# Multivariate Conditional Granger Causality Analysis for Lagged Response of Soil Respiration in a Temperate Forest

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}) above temperate regions is largely driven by the net sum of two opposing processes: carbon (C) uptake through Gross Primary Production (GPP) and carbon release from the vegetation and soil through plant and microbial respiration. The magnitude of the annual ecosystem-driven change to the global soil C pool is such that even a small increase in soil respiration (R

_{s}) rates could result in a net reduction in net terrestrial C sink strength and an increase in atmospheric CO

_{2}concentration [1,2,3].

_{2}effluxes. Understanding the underlying variability of R

_{s}responses to disturbance and canopy structural change is confounded by the covariance at multiple frequencies of the biophysical factors hypothesized to be the most important drivers of R

_{s}—soil temperature, soil moisture, radiation, and GPP [25,26].

_{s}dynamics, and particularly its outcome—R

_{s}in a forest of the Great Lakes region. To understand the motivations of using this particular statistical approach, we next frame the problem within the relevant ecological context. We describe a multivariate approach for testing the causal structure of relationships between R

_{s}and its meteorological and ecological drivers, and whether they differ between an undisturbed site and a site following prescribed disturbance in the form of stem girdling. We apply spectral G-causality analysis in a layered multi-variant approach for testing of the significance of causal relationships between one variable and another, given all other co-occurring environmental conditions, thereby choosing which variables, among an available set, are important predictors.

## 3. Methods—Field Experiment

#### 3.1. Site Description

^{2}/ha in the treatment plot and 36.87 m

^{2}/ha in the control with aspen and birch composing 40.0% of the total basal area in the treatment plot and 53.1% of the control plot. On 20 April–3 May 2008 we initiated an experiment in which we stem girdled all aspen and birch trees in a 33 ha treatment plot, totaling approximately 6700 trees. Girdling removes the bark and underlying phloem in a strip around a tree, preventing the translocation of photosynthate from leaves to roots. An unmanipulated control plot with similar site and soil characteristics is located nearby and within the footprint of an eddy-flux tower (the US-UMB Ameriflux tower, [24]). The mean annual temperature is 5.5 °C (1942–2003). The mean annual precipitation is 817 mm (including 294 cm snowfall). The site is snow covered in most years from November to April. The study area encompasses ∼140 ha on a high outwash plain and adjacent gently sloping moraine. Nearly all soils in the study area are well to excessively well drained Haplorthods of the Rubicon, Blue Lake, or Cheboygan series. About 53% of the fine-root mass is located within the upper 20 cm of the soil profile. Forest floor C mass is 5–15 Mg C/ha, and the mineral soil is ∼95% sand and ∼5% silt, with pH 4.5–5.5 in water. Soils are of low fertility, with total N capital to 40 cm depth of 2000 kg/ha, an average in situ net N-mineralization rate of 42 kg N/(ha yr) and < 2% net nitrification [24,28].

#### 3.2. Soil CO_{2} Efflux

_{2}efflux in the control and treatment plots. Air was circulated from the chamber to an infrared gas analyzer (IRGA, LI-6252, Li-Cor, Lincoln, NE, USA) and then returned to the chamber. At the treatment and control sites, eight closed system automated chambers were randomly distributed on the forest floor near the base of each flux tower. Each chamber consisted of an aluminum frame with a pneumatic actuator to control the opening and closing of the lid and a clear plexiglass cylinder (radius 14.75 cm) with a small fan near the top of the chamber to homogenize the chamber air as samples were taken (Figure 1). Within each chamber a manifold returned sampled air to the bottom of the chamber. Air from the chamber headspace was sampled through a polyethylene coated aluminum tube and pumped to the IRGA for analysis. The system was controlled by a data logger (CR23X, Campbell Scientific Inc., Logan, UT, USA) and each chamber was sampled at 10 Hz for seven minutes each hour, so that each of the eight chambers in each array was sampled once every hour. Measurements started at both sites in 18 April 2008.

