconstruction errors appear in high latitudes but decrease strongly with each additional component and nearly vanish if the third component is included.

3.2 Interpretation of the PCA

The first PC summarizes variables that are closely related to primary productivity (GPP, LE, NEE, fAPAR) and therefore are highly interrelated (see Fig. 2b). The energy for photosynthesis comes from solar radiation, and fAPAR is an indicator for the fraction of light used for photosynthesis. The available photosynthetic radiation is used by photosynthesis to fix CO₂ and to produce sugars that maintain the metabolism of the plant. The total uptake of CO₂ is reflected in GPP, which is also closely related to water consumption. The flow of water within the plant is not only essential to enable photosynthesis but also drives the transport of nutrients from the roots. The uplift of water in the plant is ultimately driven by transpiration – together with evaporation from soil surfaces one can observe the integrated latent energy needed for the phase transition (LE). However, ecosystems also respire; CO₂ is produced by plants in energy-consuming processes as well as by the decomposition of dead organic materials via soil microbes and other heterotrophic organisms. This total respiration can be observed as terrestrial ecosystem respiration (TER). The difference between GPP and TER is the net ecosystem exchange (NEE) rate of CO₂ between ecosystems and the atmosphere (Chapin et al., 2006). GPP and TER are also well represented in the first dimension.

The second component represents variables related to the surface hydrology of ecosystems (see Fig. 2b). Surface moisture, evaporative stress, root-zone soil moisture, and sensible heat are all essential indicators for the state of plant-available water. While surface moisture is a rather direct measure, evaporative stress is a modeled quantity summarizing the level of plant stress: a value of zero means that there is no water available for transpiration, while a value of 1 means that transpiration equals the potential transpiration (Martens et al., 2017). Root-zone soil moisture is the moisture content of the root zone in the soil, the moisture directly available for root uptake. If this quantity is below the wilting point, there is no water available for uptake by the plants. Sensible heat is the exchange of energy by a change in temperature; if there is enough water available, then most of the surface heat will be lost due to evaporation (latent heat), and with decreasing water availability more of the surface heat will be lost due to sensible heat, making this an indicator of dryness as well.

We observe that the third component is most strongly related to albedo (Fig. 2b). Albedo describes the overall reflectiveness of a surface. Here we refer to broadband (400–3000 nm) surface albedo; for an exact definition see Appendix A. Light surfaces, such as snow and sand, reflect most of the incoming radiation, while surfaces that have a high liquid water content or active vegetation absorb most of the incoming radiation. Local changes to albedo can be due to many causes, e.g., snowfall, vegetation greening and browning, or land use change.

The relation of PC₃ to productivity and hydrology is opposite to what we would expect from an albedo axis. Because vegetation uses radiation as an energy source, albedo is negatively correlated with the productivity of vegetation, hence the negative correlation of albedo with PC₁. Given that water also absorbs radiation, we can observe a negative correlation of albedo with PC₂ (see Fig. 2b). We observe that PC₁ and PC₂ are positively correlated with PC₃ on the positive portion of their axes (see Fig. 4d and f), which means counterintuitively that the index representing albedo is positively correlated with primary productivity and moisture content. Finally we can observe that PC₁ and PC₂ have a much higher reconstruction error in snow-covered regions, which is strongly improved by adding PC₃ (see Fig. 3f). Therefore the third component should be regarded mostly as a binary variable that introduces snow cover, as the other information that is usually associated with albedo is already contained in the first two components.

Figure 4Trajectories of some points (colored lines) and the area-weighted density over principal components one and two (the gray background shading shows the density) for (a, c, e) the raw trajectories and (b, d, f) the mean seasonal cycle. The trajectories are shown in the space of PC₁–PC₂ (first row), PC₁–PC₃ (second row), and PC₂–PC₃ (third row). The trajectories were chosen to cover a large area in the space of the first two principal components. Some of the trajectories have an arrow indicating the direction. The numbers illustrate the value of some variables; for units see Table 1. Description of the points is as follows. Red: tropical rain forest, 2.625^∘ S, 67.625^∘ W; blue: maritime climate, 52.375^∘ N, 7.375^∘ E; green: monsoon climate, 22.375^∘ N, 82.375^∘ E; purple: subtropical, 34.875^∘ N, 117.625^∘ W; orange: continental climate, 52.375^∘ N, 44.875^∘ E; yellow: arctic climate, 72.375^∘ N, 119.875^∘ E.

