Deriving state-and-transition models from an image series of grassland pattern dynamics

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Ecological Modelling 221 (2010) 433–444

Contents lists available at ScienceDirect

Ecological Modelling

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

eriving state-and-transition models from an image seriesf grassland pattern dynamics

ohan J. Sadlera,b,c,∗, Martin Hazeltond, Matthias M. Boera,c, Pauline F. Griersona,c

Ecosystems Research Group, School of Plant Biology, The University of Western Australia, 35 Stirling Highway, Crawley, 6009, AustraliaSchool of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, 6009, AustraliaBushfire Cooperative Research Centre, Level 5, 340 Alberty Street, East Melbourne, VIC 3002, AustraliaInstitute of Information Sciences and Technology, Massey University, Private Bag 11 222, Palmerston North, New Zealand

r t i c l e i n f o

rticle history:eceived 3 March 2009eceived in revised form 15 October 2009ccepted 16 October 2009vailable online 24 November 2009

a b s t r a c t

We present how state-and-transition models (STMs) may be derived from image data, providing agraphical means of understanding how ecological dynamics are driven by complex interactions amongecosystem events. A temporal sequence of imagery of fine scale vegetation patterning was acquired fromclose range photogrammetry (CRP) of 1 m quadrats, in a long term monitoring project of Themeda triandra(Forsskal) grasslands in north western Australia. A principal components scaling of image metrics calcu-lated on the imagery defined the state space of the STM, and thereby characterised the different patterns

eywords:attern dynamicslose range photogrammetryhemeda triandra grasslandsdaptive managementegetation monitoring

found in the imagery. Using the state space, we were able to relate key events (i.e. fire and rainfall) toboth the image data and aboveground biomass, and identified distinct ecological ‘phases’ and ‘transi-tions’ of the system. The methodology objectively constructs a STM from imagery and, in principle, maybe applied to any temporal sequence of imagery captured in any event-driven system. Our approach, byintegrating image data, addresses the labour constraint limiting the extensive use of STMs in managing

and

mage metricsilbara

vegetation change in arid

. Introduction

The state-and-transition model (STM) is a conceptual tool torganize our understanding of how the dynamics (or temporalhange) of vegetation communities are driven by complex inter-ctions among events (e.g. fire, grazing and flooding), processese.g. mineralisation and drainage) and biological factors (e.g. inva-ive species). STMs were originally developed to explain limitationsn Clementsian linear succession models in predicting the conse-uences of land management on rangeland vegetation dynamics,

ncluding: irreversible vegetation change; grazing catastrophe;pisodic plant recruitment; and, alternative stable vegetation statesWestoby et al., 1989). As a management tool, the STM illustrateshe potential impacts, nonlinearities and uncertainties inherentnder different environmental conditions or management sce-

arios (Bestelmeyer et al., 2004). Consequently, STMs are a keyomponent of the proposed framework for science-based landanagement (SBLM) of rangelands in the western United States

nd elsewhere (Herrick et al., 2006). However, a science-based,

∗ Corresponding author at: Ecosystems Research Group, School of Plant Biology,he University of Western Australia, 35 Stirling Highway, Crawley, 6009, Australia.el.: +61 8 6488 3445; fax: +61 8 6488 7925.

E-mail address: [email protected] (R.J. Sadler).

304-3800/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2009.10.027

semiarid rangelands.© 2009 Elsevier B.V. All rights reserved.

quantitative implementation that goes beyond current qualita-tive methods of deriving STMs is constrained by: (i) a lack oflong term data encompassing the breadth of different possibleecosystem behaviours; (ii) the need to derive STMs indirectly bycollating a range of often disparate data sources; and, (iii) sys-tem and geographical specificity, thereby creating an inequityin terms of which systems and localities will be investigatedfirst (Bestelmeyer et al., 2003, 2004). Data in the form of aerialimagery would ameliorate the above difficulties in STM modelling,as imagery is spatially extensive and may be reliably captured,georeferenced and processed at a range of geographical scales. Amethodology to arrive directly at a STM from temporal sequencesof imagery that capture vegetation pattern dynamics is thereforeproposed.