**Figure 1.**(

**A**) Automated chambers closed when pressure was applied to the pneumatic actuator which created a seal between the lid and the chamber. (

**B**) Samples were collected from the black tube near the top center of the chamber, and analyzed air was returned via the perforated manifold at the base of the chamber. (

**C**) Respiration was determined by analyzing the timeseries of concentration accumulation in the chambers—raw CO

_{2}concentration data (blue line) were graphed for 330 s while CO

_{2}accumulated in the closed automated chambers. Each accumulation curve was fit individually with a non-linear curve (black line) in MATLAB [29] (see methods). Coefficients were computed from the curve and exported to yield R

_{s}.

_{s}was calculated:

_{c}(t) is the concentration of CO

_{2}in ppm at time t (s), C

_{s}is the soil CO

_{2}concentration (ppm), C

_{0}is the CO

_{2}concentration of the air sample (ppm) at time t = 0, and k is the concentration saturation rate (1/s). The initial rate of change of carbon concentration in the chamber immediately after closing the lid can be solved as:

_{2}accumulation is true at the instant of closing the lid it is also characteristic to the rate of emission from the soil immediately before closing the lid, an thus used to calculate the diffusional flux of CO

_{2}from the soil (D

_{c}), which is equivalent to soil respiration, R

_{s}(µmol/m

^{2}/s):

^{2}), V is the chamber volume (0.0144 m

^{3}) and ρ is the molar density of air (41.6 mol/m

^{3}). To solve for C

_{s}and k an estimate of C

_{0}is needed. To obtain C

_{0}a minimizing function was employed to find the lowest 20 s moving average within the first 70 data points. Using C

_{0}, t, and C

_{c}data, a non-linear least squares fit with least absolute residuals returned estimates of C

_{s}and k. Figure 1C shows examples of raw CO

_{2}data and the fitted curve for one hourly 7 min period of data from which coefficients were calculated. C

_{0}, C

_{s}, and k values for each seven minute period for each site, day, hour, and year and chamber combination were calculated.

_{0}value was less than 380 ppm, the current ambient air CO

_{2}concentration, were removed. Estimates where the R

^{2}value for the fitted line was below 0.9 were removed. Negative D

_{c}values and values above 15 µmol/m

^{2}/s were removed as outliers. Fluxes above 15 µmol/m

^{2}/s were determined to be outliers because they did not fall within the range of values reasonable for our site [30]; outliers made up less than 1% of the data. The complete dataset is illustrated in Figure 2

**Figure 2.**Daily soil respiratory carbon loss (R

_{s}) calculated from the 1 hour soil respiration measurements (points) between Control (blue), and Treatment (red) plots in 2008, 2009, and 2010 growing seasons.

#### 3.3. Other Measurements

_{a}) and humidity from an aspirated and shielded temperature/humidity probe (model Hygroclip S3, Rotronic, Bassersdorf, Switzerland); above canopy total incoming and diffuse photosynthetic active radiation (PAR) using a pyranometer (model BF2, Delta-T Services, Cambridge, UK); Net radiation (RAD

_{n}), short and long wave downwelling and upwelling radiation streams using a 4-channel radiometer (model CNR1, Kipp & Zonen, Delft, The Netherlands); precipitation above the canopy using a tipping bucket (model TR-525-M, Texas Electronics, Dallas, TX, USA). Sensors were deployed at 46 m above ground on a tower at the control plot and at 32 m on a tower at the treatment plot. Soil temperature (T

_{s}), was measured at depths of 2, 7.5, 20, 50 and 100 cm below ground at each tower site (Type E Thermocouples, Campbell Scientific, Inc.); Soil water content (SWC) was measured at two points near each tower site and at two depths (7.5, 20 cm) at each point using a soil-moisture sensor (models CS615-L and CS616-L, Campbell Scientific, Inc.). For each site, soil moisture and temperature series were computed taking the spatial average along all depths and location. Above-canopy and soil observations were logged every minute and averaged to half-hour periods.