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3.3 Distribution of points in PCA space

The bivariate distribution of the first two principal components forms a “triangle” (gray background in Fig. 4a). At the high end of PC₁ we find one point of the triangle in which ecosystems have a high primary productivity (high values of GPP, fAPAR, LE, TER, and evaporation), mostly limited by radiation. On the lower end of the first principal component we find the other two points of the triangle describing two alternative states of low productivity. These can happen either when the second principal component coincides with temperature limitation (the negative extreme of the second principal component) as seen in the lower left corner of the distribution in Fig. 4a and b or due to water limitation (positive extreme of the second principal component, the upper left corner in Fig. 4a). This pattern reflects the two essential global limitations of GPP in terrestrial ecosystems (Anav et al., 2015).

Both components form a subspace in which most of the variability of ecosystems takes place. Component one describes productivity and component two the limiting factors to productivity. Therefore, we can see that most ecosystems with high values on component one (a high productivity) are at the approximate center of component two. When ecosystems are found outside the center of component two, they have lower values on component one (lower productivity) because they are limited by water or temperature (see Fig. 4b).

Figure 5The background shading shows the distribution of the mean seasonal cycle of the spatial points (see Fig. 4). The contour lines represent the reconstruction of the variables from the first two principal components. The reconstructed variables are (a) latent heat (LE), (b) sensible heat (H), and (c) ${\log }_{\mathrm{10}}\left(\frac{\mathrm{Sensible}\mathrm{Heat}}{\mathrm{Latent}\mathrm{Heat}}\right)$, the log ₁₀ of the Bowen ratio. Note that the LE and H have been considered in the construction of the PCs and hence are a linear function of the PCs. The Bowen ratio, instead, was not considered here and clearly responds in a nonlinear form.

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To further interpret the triangle we analyze how the Bowen ratio embeds in the space of the first two dimensions. Energy fluxes from the surface into the atmosphere can represent either a radiative transfer (sensible heat) or evaporation (latent heat). Their ratio is the “Bowen ratio”, $B=\frac{\mathrm{H}}{\mathrm{LE}}$, (Bowen, 1926; see also Fig. 5). When water is available most of the available energy will be dissipated by evaporation, B<1, resulting in a high latent heat flux. Otherwise, the transfer by latent heat will be low and most of the incoming energy has to be dissipated via sensible heat, B>1. In higher latitudes, there is relatively limited incoming radiation and temperatures are low; therefore there is not much energy to be dissipated and both heat fluxes are low. A high sensible heat flux is an indicator of water limitation.

3.4 Seasonal dynamics

The leading principal components represent most of the variability of the space spanned by the observed variables, summarizing the state of a spatiotemporal pixel efficiently. This means that the PCs track the state of a local ecosystem over time (Fig. 4a) or, in the case of the mean seasonal cycle, time of the year (Fig. 4b). For a representation of the state of the first three components in time and space, see Appendix Fig. B1.

A first inspection reveals a substantial overlap of seasonal cycles of very different regions of the world. We also see that very different ecosystems may reach very similar states in the course of the season, even though their seasonal dynamics are very different. For instance, a midlatitude pixel (blue trajectory in Fig. 4) shows very similar characteristics to tropical forests during peak growing season. This indicates that an ecosystem of the midlatitudes can reach similar levels of productivity and water availability as a tropical rain forest (see also Appendix Fig. C1). Likewise, for the first two components, many high-latitude areas show similar characteristics to midlatitude areas during winter (low latent and sensible energy release as well as low GPP), and many dry areas such as deserts show similar characteristics to areas with a pronounced dry season, e.g. the Mediterranean.

Depending on their position on Earth, ecosystem states can shift from limitation to growth during the year (Fig. 4b, e.g. Forkel et al., 2015). For example, the orange trajectory in Fig. 4, an area close to Moscow, shifts from a temperature-limited state in winter to a state of very high productivity during summer. Other ecosystems remain in a single limitation state with only slight shifts, such as the red trajectory in Fig. 4. In the corner of maximum productivity of the distribution, we find tropical forests characterized by a very low seasonality. We also observe that very different ecosystems can have very similar characteristics during their peak growing season; e.g. green (located in northeast India), blue (northwest Germany), and orange (located close to Moscow) trajectories have very similar characteristics during peak growing season compared to the red trajectory.

