# Information Driven Ecohydrologic Self-Organization

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{a},

^{o}C), estimated gross ecosystem respiration (GER, µmol CO

_{2}m

^{-2}s

^{-1}), soil temperature in the surface layer (Θ

_{s},

^{o}C), estimated gross ecosystem productivity (GEP, µmol CO

_{2}m

^{-2}s

^{-1}), latent heat flux (γ

_{LE}, W m

^{-2}), vapor pressure deficit (VPD, Pa), soil moisture (θ, m

^{3}/ m

^{3}) sensible heat flux (γ

_{H}, W m

^{-2}), total incoming shortwave radiation (R

_{g}, W m

^{-2}), and precipitation (P, mm).

**Figure 1.**The seven Fluxnet study sites used in the study, including Atqasuk (ATQ, North Slope of Alaska), Audubon Research Ranch (ARR, Arizona Semi-Arid Grassland), UCI 1964 Burn Site (UCI, Canadian Boreal Forest), Bondville Original Site (BV1, Illinois Corn & Soybeans), Goodwin Creek (GCR, Mississippi Semitropical Hardwood Forest), Kennedy Space Center Scrub-Oak (KSC, Florida Semitropical Marine Scrub), and Tonzi Ranch (TZR, Mediterranean Central California). The gradient of mean annual precipitation from wet (> 1,000 mm/yr shown in green) to dry (< 100 mm/yr shown in red) shows the diversity of climate variability captured by the selection of Fluxnet sites. The insets show the normalized (zero mean and unit standard deviation) variation of mean annual patterns of monthly precipitation, enhanced vegetation index (EVI) from MODIS, and $TS{T}_{V}^{m}$ (see Table 1 for a summary of climate data for each site) (vertical tick marks are 0.5 standard deviation increments above and below the mean and horizontal tick marks indicate the month of the year).

Fluxnet Site | Code | Mean Annual Precipitation (mm) | Mean Annual Evapo-transpiration (ET) (mm) | Mean Annual Air Temperature θ_{a}(C) | ET Response Adaptation Factor, c | Thermal Offset Adaptation Temperature, θ_{a} (K) |

Atqasuk | ATQ | 112 | 178 | −8 | 0.881 | 9 |

UCI (1964 Burn Site) | UCI | 202 | 261 | 2 | 2.601 | 19 |

Audubon Research Ranch | ARR | 389 | 290 | 17 | 1.284 | −8 |

Tonzi Ranch | TZR | 574 | 405 | 17 | 0.805 | −9 |

Bondville (original site) | BV1 | 839 | 603 | 11 | 0.294 | −14 |

Kennedy Space Center (Scrub oak) | KSC | 1,120 | 808 | 22 | 0.580 | −6 |

Goodwin Creek | GCR | 1,494 | 690 | 17 | 0.554 | −7 |

## 2. Methods

_{t}as:

_{t}(see [5] for a justification for this choice). Noting that transfer entropy is asymmetric both in strength and lag, it provides a two-way measure of information flow, or coupling strength. The information flow process network consists of the asymmetric pair wise transfer entropy between the i

^{th}and j

^{th}variable from the set of n

_{V}observed variables and can be represented as an adjacency matrix $A(i,j,\tau )={T}^{`}({X}_{t}^{(i)}\to {X}_{t}^{(j)},\tau )$ [5]. Process networks are computed for each of thirty-six sub-daily time lags between half an hour and eighteen hours. This range captures the primary scales of interaction between the atmospheric boundary layer (ABL) and the terrestrial processes [16]. Estimation and methodological issues, including robustness of the method in presence of noise, and validation using noisy chaotic data are discussed in [5,6].

_{s}variables. We use several metrics to measure flow of information [5,6]. The mean relative entropy for a subsystem S ⊆ V consisting of ${n}_{S}\le {n}_{V}$ variables is computed as:

^{m}

_{V}as a special case of ${T}_{S}^{[+]}$ when S = V. An increase in TST

^{m}

_{V}is an indicator of increased feedback between system components.

**Figure 2.**Mean annual phase diagrams for ${T}_{GEP}^{[+]}(0.5h)$ for all seven sites. The size of each circle scales in proportion to ${T}_{GEP}^{[+]}(0.5h)$, and relates ecosystem information production to the mean monthly latent heat flux (γ

_{LE}) and air temperature (Θ

_{a}) at each site. The month and arrow on each subplot indicate the timing of the annual peak in ${T}_{GEP}^{[+]}(0.5h)$ and the direction of chronological rotation of the phase diagram at that point (clockwise or counterclockwise). Regardless of climate and ecosystem type the peak ecosystem information production coincides with the maximum latent heat production indicating that the moisture and energy balance controlled by vegetation growth mediates the feedback between all system components.

## 3. Results

_{LE}and the air temperature Θ

_{a}for all sites can be collapsed to single curves provided we account for the site specific dependencies. These take the form ${T}_{GEP}^{[+]}(0.5h)=c\cdot \alpha \cdot {\gamma}_{LE}$ (R

^{2}=0.63, Figure 3a) and ${T}_{GEP}^{[+]}(0.5h)=\beta {({\Theta}_{a}+{\Theta}_{a}^{`}+k)}^{\lambda}$ (R

^{2}=0.54, Figure 3b) where α (=1.88 x 10

^{-4}mm

^{-1}month

^{-1}), β (=1.8 x 10

^{-7}K

^{-1}month

^{-1}), and λ (=2.78) are site independent parameters.