_{2}between atmosphere and forest (net ecosystem exchange, NEE), the surface sensible (H) and latent (LE) heat fluxes were measured using the flux-covariance approach. Wind velocity and temperature fluctuations were sampled at 10 Hz using three-dimensional sonic anemometers (model CSAT3, Campbell Scientific, Inc.); CO

_{2}and water vapor concentrations were sampled at 10 Hz using closed-path infrared gas analyzers (IRGAs) (model LI-7000, Li-Coer Bioscience). Measurement outliers and measurements marked as faulty by the sensors' quality control variables were removed (despiked). Wind data were processed using a 3-D coordinate rotation assuming half-hourly mean vertical wind velocity is 0 and rotating the horizontal wind components toward the mean wind direction [31]. Lag time between the anemometer and scalar concentrations was calculated using the maximal covariance method [32]. The Schotanus correction [33,34] and conversion to “real” temperature [35] were applied to sonic anemometer measurements of temperature. Wind and concentrations observations were compiled into half-hour block averages of net fluxes following the AmeriFlux recommendations [36]. Water vapor and CO2 concentrations were adjusted using the Webb, Pearman, and Leuning correction [37] in a modified form derived by Detto and Katul [38] as a correction for the 10 Hz time series of the scalar. Observations were divided into seasons according to the carbon flux phenology in the site [39] and only observations during the growing season were used here. Data were filtered based on a seasonal frictional velocity (u*) threshold criterion [40,41], with a prescribed maximum u* threshold of 0.35 m/s [42], typical growing season u* threshold values were between 0.29 and 0.35. We used a bilinear periodic method to fill gaps in temperature, moisture, humidity, and radiation observations and assumed that CO

_{2}fluxes during nighttime were driven entirely by ecosystem respiration (R

_{e}). R

_{e}was calculated using site-specific empirical formulas developed for our site [30], which relate nighttime NEE to soil moisture and temperature [43]. We used this R

_{e}equation to gap-fill NEE during the nighttime. Gaps in GPP were filled using the mean of 100 neural network simulations [44]. Gap-filled GPP was added to R

_{e}to provide gap-filled NEE. Since the FASET-plot tower sampled a footprint that was sometimes larger than the 33 ha of contiguous experimental girdling area, we used a 2-D footprint model [45] that we modified to automatically integrate the flux-source probability over the treatment area and thus provide an index for the treatment footprint probability in each 30 min block average period. We then used the probabilistic flux footprint climatology [46] to scale our conclusions to fluxes originating only from the FASET plot. More details about the flux data analysis in our site are listed in [24,39]. Time series are shown in Figure 3.

**Figure 3.**Time series of the four variables measured at control (AmeriFlux in blue) and treatment (FASET in green) sites.

## 4. Granger Causality

_{1}, …, Z

_{k}) which admits spectral representation and factorization in the form [47]:

**S**and

**U**are spectral matrices respectively, of the complete system and of a system which does not include the variable Y, whose causality is tested, * indicates matrix adjoint.

_{1}, Z

_{2}, …, Z

_{k}is computed as [48]:

**P**are normalization matrices needed to recast the multivariate systems in the canonical form (with uncorrelated errors).

#### 4.1. Estimation of G-Causality

**S**(ω) can be estimated directly from the data using classic routines (multitaper, wavelet transform or analogous spectral methods). The factorization theorem of a spectral matrix [50] ensures that, under fairly general conditions, any spectral matrix, can be decomposed, or factorized, into

**S**(ω) =

**H**(ω)

**Σ**

**H**(ω). Iterative algorithms have been proposed to evaluate

^{*}**H**and

**Σ**from

**S**. Here, we used the algorithm formulated by Wilson [51]. Since

**S**,

**H**and

**Σ**can be inferred from the time series, the method does not require any assumption regarding the autoregressive order of the model representing the data and the estimation of

**G**(ω) is considered non-parametric [19].

#### 4.2. Direct Connectivity Diagram

- (1)
- Only direct causal interactions are depicted. This implies that if the bivariate interaction is significant, but the multivariate is not, an arrow is not drawn. However, if the bivariate G-causality is not significant, no further action is taken.
- (2)
- If both directional interactions are detected among two nodes, only the greater is retained.