The third component shows a different picture. Due to a consistent winter snow cover in higher latitudes, the albedo is much higher and the amplitude of the mean seasonal cycle is much larger than in other ecosystems. Other areas show comparatively little variance on the third component and their relation to productivity and moisture content is even positively correlated to the third component, which is the opposite of what is expected from an albedo axis.

Figure 6Mean seasonal cycle of the first three principal components (in columns) during the seasons (in rows). Left column: first principal component. Middle column: second principal component. Right column: third principal component. Rows from top to bottom: equally spaced intervals during the year. Values have been clamped to 0.7 times their range to increase contrast.

The global pattern of the first principal component follows the productivity cycles during summer and winter (Fig. 6, left column) of the Northern Hemisphere, with positive values (high productivity, green) during summer and negative values (low productivity, brown) during winter. The tropics show high productivity all year. The global pattern shows the well-known green wave (Schwartz, 1994, 1998) because the first dimension integrates over all variables that correlate with plant productivity.

The second principal component (Fig. 6, middle column) tracks water deficiency: red and light red areas indicate water deficiency, light blue areas excess water, and dark blue areas water growth limitation due to cold. Areas which are temperature limited during winter but have a growing season during summer, such as boreal forests, change from dark blue in winter to light blue during the growing season. Areas which have low productivity during a dry season change their coloring from red to light red during the growing season, e.g the northwest of Mexico and southwest of the United States.

The third principal component (Fig. 6, right column) tracks surface reflectance. Therefore we can see the highest values in the arctic region during winter, and other areas vary much less in their reflectance throughout the year. Again, the third component shows a counterintuitive behavior in the midlatitudes, as it is positively correlated with productivity and therefore shows the opposite behavior of what would be expected from an indicator tracking albedo.

Although the principal components are globally uncorrelated, they covary locally (see Fig. D1). Ecosystems with a dry season have a negative covariance between PC₁ and PC₂, while ecosystems that cease productivity in winter have a positive covariance. Cold arid steppes and boreal climates show a negative covariance between PC₁ and PC₃. While other ecosystems that have a strong seasonal cycle show a positive correlation, many tropical ecosystems do not show a large covariance. A very similar picture is painted between the covariance of PC₂ and PC₃: boreal and steppe ecosystems show a negative covariance, while most other ecosystems show a more or less pronounced positive covariance, again depending on the strength of the seasonality.

Observing the mean seasonal cycle of the principal components gives us a tool to characterize ecosystems and may also serve as a basis for further analysis, such as a global comparison of ecosystems (Metzger et al., 2013; Mahecha et al., 2017).

Figure 7The area inside the mean seasonal cycles of (a) PC₁–PC₂, (b) PC₁–PC₃, and (c) PC₂–PC₃. The area is positive if the direction is counterclockwise and negative if the direction is clockwise. Most of the trajectories need a strong seasonal cycle to show a pronounced hysteresis effect. If the mean seasonal cycle intersects, the areas cancel each other out, e.g. the green trajectory of Fig. 4b.

3.5 Hysteresis

The alternative return path between ecosystem states forming the hysteresis loops arises from the ecosystem tracking seasonal changes in the environmental condition, e.g. summer–winter or dry–rainy seasons (Fig. 4b). Hysteresis is a common occurrence in ecological systems (Folke et al., 2004; Blonder et al., 2017; Renner et al., 2019). For instance, a hysteresis loop can be found when plotting soil respiration against soil temperature (Tang et al., 2005). The sensitivity of soil respiration to soil temperature changes seasonally due to changing soil moisture and photosynthesis (by supplying carbon to the rhizosphere), producing a seasonally changing hysteresis effect (Gaumont-Guay et al., 2006; Richardson et al., 2006; Zhang et al., 2018). Biological variables also show a hysteresis effect in their relations with atmospheric variables; e.g. Mahecha et al. (2007b) found a hysteresis effect between seasonal NEE, temperature, and a number of other ecosystem and climate-related variables. Here we look at the mean seasonal cycles of pairs of indicators and the area they enclose.

