# New Insights on Land Surface-Atmosphere Feedbacks over Tropical South America at Interannual Timescales

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## Abstract

**:**

_{925}), volumetric soil water content (Θ), precipitation (P), and evaporation (E), at monthly scale during 1979–2010. Applying a Maximum Covariance Analysis (MCA), we identify the modes of greatest interannual covariability in the datasets. Time series extracted from the MCAs were used to quantify linear and non-linear metrics at up to six-month lags to establish connections among variables. All sets of metrics were summarized as graphs (Graph Theory) grouped according to their highest ENSO-degree association. The core of ENSO-activated interactions is located in the Amazon River basin and in the Magdalena-Cauca River basin in Colombia. Within the identified multivariate structure, Θ enhances the interannual connectivity since it often exhibits two-way feedbacks with the whole set of variables. That is, Θ is a key variable in defining the spatiotemporal patterns of P and E at interannual time-scales. For both the simultaneous and lagged analysis, T activates non-linear associations with q

_{925}and Θ. Under the ENSO influence, T is a key variable to diagnose the dynamics of interannual feedbacks of the lower troposphere and soil interfaces over tropical South America. ENSO increases the interannual connectivity and memory of the feedback mechanisms.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data

_{925}), and soil moisture (Θ) (as state variables), precipitation (P) and evaporation (E) (as process variables) [7].

_{925}, and E, the latter represented by the Instantaneous Moisture Flux. Volumetric Soil Water content (Θ) is obtained from the ERA-Interim/Land Reanalysis V.2 [60,61]. This latter product is focused on those land processes not well represented in the ERA-Interim Reanalysis and is just available for the period 1979–2010 [62], which justifies this time span for our study. The monthly climatic fields were originally downloaded at 1° × 1° spatial resolution.

#### 2.2. Maximum Covariance Analysis

_{XY}, as:

_{XY}was factorized employing a singular value decomposition (SVD) to extract those patterns explaining the maximum covariance fraction between X and Y at the interannual timescale. This method is also known as Maximum Covariance Analysis (MCA) [54,64,65]. Through the MCA, C

_{XY}can be decomposed as:

_{1}, indicates the amount of covariance between both variables, which is captured by the leading mode. The second singular factor, ${u}_{2}{\sigma}_{2}{v}_{2}^{\mathrm{T}}$, is orthogonal to the first one, and maximizes the remaining portion of the covariance, and so on until explaining all the covariance contained in C

_{XY}. A measure of the degree to which both variables maximize their temporal covariance within each mode, is provided through the estimation of the MCA-series x

_{k}and y

_{k}, by projecting the matrices u

_{k}and v

_{k}(containing the singular vectors of C

_{XY}) on the original matrices X and Y (Equation (3)), as:

_{k}and y

_{k}and the matrices of standardized data X and Y.

#### 2.3. A Graph Model and Feedbacks at Interannual Timescales

_{k}and y

_{k}as a schematic view in two-time steps: x(t − 1) influences y(t) and y(t − 1) influences x(t). The red arrow in Figure 3 connects the series when x leads y, and the blue arrow when y leads x. Hence, in both cases, we systematically estimate correlations, $\rho \left(x\left(t\right),y\left(t-1\right)\right)$ and $\rho \left(x\left(t-1\right),y\left(t\right)\right)$, and causalities, ${\tau}_{\left(x\left(t\right),y\left(t-1\right)\right)}$ and ${\tau}_{\left(x\left(t-1\right),y\left(t\right)\right)}$.

_{1}value (emitter) transfers the highest amount of information to the whole set of variables, while the maximum vector v

_{1}value is the best information receiver.

## 3. Results

#### 3.1. Maximum Covariance Analysis among Variables and Strength of ENSO Influence

_{925}interaction presents the highest fraction in the leading mode corresponding to 73% of its total covariance. This is because both variables are strongly linked by the Clausius-Clapeyron formulation between temperature and saturation water vapor pressure, especially in the humid tropics [1].

_{925}), Θ contributes to determining the spatiotemporal pattern of P [70]. Regularly, those pairs involving T exhibit the highest correlation values with the Niño 3.4 index (Table 2); this means that surface temperature is essential to propagate the ENSO-effect on studied variables.