In its simplest form the STM identifies the possible vegeta-tion communities that can occur in a particular system or locale,labelling them as different meta-stable phases (i.e. these vegetationcommunities may vary from being transient to persistent; Westobyet al., 1989; Stringham et al., 2003). The system may then experi-ence a transition (or pathway) from one phase to another that was

triggered (or driven) by the interacting effect of different events andprocesses. Hence, some of the nonlinearities observed in the fieldthat confound linear succession models may be described, a listincluding: ecological thresholds and irreversible transitions; mul-tiple stable states; historical contingency; cross-scale interactions,

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434 R.J. Sadler et al. / Ecological Mode

Fig. 1. State-and-transition model. The state space contains ecological states (orvegetation communities) separated by transitions across irreversible ecologicalthresholds. Each state may be described by its own characteristic set of phasedre

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and Bartolome, 2002), who were first to realize the feasibility of

ynamics. Transitions between these phases are reversible, either directly or indi-ectly. Transitions, whether reversible or irreversible, are triggered by interactingcological events. Modified from Stringham et al. (2003).

nd lagged effects (Friedel, 1991; Rietkerk and van de Koppel,997; Peters et al., 2004). The dynamics of a system are thereforen observed sequence of nonlinear transitions between differenthases (Fig. 1).

A key concept of the STM is the ecological threshold, a bound-ry in space and time between different phases of the system (oromains of stable system behaviour; May, 1977; Friedel, 1991).ransitions across ecological thresholds can occur along a con-inuum from irreversible to immediately reversible pathwaysStringham et al., 2003; Briske et al., 2005). Irreversible transi-ions are triggered by events that are typically unprecedented andhich lead to a degraded state that is constrained by a different

et of biological and soil processes (e.g. soil erosion, overgrazing,ntroduction of exotic plant species, altered fire regimes or a com-ination of such events). The typology of Stringham et al. (2003)ses irreversible transitions to define ‘states’ of the system, whereaseversible transitions separate ‘phases’ or ‘communities’. Collec-ions of phases, connected by a network of transitions are visualizeds the phase dynamics nested within each system state (Fig. 1;tringham et al., 2003).

Science-based land management (SBLM) is defined by four corelements: (1) a method of land classification to describe ‘eco-ogical’ sites; (2) a data storage and management facility; (3)onceptual models of ecosystem dynamics, including site spe-ific STMs; and (4) a methodology to evaluate the status ofhe ecosystem, be it qualitative or quantitative (Herrick et al.,006). To date, SBLM has been applied to arid lands of westernnited States, where episodic events such as rainfall drive oftenramatic shifts in ecological processes. In arid lands, intensivessessment is necessary to match and capture the frequency ofriving events, otherwise important factors leading to vegetationhange will likely be missed and long term trends will be con-ounded by temporary responses to recent events. Consequently,he need for long term monitoring is implicit in SBLM, and is akino adaptive management (Holling, 1978): data from long term

onitoring is used to update models and knowledge that assistsanagement decision making in dynamic and uncertain ecological

ystems.The main constraint to SBLM is the ability to capture data. Cur-

ently, SBLM defines a protocol of 17 qualitative indicators for

he evaluation of three key ecosystem attributes: soil and sitetability, hydrologic function and biotic integrity (e.g. Pellant etl., 2005). If more precise information is required then a fur-her protocol composed of quantitative indicators may be applied

lling 221 (2010) 433–444

(Herrick et al., 2005). Both protocols are similar to landscapefunctional analysis (LFA) in the type of indicators they employ(Ludwig et al., 1997; Tongway and Hindley, 2005). However,acquiring data by both LFA and SBLM is labour intensive, result-ing in limited spatial sampling (i.e. a sampling bias, Watsonand Novelly, 2004), and risking significant observer bias (differ-ent observers may provide different assessments, and the sameobserver may provide different assessments under different con-ditions, Burrough and McDonnell, 1998; Hunt et al., 2003; Boothet al., 2006). The limitations inherent in labour-intensive, qual-itative assessments is exacerbated by the need for repeatedlong term monitoring in capturing ecological dynamics, and islikely to limit the application of STMs beyond intensively studiedsites.