The orange trajectory (area close to Moscow) in Fig. 4b shows that the paths between maximum and minimum productivity can be very different, in contrast to the blue trajectory located in the northwest of Germany which also has a very pronounced yearly cycle but shows no such effect. Figure 4 also indicates that the area inside the mean seasonal cycles of PC₁–PC₂ and PC₁–PC₃ shows important characteristics while hysteresis in PC₂–PC₃ is a much less pronounced feature; i.e., we can only see a pronounced area inside the yellow curve in Fig. 4f.

The trajectories that show a more pronounced counterclockwise hysteresis effect in PC₁–PC₂ (Fig. 7a) are areas with a warm and temperate climate and partially those that have a snow climate with warm summers, i.e., areas that have pronounced growing, dry, and wet seasons and therefore shift their limitations more strongly during the year. That means the moisture reserves are depleted during growing season, and therefore the return path has higher values on the second principal component (the climatic zones are taken from the Köppen–Geiger classification; Kottek et al., 2006). We can also see that areas with dry winters tend to have a clockwise hysteresis effect, e.g. many areas in East Asia. Due to the humid summers there is no increasing water limitation during the summer months which causes a decrease for PC₂ instead of an increase. Other areas with clockwise hysteresis can be found in winter dry areas in the Andes and the winter dry areas north and south of the African rain forests. Tropical rain forests do not show any hysteresis effect due to their low seasonality. In general we can say that the area inside the mean seasonal cycle trajectory of PC₁–PC₂ depends mostly on water availability in the growing and non-growing seasons, i.e., the contrast of wet summer and dry winter vs. dry summer and wet winter.

The hysteresis effect on PC₁–PC₃ (Fig. 7b) shows a pronounced counterclockwise MSC trajectory mostly in warm temperate climates with dry summers, while it shows a clockwise MSC trajectory in most other areas; again tropical rain forests are an exception due to their low seasonality. The most pronounced clockwise MSC trajectories can be found in tundra climates in arctic latitudes, where we have a consistent winter snow cover and a very short growing period. A counterclockwise rotation can be found in summer dry areas, such as the Mediterranean and California, but also some more humid areas, such as the southeast United States and the southeast coast of Australia. In these areas we can find a decrease for PC₃ during the non-growing phase which probably corresponds to a drying out of the vegetation and soils.

The hysteresis effect on PC₂–PC₃ (Fig. 7c) mostly depends on latitude. There is a large counterclockwise effect in the very northern parts, due to the large amplitude of PC₃. The amplitude gets smaller further south until the rotation reverses in winter dry areas at the northern and southern extremes of the tropics and disappears at the equatorial humid rain forests.

We can see that the hysteresis of pairs of indicators represents large-scale properties of climatic zones. The enclosed area and the direction of the rotation provide interesting information. Hysteresis can provide information on the seasonal availability of water, seasonal dry periods, or snowfall. With the method presented here, we can not observe intersecting trajectories, which would probably provide even more interesting insights (e.g. the green trajectory in Fig. 4b).

Figure 8Anomalies of the first three principal components. The brown–green contrast shows the anomalies on PC₁, a relative low productivity or greening, respectively. The blue–red contrast shows the anomalies on PC₂, a relative wetness or dryness, respectively. The brown–purple contrast shows the anomaly on PC₃, a relative deviation in albedo. Panels (a), (e), and (i) are maps showing the anomalies of PC₁–PC₃, respectively, on 1 January 2001. Panels (b), (c), and (d) show longitudinal cuts of PC₁–PC₃, respectively, at the red vertical line in (a). The effects of the floods on the Horn of Africa (2006) and the Russian heat wave (2010) are highlighted by circles. Panels (f), (g), and (h) show longitudinal cuts of PC₁–PC₃, respectively, at the red vertical line in (e). Strong droughts in the Amazon during 2005 and 2010 can be observed as large red spots on the fringes of the Amazon basin (highlighted by circles). Panels (j), (k), and (l) show longitudinal cuts of PC₁–PC₃, respectively, at the red vertical line in (i). A strong snowfall event affecting central and southern China is marked as circles.