_{XY}including P (P-q

_{925}, P-T, E-P, and P-Θ) consistently shows the highest correlations with N3.4 in the second MCA-mode (Table 2). The P-q

_{925}pair shows a clear difference of correlation with N3.4 among the three modes; in that case, the second mode of P-q

_{925}is best correlated with N3.4.

#### 3.2. Non-Linear Analysis of the MCA-Series and Feedbacks

_{925}-T shows the largest relative causalities from the set of estimated values. Specifically, T reaches up to −23.3% of causality on q

_{925}. The negative value indicates that T favors the predictability of q

_{925}under the influence of ENSO [70]. After q

_{925}-T, the P-Θ interaction shows the largest causality value. Θ consistently subordinates P (${\mathsf{\tau}}_{y\to x}=$ 21.6%), while the reverse causality is 15.2%, implying a coupling ${\mathsf{\Phi}}_{VSW\u27f7PRC}=1.43$. These non-linear connections suggest that Θ is fundamental in modulating the interannual variability of P over the studied region under the ENSO influence.

_{925}-T, and T-Θ). Additionally, the interaction T-Θ shows a perfect bidirectional feedback (Table 3). T is essential in capturing the non-linear interactions at interannual timescale within the studied variables in the simultaneous analysis.

#### 3.3. Visualizing Linear and Non-Linear Dependences Using Graph Theory

_{1}and v

_{1}estimated from the SVD applied to the graph adjacency matrix. We also show those variables as red horizontal bars in Figure 6 and Figure 7. Red paired-edges in each graph represent correlations and causality ratios (Equation (6)) ranging between 0.9 and 1.1.

#### 3.3.1. Linear Graphs

_{925}) corresponds to the highest entries of leading singular vectors -u

_{1}and v

_{1}- of the graph matrix (Figure 6a). The high degree of response of the atmosphere to anomalies in the land surface state is taking into account all the interdependences among the five selected variables. Our results point out that ENSO favors an important chain of linear feedbacks among T, q

_{925}, and E. This means that land-atmosphere interactions are fundamental driving mechanisms of interannual variability and delayed effects on hydrologic response over the study region [42,45]. Led by the ENSO-forcing, Θ presents the highest correlation value with P among the variables (ρ = 0.69, Figure 6a) implying that the interannual anomalies of soil moisture are strongly related with the subsequent anomalies at one month of precipitation over the region [42]. Surface temperature anomalies are also directly regulated by soil moisture anomalies and through linear feedbacks with E and q

_{925}(Figure 6a). From this same graph, we infer that antecedent anomalies of q

_{925}do not influence directly the subsequent anomalies of Θ but through changes in T; instead, one-month lagged anomalies of Θ linearly controls the anomalies in q

_{925}.

_{1}); but, in this case, evaporation (E) is the best receiver (maximum value of v

_{1}). For two-month lags, T establishes linear feedbacks to connect q

_{925}with Θ. Again, the highest correlation of the entire set shows the predominance of Θ on P (ρ = 0.53). Soil moisture anomalies regulate the interdependence of P anomalies with T, q

_{925,}and E anomalies at interannual timescale.

_{925}). As shown by one- and two-month lags, the structure of the graph represented in a hierarchical form demonstrates that Θ constitutes a heterogeneous node connecting the anomalies of E, q

_{925,}and T with the subsequent anomalies of precipitation (Figure 6c). However, in this case the correlations P-Θ are not the highest among the variables, and thus the structure of the graph is inverted with respect to Figure 6a,b. The linear feedbacks are identified for the pairs E-Θ, Θ-q

_{925}and q

_{925}-T (Figure 6c). Among the studied variables, T and Θ exhibit the highest amount of linear feedbacks, including the correlation among them at one and two month-lag, and with E.