**Figure 3.**For all sites information production ${T}_{GEP}^{[+]}(0.5h)$ (a) as a function of latent heat flux follows ${T}_{GEP}^{[+]}(0.5h)=c\cdot \alpha \cdot {\gamma}_{LE}$, and (b) as a function of air temperature follows ${T}_{GEP}^{[+]}(0.5h)=\beta {({\Theta}_{a}+{\Theta}_{a}^{`}+k)}^{\lambda}$ (see Table 1). The whole system mean information production for all time lags $TS{T}_{V}^{m}(\tau =0.5h...18h)$ increases rapidly with increase in the entropy of the system as $TS{T}_{V}^{m}(\tau )=a(\tau ){({H}_{V}^{m})}^{b}$ (each point on the figures represents results for one month at one site and time lag).

_{a}

^{`}(Table 1) are termed the “evapotranspiration response adaptation factor” and “thermal offset adaptation temperature.” The former captures the property that information production in humid ecosystems (eg., BV1) responds more slowly to increased moisture (lower c) as compared to drier regions (eg., ARR), while the latter captures the behavior that ecosystems in colder regions (eg., UCI, ATQ) begin producing information at much lower temperatures (higher Θ

_{a}

^{`}) than more temperate ecosystems (e.g., GCR, BV1). Therefore, colder and drier ecosystems are adapted to achieve higher levels of information production, that is, increased coupling, per unit of water or energy use. In other words, each ecohydrologic system is adapted to produce information, i.e., process coupling, in a unique way, such that drier and colder systems have a more intense and immediate response to smaller amounts of moisture and energy. These results are computed using “global” bounds of variability, such that the minimum and maximum bounds on each variable X are those of the entire long-term dataset across all sites. In this case, this means that entropies are computed relative to the full spectrum of variable states encountered over the entire observation period for the site.

_{X}in Figure 4, along with the distribution of H`

_{X}for each of the ten variables across seven sites and all months in the data.

**Figure 4.**Information production ${T}_{X}^{[+]}(\tau )$, consumption ${T}_{X}^{[-]}(\tau )$, and net production ${T}_{X}^{net}(\tau )$ are plotted against each variable X’s entropy H`

_{X}, for each site, time lag, and month in this dataset. The synoptic (blue) subsystem includes weather-forcing variables, the turbulent (green) subsystem includes variables directly influenced by the ecosystem, and the ABL (red) subsystem includes precipitation and radiation. Due to an imbalance in information production and consumption as H`

_{X}increases, the net production is negative for the turbulent subsystem and is positive for the synoptic subsystem. Also observe that the smaller scale turbulent and ABL subsystems rarely take values of H`

_{X}> 0.7 but the large-scale synoptic subsystem takes values closer to the upper bound of the Shannon entropy.

_{a}, GER, Θ

_{s}, VPD, and θ) to the net-importing “turbulent subsystem” (consisting of GEP, γ

_{LE}, and γ

_{H}) with the “ABL subsystem” (consisting of P and R

_{g}) generally being net-neutral. The synoptic system varies with weather patterns on a timescale of hours to days, while turbulent systems varies with atmospheric mixing processes on a timescale of seconds to minutes, and the ABL system varies on a timescale of hours and couples the synoptic and turbulent scales [16]. Net information flows from high-entropy larger-scale subsystems to low-entropy smaller-scale subsystems.

## 4. Discussion

_{X}which serves as a control parameter. To explain these results, the Moderate Entropy Hypothesis (MEH) is proposed: variables that participate the most in the self-organizing feedback have moderate entropy or variability (between roughly 40% and 70%, for the systems studied). The statistical results presented for the MEH in Figure 4 are computed using “local” bounds of variability, such that the minimum and maximum bounds on each variable X are set independently for each local time period at each site. In this case, this means that Shannon entropies are computed relative to the full spectrum of variable states encountered over exactly one month. Therefore, while the “global” entropies allow us to capture the long-term average dynamics of the system, the “local” entropies allow us to capture the behavior of individual subsystems as they adapt to short-term variability. While the IPH ascertains that larger variability leads to stronger feedback in the system as a whole, the MEH indicates that increased feedback between individual components causes moderated variability in those components. This suggests a tug-of-war between the variability in the system as a whole with a tendency toward maximum variability, and the variability of individual components with a tendency toward moderate variability. This tension appears to be the causal reason for the emergence of order. Self-organizing systems evolve to maximize order (measured as information production), but this comes at the cost of increased variability (measured as entropy). Beyond relative entropy values of 70% (a rough value based on the systems studied here), it may be inferred that increased variability does not return increased order. Perhaps a further increase in the entropy of a subsystem will result in a breakdown of self-organization.

## Acknowledgement

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Kumar, P.; Ruddell, B.L.
Information Driven Ecohydrologic Self-Organization. *Entropy* **2010**, *12*, 2085-2096.
https://doi.org/10.3390/e12102085

**AMA Style**

Kumar P, Ruddell BL.
Information Driven Ecohydrologic Self-Organization. *Entropy*. 2010; 12(10):2085-2096.
https://doi.org/10.3390/e12102085

**Chicago/Turabian Style**

Kumar, Praveen, and Benjamin L. Ruddell.
2010. "Information Driven Ecohydrologic Self-Organization" *Entropy* 12, no. 10: 2085-2096.
https://doi.org/10.3390/e12102085