3.6 Anomalies of the trajectories

The deviation of the trajectories from their mean seasonal cycle should reveal anomalies and extreme events. These anomalies have a directional component which makes them interpretable the same way the original PCs are. Therefore one can infer the state of the ecosystem during an anomaly. For instance the well-known Russian heat wave in summer 2010 (Flach et al., 2018) appears in Fig. 8 as a dark brown spot in the southern part of the affected area, indicating lower productivity, and as a thin green line in the northern parts, indicating increased productivity. This confirms earlier reports in which only the southern agricultural ecosystems were negatively affected by the heat wave, while the northern predominantly forest ecosystems rather benefited from the heat wave in terms of primary productivity (Flach et al., 2018).

Another example of an extreme event that we find in the PCs is the very wet November rainy season of 2006 in the Horn of Africa after a very dry rainy season in the previous year. This event was reported to bring heavy rainfall and flooding events which caused an emergency for the local population but also increased ecosystem productivity (Nicholson, 2014). The rainfall event appears as green and blue spots in Fig. 8b and c, preceded by the drought events which appear as red and brown spots.

Figure 8f and g also show the strong drought events in the Amazon, particularly the droughts of 2005 and 2010 (Doughty et al., 2015; Feldpausch et al., 2016) appear strongly north and south of the Amazon basin. The central Amazon basin does not show these strong events, because the observable response of the ecosystem was buffered due to the large water storage capacity in the central Amazon basin.

Another extreme event that can be seen is the extreme snow and cold event affecting central and south China in January 2008, causing the temporary displacement of 1.7 million people and economic losses of approximately USD 21 billion (Hao et al., 2011). This event shows up clearly on PC₂ and PC₃ as cold and light anomalies, respectively (see Fig. 8k and f).

Figure 9Trajectories of the first two principal components for single pixels. (a) Deforestation increases the seasonal amplitude of the first two PCs (Brazilian rain forest, 9.5^∘ S, 63.5^∘ W). The red line shows the trajectory before 2003 and the blue line the trajectory 2003 and later. A strong increase in seasonal amplitude can be observed after 2003. (b) The heat wave is clearly visible in the trajectory (red, Russian heat wave, summer 2010, 56^∘ N, 45.5^∘ E). (c) Rainfall in the short rainy season (November–December) influences agricultural yield and can cause flooding (extreme flooding after drought, November 2006, 3^∘ N, 45.5^∘ E). (d) The European heat wave in summer 2003 was one of the strongest on record (France, 47.2^∘ N, 3.8^∘ E). The mean seasonal cycle of the trajectories is shown in purple.

3.7 Single trajectories

Observing single trajectories can give insight into past events that happened at a certain place, such as extreme events or permanent changes in ecosystems. The creation of trajectories is an old method used by ecologists, mostly on species assembly data of local communities, to observe how the composition changes over time (e.g. Legendre et al., 1984; Ardisson et al., 1990). In this context, we observe how the states of the ecosystems inside the grid cell shift over time, which comprises a much larger area than a local community but is probably also less sensitive to very localized impacts than a community-level analysis. One of the main differences of the method applied here from the classical ecological indicators is that the trajectories observed here are embedded into the space spanned by a single global PCA, and therefore we can compare a much broader range of ecosystems directly.

Figure 10(a, c, e) Trends in PC₁–PC₃, respectively (2001–2011). (b, d, f) Bivariate distribution of trends. Trends were calculated using the Theil–Sen estimator. Panels (a), (c), and (e) show significant trends only (p<0.05, Benjamini–Hochberg adjusted).

The seasonal amplitude of the trajectory in the Brazilian Amazon increases due to deforestation and crop growth cycles. Figure 9a shows an area in the Brazilian Amazon in Rondônia (9.5^∘ S, 63.5^∘ W) which was affected by large-scale land use change and deforestation. It can be seen that the seasonal amplitude increases strongly after the beginning of 2003. This increased amplitude could be due to any of the following reasons or a combination of them: deforestation decreases water storage capability and dries out soils, causing larger variability in ecosystem productivity. Therefore, during periods of no rain, large-scale deforestation can cause a shift in local-scale circulation patterns, causing lower local precipitation (Khanna et al., 2017). Crop growth and harvest cause an increased amplitude in the cycle of productivity. An analysis of the trajectory can point to the nature of the change; however finding the exact causes for the change requires a deeper analysis.