#### 3.3.2. Non-Linear Graphs

_{925,}and T (Figure 7a). The highest causality of this graph of concurrent interactions corresponds to the T → q

_{925}relationship (23.3%). In addition, the structure of this graph is nearly equally dominated by Θ as the best emitter (u

_{1}), while the best receptor is q

_{925}(Figure 7a). This means that Θ dominates the non-linear relationships with P, q

_{925,}and T at interannual timescales and without lagging. It is remarkable that under this structure of causalities, evaporation (E) is the only variable which non-linearly controls Θ. This agrees with the essential role of evaporation in the surface energy partitioning in wetter environments [74], taking into account that under this non-linear framework, precipitation presents a relative causality on soil moisture (Figure 7a). In general, Θ is essential over Tropical South America to structurally feedback on q

_{925}under the highest influence of ENSO at concurrent series (non-linear links).

_{925}(Figure 7a), T exerts a non-linear control on Θ at one- and two-month lags (Figure 7b,c). In this way, non-linear associations reveal that T is essentially strengthening the identified feedbacks under the ENSO effect [6]. For both the simultaneous and lagged analyses, surface temperature (T) exhibits non-linear associations with the atmospheric moisture (q

_{925}) and soil moisture (Θ) causality graphs. Under the ENSO influence, T is not only a key variable in the diagnostics of the dynamics of interannual LAFs but also has a substantial role in the dynamics and thermodynamics of the lower troposphere and soil interfaces over TropSA [2,3,40,44,69,75]. Besides, interannual anomalies in P are associated with perturbations of the surface water balance at the regional scale by controlling T interactions with Θ and E [10,17,30,31,45,49,76,77,78].

_{925}) with a highest value coming from temperature (${\tau}_{\mathrm{T}\to \mathsf{\Theta}}=$ 11.4%, Figure 7b). Another important fact is the causality that precipitation has on temperature (${\tau}_{\mathrm{P}\to \mathrm{T}}=$ 5.45%). This implies an important non-linear connection of the antecedent anomalies of temperature and precipitation on the one-month subsequent anomalies of soil moisture. This also contributes to understanding that the linear (1-month lag, Figure 6a) and non-linear (simultaneous, Figure 7a) structure of relationships are connected over TropSA.

## 4. Discussion

_{925}) in Tropical South America at an interannual time scale.

_{925}) is mediated through non-linear interdependences of the surface temperature (Θ-T and T-q

_{925}) as defined by Equation (6) (bi-directional causalities). At the same time, the anomalies of precipitation are highly dependent on soil moisture anomalies (Figure 7a). Results from the linear analysis show the fundamental role of soil moisture as the best emitter within the multivariate dynamics of LAFs over TropSA (Figure 6). This role is not completely demonstrable in the non-linear graphs where temperature takes more relevance (Figure 7). The greater differences between linear and non-linear analysis correspond to the one-month lag graphs (Figure 6a and Figure 7b). However, results indicate a connection between the structure of the linear one-month lagged and non-linear simultaneous analyses (Figure 6a and Figure 7a). In summary, we show that linear and non-linear structural relationships are complementary to capture the dynamics of LAFs over tropical South America. We defined and estimated metrics of linear and non-linear feedbacks among several variables in a common interannual mode mainly dominated by ENSO.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LAFs | Land-Atmosphere Feedbacks |

ENSO | El Niño-Southern Oscillation |

TropSA | Tropical South America |

SAM | South American Monsoon |

TNA | Tropical North America |

SST | Sea Surface Temperature |

T | Surface Temperature |

q_{925} | Specific Humidity at 925 hPa |

Θ | Volumetric Soil Water |

P | Precipitation |

E | Evaporation |

GPCC | Global Precipitation Centre |

SVD | Singular Value Decomposition |

C_{XY} | Covariance Matrix of Variables X and Y |

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**Figure 1.**Conceptual scheme of interconnections among the main variables involved in the studied land surface-atmosphere feedbacks (LAF), adapted as a graph from [48] and hierarchized by a number of connections of each variable. The state (process) variables are denoted as circles (diamonds) nodes.

**Figure 2.**Study region with major river basins included in the analysis: Amazon, Tocantins, Orinoco, and Magdalena. The Intertropical Convergence Zone (ITCZ), and the South Atlantic Convergence Zone (SACZ) are important climate modulators of the region.

**Figure 3.**The basic scheme of the interaction between two singular vectors resulting from a Maximum Covariance Analysis (MCA) between two variables x and y assuming 1 lag-month.