## 5. Results

_{s}than the control site for all years (Figure 3).

_{s}over R

_{s}showed a large effect of temperature and a much smaller effect of GPP, especially at low frequencies (Figure 4). At daily frequency, and in minor extent at the corresponding half day sub-harmonic, the effect of GPP on R

_{s}appears to be important, although it is largely reduced when T

_{s}is included in the analysis. There are no apparent differences between the two sites in response to temperature and/or GPP. This hypothesis was tested computing the difference . Confidence intervals obtained upon bootstrapping showed that the differences were not significant (Figure 5).

^{−1}) and short time scales (0.7625 < frequency < 1.7 day

^{−1}). Given the fact that the results in both sites were very similar, we also aggregated the data without differentiating among sites. The connectivity graphs (Figure 5) depict only significant direct interactions, i.e., not mediated by other variables.

**Figure 4.**Bivariate (dashed lines) and conditional (thick lines) G-causalities of GPP and T

_{s}over R

_{s}. Top panels represent control (AmeriFlux) and bottom panels, treatment (FASET). The thick lines represent the desired goal of the analysis because they isolate direct effects, while the dashed lines indicate the apparent connections which are due to a combination of direct and indirect effects.

**Figure 5.**Difference of conditional G-causalities of GPP given T

_{s}over R

_{s}between control (AmeriFlux) and treatment (FASET). Dashed lines are the 95th quantiles obtained boostrapping the 60 days blocks of the pulled dataset and divide them in two random groups.

**Figure 6.**Connectivity diagram based on the significant direct G-causalities using (

**A**) three, (

**B**) four or (

**C**) five multivariate systems. The widths and values on the arrows indicate average G-causalities for long time scale (0.2 < frequency < 0.7625 day

^{−1}) and short time scales (0.7625 < frequency < 1.7 day

^{−1}

_{).}

_{s}and T

_{s}are included. This diagram supports the hypothesis of GPP control over R

_{s}at long time scales, though at short time scales a significant interaction cannot be detected (Figure 6A). Note the biologically improbable interaction of GPP over T

_{s}especially at short time scales.

_{n}also is included in the analysis (Figure 6B). Introducing this variable changes the causality pattern slightly. The causal effect of GPP over R

_{s}can still be detected at long scales, although considerably reduced. The causality of GPP over T

_{s}, however, disappears and RAD

_{n}influences all variables at all time-scales, except for GPP at short scale. This makes sense, as RAD

_{n}controls all energy processes and is closely coupled with biological processes such as GPP. The disappearance of the influence of GPP on T

_{s}, which was previously detected, indicated that GPP influences T

_{s}indirectly through an unmeasured variable not included in the 3-variable analysis, and most probably RAD

_{n}. The lack of G-causality of RAD

_{n}over GPP at short scales can be explained by the fact that radiation influences GPP almost instantaneously, while G-causality only detects lagged responses at the time-scale of the measurements, here—one hour.

_{n}and SWC largely control all other variables and the influence of GPP over R

_{s}is lost. Interestingly, SWC is not influenced by any of the other variables, i.e., SWC is Granger autonomous while RAD

_{n}is influenced by SWC.

## 6. Discussion

_{s}varies considerably among forest types, though the fundamental variables regulating carbon cycling process rates are generally conserved across ecosystems [53]. In the mixed deciduous forests of Michigan, USA, respiration from soils is the largest component of ecosystem respiration, comprising approximately 70% of the total flux of CO

_{2}[30]. Ecological transitions and regional climatic changes can affect Rs. Carbon sequestration by an ecosystem is determined as a small difference between two large fluxes: respiration, which emits CO2 to the atmosphere and gross primary productivity (GPP) that takes up CO

_{2}from the atmosphere through the process of photosynthesis. The small difference between these two opposing fluxes makes the net CO

_{2}sequestration rate highly sensitive to any change in either respiration of GPP.