Ecosystem assessments that serve multiple management goals,such as that promoted by SBLM, employ a large number of indi-cators, as individual indicators represent only singular aspects ofecosystem behaviour. For example, a number of metrics that quan-tify pattern in imagery may be applied, be they structural (e.g. patchbased) or textural (i.e. pixel based). These image metrics frequentlydescribe similar aspects of image pattern, such as average patchradius and the perimeter–area ratio (Riitters et al., 1995). Dimen-sion reduction techniques, such as principal components, can beused to simplify the multivariate information of a large numberof candidate metrics into a small number of summary variables(Riitters et al., 1995). In effect, the summary variables define a‘state space’ (or blank canvas) on to which individual images canbe plotted.

An ordination of image metrics calculated on time sequencesof imagery will generate a trajectory of a system’s dynam-ics in the ordination defined state space. Time trajectories ofa system’s behaviour have previously been applied to study-ing changes in plant community composition, with ordinationplots used to elucidate the difference in dynamics betweengood and poor condition shrubby grasslands of arid Australia(Friedel, 1991). Nonmetric multidimensional scaling has alsodescribed the regeneration of grassland at polluted sites in com-parison to semi-natural meadows in East Germany (Voigt andPerner, 2004). A partitioning of the state space into variousecological phases can then be linked to ecological events thatdrive transitions between the ecological phases, thus deriving aSTM.

The flexibility of imagery in terms of scale, geographical extentand relative ease of capture makes imagery an ideal data source,with potential to alleviate both the sampling and observer bias ofmanual methods, and to reduce costs associated with long termmonitoring. However, it remains uncertain how to best incorporateimagery within existing assessment frameworks, despite remotesensed imagery being viewed as an important facet of ecosystemmonitoring in spatially extensive systems (Ludwig et al., 2004;Herrick et al., 2006). Our challenge is to derive models of ecosys-tem behaviour directly from imagery sourced data, a step criticalin furthering the utility of both imagery and the STM in the SBLMframework (Bestelmeyer et al., 2004). Consequently, our objec-tives were to: (i) generate a state space by applying image metricsto an image series; (ii) partition the state space to represent dif-ferent ‘phases’ of the system’s dynamics; (iii) associate sets ofsystem events to the different phases, and thus define the eco-logical triggers for transitions between phases; and, (iv) constructa state-and-transition model by integrating the above informa-tion. As discussed below our approach extends that of (Jackson

applying data driven methods in developing STMs. Further, ourapproach uses only off-the-shelf statistical tools, readily availableto both managers and applied scientists, in fulfilling our statedobjectives.

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R.J. Sadler et al. / Ecological Modelling 221 (2010) 433–444 435

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ig. 2. Monthly and annual rainfall at Hamersley Station (October 1996–October 2f fires and image capture; (b) variability in monthly rainfall is illustrated by the baximum temperature and average monthly minimum temperature are given by t

n this figure legend, the reader is referred to the web version of the article.)

. Method

The methodology for deriving a state-and-transition model fromn image series was developed on a grazing exclosure experimenthat focuses on the long-term biomass and nutrient cycling ofhemeda triandra (kangaroo grass) tussock (bunch) grasslands inhe sub-tropical and semi-arid Pilbara region of north western Aus-ralia. The data here represent a case study of how the methodology

ay be applied to other systems and image series.