**Figure 4.**Leading MCA-mode between evaporation (X = E) and soil moisture (Y = Θ). The spatial-pattern shows a core over the Amazon River basin. (

**a**) MCA-series associated with this leading mode; (

**b**,

**c**) Spatial correlations-pattern of the MCA-mode as the influence of each variable in the other one (

**d**) Lagged cross-correlations between MCA-series.

**Figure 5.**Maps of simultaneous correlations between SST anomalies and the time series associated with the leading MCA-mode between E(x) and Θ(y).

**Figure 6.**Correlation graphs constructed with the highest ENSO-related modes from each pair of variables. The weights of the linkages between nodes are estimated as the lagged correlation among MCA-series. Red linkages represent feedbacks between variables. Leading SVD vectors u

_{1}and v

_{1}are estimated from the adjacency matrix associated with the graph and represented with red nodes. Red bars also represent the highest values of these vectors. (

**a**) one-month lag graph; (

**b**) two-month lags graph; (

**c**) three-month lags.

**Figure 7.**Causality Graphs constructed with the highest ENSO-related mode time series from each pairwise of variables. The weights of the edges between the nodes are estimated as the simultaneous causality ($\left|{\tau}_{x\u27f7y}\right|$ and $\left|{\tau}_{y\u27f7x}\right|$) between the MCA-series. Red edges represent feedbacks between variables. Bar panel represents leading singular vectors u

_{1}and v

_{1}estimated from the the adjacency matrix; (

**a**) Simultaneous (No Lags); (

**b**) one-month lag; (

**c**) two-month lags.

**Table 1.**Three leading MCA-modes of all possible covariance matrices C

_{XY}. Fraction of relative (${f}_{k}^{\mathrm{XY}}$), cumulative square covariance (${F}_{k}^{\mathrm{XY}}$), and simultaneous correlation between MCA-series, $\rho \left({x}_{k},{y}_{k}\right)$. Pairs of variables are sorted according with the value of the first fraction.

C_{XY} | Mode 1 | Mode 2 | Mode 3 | ||||
---|---|---|---|---|---|---|---|

X | Y | ${f}_{1}^{\mathrm{XY}}({F}_{1}^{\mathrm{XY}})$ (%) | $\rho \left({x}_{1},{y}_{1}\right)$ | ${f}_{2}^{\mathrm{XY}}({F}_{2}^{\mathrm{XY}})$ (%) | $\rho \left({x}_{2},{y}_{2}\right)$ | ${f}_{3}^{\mathrm{XY}}({F}_{3}^{\mathrm{XY}})$ (%) | $\rho \left({x}_{3},{y}_{3}\right)$ |

q_{925} | T | 73 (73) | 0.96 | 16 (88) | 0.97 | 5 (93) | 0.96 |

E | Θ | 53 (53) | 0.77 | 17 (70) | 0.64 | 11 (81) | 0.64 |

P | q_{925} | 51 (51) | 0.78 | 27 (78) | 0.6 | 9 (87) | 0.64 |

P | T | 50 (50) | 0.72 | 29 (79) | 0.6 | 9 (86) | 0.64 |

T | Θ | 50 (50) | 0.67 | 29 (78) | 0.66 | 7 (85) | 0.66 |

E | T | 49 (49) | 0.58 | 18 (67) | 0.67 | 8 (76) | 0.64 |

q_{925} | Θ | 48 (48) | 0.7 | 30 (78) | 0.57 | 8 (86) | 0.68 |

E | q_{925} | 45 (45) | 0.58 | 23 (68) | 0.65 | 9 (76) | 0.65 |

P | Θ | 39 (39) | 0.78 | 32 (71) | 0.78 | 12 (83) | 0.79 |

E | P | 39 (39) | 0.71 | 32 (71) | 0.73 | 10 (81) | 0.71 |

**Table 2.**Correlation between MCA-series with the Niño 3.4 index,$\rho \left({x}_{k},N3.4\right)$ and $\rho \left({y}_{k},N3.4\right)$ for the three leading MCA-modes. Highest correlation value of each pair is denoted in bold.