The 2010 Russian heat wave has a very clear signal in the trajectories. Figure 9b shows the deviation of the trajectory during the Russian heat wave (red line) in an area east of Moscow (56^∘ N, 45.5^∘ E). In the southern grass- and croplands, the heat wave caused the productivity to drop significantly during summer due to a depletion of soil moisture. In the northern forested parts affected, the heat wave caused an increase in ecosystem productivity during spring due to higher temperatures combined with sufficient water availability. This shows the compound nature of this extreme event (see Fig. 8a and Flach et al., 2018). The analysis of the trajectory points directly towards the different types of extremes and responses that happened in the biosphere during the heat wave.

Variability of rainfall during the November rainy season in the Horn of Africa (3^∘ N, 45.5^∘ E, Fig. 9c) shows the trajectory and points in November of the observed time. The November rain has implications for food security because the second crop season depends on it. In 2006, the rainfall events were unusually strong and caused widespread flooding and disaster but also higher ecosystem productivity (see also Fig. 8). This was especially devastating because it followed a long drought that caused crop failures. Note also the two rainy seasons in the mean seasonal cycle (purple line in Fig. 9c).

The 2003 European heat wave is reflected in the trajectories just like the 2010 Russian heat wave. Figure 9d shows the trajectory during the August 2003 heat wave in Europe (France, 47.2^∘ N, 3.8^∘ E). The heat wave was unprecedented and caused large-scale environmental, health, and economic losses (Ciais et al., 2005; García-Herrera et al., 2010; Miralles et al., 2014). The 2010 heat wave was stronger than the 2003 heat wave but the strongest parts of the 2010 heat wave were in eastern Europe (see Fig. 8), while the center of the 2003 heat wave was located in France.

As we have seen here, observing single trajectories in reduced space can give us important insights into ecosystem states and changes that occur. While the trajectories can point us towards abnormal events, they can only be the starting points for deeper analysis to understand the details of such state changes.

3.8 Trends in trajectories

The accumulation of CO₂ in the atmosphere should cause an increase in global productivity of plants due to CO₂ fertilization, while larger and more frequent droughts and other extremes may counteract this trend. Satellite observations and models have shown that during the last decades the world's ecosystems have greened up during growing seasons. This is explained by CO₂ fertilization, nitrogen deposition, climate change, and land cover change (Zhu et al., 2016; Huang et al., 2018; Anav et al., 2015). Tropical forests especially showed strong greening trends during the growing season.

General patterns of trends that can be observed are a positive trend (higher productivity) on the first principal component in many arctic regions. Many of these regions also show a wetness trend, with the notable exception of the western parts of Alaska, which have become drier. This is important, because wildfires play a major role in these ecosystems (Jolly et al., 2015; Foster et al., 2019). These changes are also accompanied by a decrease for PC₃ due to a loss in snow cover. A large-scale dryness trend can also be observed across large parts of western Russia. Increasing productivity can also be observed for large parts of the Indian subcontinent and eastern Australia. Negative trends in the first component can also be observed: they are generally smaller and appear in regions around the Amazon and the Congo Basin, but also in parts of western Australia. The main difference from previous analyses on the observations presented here is that Zhu et al. (2016), for example, looked only at trends during the growing season, while this analysis uses the entire time series to calculate the slope.

In the Amazon basin, we find a dryness trend accompanied by a decrease in productivity and a slight increase in PC₃. In the Congo Basin, we find a wetness trend and an increasing productivity in the northern parts, while the southern part and woodland south of the Congo Basin show a strong dryness trend with decreased productivity. This is different to the findings of Zhou et al. (2014), who found a widespread browning of vegetation in the entire Congo Basin for the April–May–June seasons during the period 2000–2012. The findings of Zhou et al. (2014) are not reflected in our data, especially compared to the areas surrounding the Congo Basin. We can find only minor browning effects inside the basin, and our findings are more in line with the global greening (Zhu et al., 2016), which shows a browning mostly outside the Congo Basin.