.1. Study sites

The study sites were located on Hamersley Station, a cattle prop-rty ∼1700 km north of Perth, where T. triandra grasslands areonfined to deep self-mulching red Vertisol soils on very gentlyloping alluvial plains. Soils are extremely low in organic matterless than 1%) and nutrients (Bentley et al., 1999). The T. trian-ra grasslands studied are monospecific swards, with less than 3%nnuals and forbs by biomass (Bennett et al., 2002). On drier slopesdjacent to the plains, vegetation is generally a mix of open mulgaAcacia anuera complex) woodland and spinifex (Triodia spp.) hum-

ock grasslands.The dynamics of tussock grassland biomass in the Pilbara region

re largely driven by episodic events such as flood and fire. In

articular, growth of T. triandra is highly responsive to seasonaleluges associated with summer cyclonic activity, and senescesnder drought conditions that may last for several years. Meannnual rainfall is 350 mm y−1, but varies from 50 to 800 mm y−1

Fig. 2a). Fire generally consumes all standing biomass and sev-

(a) The monthly rainfall record shows a seasonal summer rain pattern, and datests, with median monthly rainfall given by a thick horizontal bar. Average monthlylid and dashed red lines, respectively. (For interpretation of the references to color

eral good growing seasons (i.e. high rainfall) are required beforepre-fire biomass is regained. Consequently, fuel accumulation afterfire is inextricably linked to the unpredictable timing and extent ofdrought and large rainfall events. The fire return interval for T. trian-dra grasslands in the Pilbara is in the order of 4–10 years, dependingon the occurrence of rainfall, compared to 1–3 years for T. triandrain temperate and tropical regions elsewhere (Bennett et al., 2002;Lunt and Morgan, 2002).

2.2. Data acquisition

We acquired data from two fenced grazing exclosures located12 km apart (‘Ridge’ and ‘Cattlewell’ paddocks). Exclosures wereestablished in 1995 to assess the productivity and dynamics ofT. triandra grasslands in the absence of cattle grazing. Changesin cover and biomass in response to climate, fire and nutrientadditions have been measured at these sites since October 1996(Bennett et al., 2002).

Imagery captured through close range photogrammetry (CRP)have spatial resolutions as fine as 1 mm, and includes nadir pho-tography captured by on-ground frame mounted cameras (e.g.Cooper, 1924; Bennett et al., 2000; Laliberte et al., 2007), or morerecently digital sensors mounted on ultralight airplanes (Hunt etal., 2003; Booth et al., 2006). CRP methods have small fields of

view, typically ranging from 1 to 100 m, and do not provide con-tinuous photographic coverage over extensive landscapes. Instead,images are sampled intermittently across the landscape (Booth etal., 2006). Combined with image processing, CRP in rangelands is atleast as accurate as non-image assessment methods in estimating

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436 R.J. Sadler et al. / Ecological Modelling 221 (2010) 433–444

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ig. 3. Image series of Cattlewell plot. The image series are of a single 1 m × 1 m coypes of pattern seem to occur (above vs. below rows), the separation of which coin

egetation variables such as percentage foliage cover (Bennett etl., 2000; Booth et al., 2006). Here, we use CRP to regularly capturehe fast system dynamics of biomass turnover in T. triandra grass-ands. As biomass will likely be highly correlated with vegetationtructure or “texture”, we propose a STM mapping of the “texture”ynamics. In principle, the same methodology may be applied to

mage series captured at different scales for event driven vegetationystems exhibiting slower dynamical cycles and/or shifts.

At each site a sub-plot measuring 1 m × 1 m central to a largerm × 5 m plot was photographed from above using a cameraounted on a 2 m high tripod (Bennett et al., 2000). Initially,

rom October 1996 to October 2002, a film SLR camera was used,ut was replaced with a digital SLR from March 2003 to October005. Sites were sampled twice a year, generally corresponding toefore (October) and after (near April) the summer growing season,lthough at times major rainfall events that resulted in extensiveheet flooding of the sites limited accessibility to the sites. Conse-uently, in 2000 and 2001 plots were sampled only once. Half of thelots were experimentally burnt and photographed in November996. The complete data set therefore consists of four replicationsf two time series (experimentally burnt and unburnt) of 18–19mages at each of two sites, giving a total of 295 images (with onemage treated as a missing value).