_{s}and to better understand the factors controlling it, particularly during and after disturbance [54]. Primary climatic controls on R

_{s}are temperature, solar radiation, and precipitation [55]

**.**Soil respiration increases as soil temperatures rise, because microbial community abundance and activity both respond positively to warming [27,56]. Drought and extremely wet conditions limit microbial activity and decrease respiration [57]. Over periods of hours to days, photosynthetic activity also controls carbohydrate supply from leaves to roots resulting in increased C availability to mycorrhizal fungi and microorganisms in the root environment (rhizosphere), thereby increasing R

_{s}[58,59,60,61].

_{s}, modifying light and nutrient use efficiency [62], roughness length, canopy-atmosphere coupling, and increasing turbulence mixing above the soil [63,64]. In both 2008 and 2009 respiration was lower in the treatment site than in the control. Due to the lack of respiration measurements prior to the girdling treatment we cannot rule out that this difference is a result of the treatment.

_{s}that reaches its lowest point within days following stem girdling [54,66], suggesting that the exhaustion of stored labile carbohydrates progressively limits root and rhizosphere respiration.

_{s}in either treatment or control site, and that RAD

_{n}was a primary environmental constraint on R

_{s}through its effects on T

_{s}at short time scales. At longer scale, both SWC and RAD

_{n}controlled R

_{s}. This finding is similar to other studies suggesting no diurnal relationship between GPP and R

_{s}[67] and in contrast to others showing that GPP or photosynthesis constrains R

_{s}over diurnal timescales [15,58,59,68]. Respiration from our sites is principally limited by soil moisture and radiation (heat to the soil), suggesting that RAD

_{n}, which is remotely sensed with high accuracy, may serve as a useful predictor of R

_{s}. In another site, the Duke Forest, where a significant link between GPP and R

_{s}was established [15], moisture and temperature were less limiting factors.

_{s}interactions were reviewed [60]. Among non-manipulative approaches, methods based on time series analyses, such as cross-correlation, pulse-response and Fourier decomposition, were suggested to detect lagged response. However, we note that only G-causality is able to provide a directionality of the interactions, which is the real signature of a causal mechanism. Furthermore, the multivariate approach investigated in this study showed that including more biophysical factors may reveal indirect or apparent interactions that, in lower dimension systems, may appear as direct and significant. However, the proposed method has also some limitations. Long, continuous and multiple time series are required to construct robust spectral matrices and reduce noise. Although G-causality is less sensitive to random Gaussian noise, other non-Gaussian source of error (e.g., spikes) may cause unpredictable effects. Any additional variable that contains such errors may reduce the statistical power to the extent that weak interactions may not be detectable.

## Acknowledgments

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Detto, M.; Bohrer, G.; Nietz, J.G.; Maurer, K.D.; Vogel, C.S.; Gough, C.M.; Curtis, P.S. Multivariate Conditional Granger Causality Analysis for Lagged Response of Soil Respiration in a Temperate Forest. *Entropy* **2013**, *15*, 4266-4284.
https://doi.org/10.3390/e15104266

**AMA Style**

Detto M, Bohrer G, Nietz JG, Maurer KD, Vogel CS, Gough CM, Curtis PS. Multivariate Conditional Granger Causality Analysis for Lagged Response of Soil Respiration in a Temperate Forest. *Entropy*. 2013; 15(10):4266-4284.
https://doi.org/10.3390/e15104266

**Chicago/Turabian Style**

Detto, Matteo, Gil Bohrer, Jennifer Goedhart Nietz, Kyle D. Maurer, Chris S. Vogel, Chris M. Gough, and Peter S. Curtis. 2013. "Multivariate Conditional Granger Causality Analysis for Lagged Response of Soil Respiration in a Temperate Forest" *Entropy* 15, no. 10: 4266-4284.
https://doi.org/10.3390/e15104266