In eastern Australia we find a strong wetness and greenness trend which is due to Australia having a “millennium drought” since the mid-1990s with a peak in 2002 (Nicholls, 2004; Horridge et al., 2005) and extreme floods in 2010–2011 (Hendon et al., 2014).

Large parts of the Indian subcontinent show a trend towards higher productivity and an overall wetter climate. The greening trend in India happens mostly over irrigated cropland. However browning trends over natural vegetation have been observed but do not emerge in our analysis (Sarmah et al., 2018). A very notable greening and wetness trend can be observed in Myanmar due to an increase in intense rainfall events and storms, although the central part experienced some strong droughts at the same time (Rao et al., 2013). In Myanmar we also find one of the strongest trends in PC₃ outside of the Arctic.

In large parts of the Arctic, a trend towards higher productivity can be observed. Vegetation models attribute this general increase in productivity to CO₂ fertilization and climate change. The changes also cause changes to the characteristics of the seasonal cycles (Forkel et al., 2016). Stine et al. (2009) found a decreased seasonal amplitude of surface temperature over northern latitudes due to winter warming.

The seasonal amplitude of atmospheric CO₂ concentrations has been increasing due to climate change, causing longer growing seasons and changing vegetation cover in northern ecosystems (Forkel et al., 2016; Graven et al., 2013; Keeling et al., 1996). Therefore we checked for trends in the seasonal amplitude, but because each time series only consists of 11 values (one amplitude per year), after adjusting the p values for false discovery rate, we could not find a significant slope. However, there were many significant slopes with the unadjusted p values; see the appendix, Fig. E1.

Another way to detect changes to the biosphere consists in the detection of breakpoints, which has been applied successfully to detect changes in global normalized difference vegetation index (NDVI) time series (de Jong et al., 2011; Forkel et al., 2013) or generally to detect changes in time series (Verbesselt et al., 2010). A proof-of-concept analysis can be found in Fig. F1. We hope that applying this method to indicators instead of variables can detect a wider range of breakpoints analyzing a single time series.

3.9 Relations to other PCA-type analyses

One of the most popular applications of PCA in meteorology are EOFs, which typically apply PCA to a single variable, i.e., on a dataset with the dimensions $\mathrm{lat}\times \mathrm{long}\times \mathrm{time}$, although EOFs can be calculated from multiple variables. EOFs can be calculated in S mode and R mode. If we matricize our data cube so that we have time in rows and $\mathrm{lat}\times \mathrm{long}\times \mathrm{variables}$ in columns, then S mode PCA works on the correlation matrix of the combined variable and space dimension. In T mode, the PCA works on the correlation matrix formed by the time dimension (Wilks, 2011). The PCA presented here works slightly differently. (1) We performed a different matricization ($\mathrm{lat}\times \mathrm{long}\times \mathrm{time}$ in rows and variables in columns) and then (2) the PCA works on the correlation matrix formed by the variables. Therefore in this framework we could call this a V mode PCA.

Ecological analyses usually use PCA with matrices of the shape object×descriptors. When calculating the PCA on the correlation matrix formed by the objects, then it is called a Q mode analysis. When the PCA is applied to the correlation matrix formed by the variables, then it is called an R mode analysis (Legendre and Legendre, 1998). The PCA carried out in this study is closest to an R mode analysis. In the present case the descriptors are the various data streams and the objects are the spatiotemporal pixels.

Using PCA as a method for dimensionality reduction means that we are assuming linear relations among features. A nonlinear method could possibly be more efficient in reducing the number of variables but would also have significant disadvantages. In particular, nonlinear methods typically require tuning specific parameters, objective criteria are often lacking, a proper weighting of observations is difficult, the methods are often not reversible, and it is harder to interpret the resulting indicators due to their nonlinear nature (Kraemer et al., 2018). The salient feature of PCA is that an inverse projection is well defined and allows for a deeper inspection of the errors, which is not the case for nonlinear methods which learn a highly flexible transformation that is hard to invert. Therefore interpretability of the transform in meaningful physical units in the input space is often not possible. In the machine-learning community, this problem is known as the “pre-imaging problem” (Mika et al., 1999; Arenas-Garcia et al., 2013) and is a matter of current research.