C_{XY} | Mode 1 | Mode 2 | Mode 3 | ||||
---|---|---|---|---|---|---|---|

X | Y | $\rho \left({x}_{1},N3.4\right)$ | $\rho \left({y}_{1},N3.4\right)$ | $\rho \left({x}_{2},N3.4\right)$ | $\rho \left({y}_{2},N3.4\right)$ | $\rho \left({x}_{3},N3.4\right)$ | $\rho \left({y}_{3},N3.4\right)$ |

q_{925} | T | −0.44 | −0.50 | 0.15 | 0.14 | 0.19 | −0.19 |

E | Θ | 0.40 | 0.45 | 0.33 | 0.36 | 0.14 | 0.19 |

P | q_{925} | 0.25 | 0.33 | −0.40 | −0.48 | −0.19 | −0.01 |

P | T | −0.33 | −0.47 | −0.33 | −0.48 | −0.15 | −0.01 |

T | Θ | 0.58 | −0.58 | −0.21 | −0.12 | −0.13 | −0.07 |

E | T | −0.50 | −0.54 | −0.09 | −0.12 | −0.16 | −0.22 |

q_{925} | Θ | 0.52 | 0.53 | −0.30 | −0.25 | −0.13 | −0.08 |

E | q_{925} | −0.46 | −0.47 | −0.14 | −0.16 | −0.20 | 0.05 |

P | Θ | 0.30 | 0.37 | 0.35 | 0.4 | −0.22 | −0.21 |

E | P | 0.07 | 0.18 | −0.49 | −0.42 | −0.22 | 0.31 |

**Table 3.**Estimated values of simultaneous (no-lags) relative causality, τ (%), and feedback ${\mathsf{\Phi}}_{y\u27f7x}$ for the highest ENSO-related covariance modes for all pairs of variables.

X | Y | ${\mathit{\tau}}_{\mathit{x}\to \mathit{y}}$ | ${\mathit{\tau}}_{\mathit{y}\to \mathit{x}}$ | ${\mathsf{\Phi}}_{\mathit{y}\mathbf{\leftrightarrow}\mathit{x}}$ |
---|---|---|---|---|

q_{925} | T | 21.9 | −23.3 | −1.06 |

E | Θ | 12.1 | 9.4 | 0.78 |

P | q_{925} | 7.7 | −2 | −0.26 |

P | T | 10.4 | −6 | −0.58 |

T | Θ | 13.4 | 13.4 | 1.00 |

E | T | 14 | 11.8 | 0.84 |

q_{925} | Θ | 11.4 | 14 | 1.22 |

E | q_{925} | 11.9 | 9.6 | 0.81 |

P | Θ | 15.2 | 21.6 | 1.43 |

E | P | −6.1 | 5.8 | −0.95 |

**Table 4.**Threshold of edges magnitude to construct correlation and causality graphs (percentile of 50% of the empirical probability distribution of causalities and correlations per lag).

Metric Type | Lag | ||||||
---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | |

Correlation ($\rho $) | 0.68 | 0.43 | 0.35 | 0.315 | 0.3 | 0.265 | 0.22 |

Relative Causality ($\tau $) | 11.85 | 5.33 | 4.14 | 4.54 | 3.53 | 1.75 | 1.89 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bedoya-Soto, J.M.; Poveda, G.; Sauchyn, D.
New Insights on Land Surface-Atmosphere Feedbacks over Tropical South America at Interannual Timescales. *Water* **2018**, *10*, 1095.
https://doi.org/10.3390/w10081095

**AMA Style**

Bedoya-Soto JM, Poveda G, Sauchyn D.
New Insights on Land Surface-Atmosphere Feedbacks over Tropical South America at Interannual Timescales. *Water*. 2018; 10(8):1095.
https://doi.org/10.3390/w10081095

**Chicago/Turabian Style**

Bedoya-Soto, Juan Mauricio, Germán Poveda, and David Sauchyn.
2018. "New Insights on Land Surface-Atmosphere Feedbacks over Tropical South America at Interannual Timescales" *Water* 10, no. 8: 1095.
https://doi.org/10.3390/w10081095