Covariates assigned to each image were: site (‘Ridge’, ‘Cat-lewell’); biomass; time since last fire (months); and rainfall (mm).otal aboveground biomass was harvested from half of the repli-ated plots at any one sampling date, from a sub-plot on theerimeter of each 5 m × 5 m plot to minimise disturbance to the

able 1mage metrics

Metric Description

White Fraction The proportion of the image occupied by the ‘white’ fraand white image.

Number of Patches Counts the number of discrete “white” patches.Twist Number A measure of overall shape complexity. Calculated by c

twists in the perimeter of a shape.Effective Mesh Size Associated with the probability of two locations belong

and determines the size of uniform patches that corresprobability.

Box Counting Slope Complex systems may display some form of multi-fracscaled regions may display different space filling propecounting plot are therefore used as separate metrics us1, 2, 5, 10, 25, and 50 pixels in length.

Contrast Measures local variability of pixel values.Contagion Based on the relative frequency of finding a pixel of on

another type.Recursivity Measures uniformity of pixel pair combinations. Desig

of contrast and scaled to between 0 and 1.Compactness Counts the number of internel pixel edges contained in

expressed as a fraction of the maximum number of inta shape of the same number of pixels.

treatment plot at the Cattlewell site (i.e. not burnt in November 1996). Two mainwith the wildfire of April 2001.

central unharvested photo/monitoring sub-plot. The sub-plot sam-pled for biomass therefore differed at each sampling date. Totalaboveground biomass was oven dried (75 ◦ C for 48 h) and weighed(Bennett et al., 2000, 2002).

Wildfire, ignited by lightning strikes, was documented for eachof the two sites (Fig. 2a). However, an arbitrary value of 100 monthsfor time since last fire was assigned to the first sampling date inOctober 1996, as fire had not been previously observed by the sta-tion managers at either site for at least a decade. Rainfall data for theperiod were acquired from Hamersley Station, with temperaturedata from the nearest town (Tom Price, 35 km distant). Annual rain-fall was defined as two variables, summed over the early wet season(November–January), and middle wet season (February–March).The early season and middle season rainfall variables correspondrespectively to rising and peak ambient temperatures and evapo-transipiration demand over the summer season (Fig. 2b).

2.3. Image processing and metrics

A complete sequence of digital RGB (red–green–blue) imageswas assembled by scanning photographs acquired by the SLRcamera during the first part of the monitoring program at high res-olution (2000 dpi). Images were processed in Adobe® Photoshop®

7.0 (ADOBE®, 2002) and were firstly color adjusted using a blackand white color tile placed in each image. The color tile permittedstandardisation of color between sampling periods, as the sub-plotswere shaded with a purpose built shade cloth when photographed.Georeferencing was enabled by placing a 1 m × 1 m metal sub-

Reference

ction of a binary black

See Frohn (1998) for a discussion.ounting the number of Bogaert et al. (1999)

ing to the same patch,ponds to that

Jaeger (2000)

tality, i.e. differentlyrties. Slopes of the boxing box dimensions of

Li (2000)

Haralick et al. (1973)e type next to a pixel of O’Neill et al. (1988); Li and Reynolds (1993)

ned to be independent Baraldi and Parmiggiani (1995)

a shape, and isernel edges possible for

Bribiesca (1997)

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R.J. Sadler et al. / Ecologica

lot frame over marker pegs, and the camera tripod inserted ontopole, both permanently situated in the ground at the start of

he monitoring program. Images were warped and clipped to a000 × 1000 pixel template to compensate for any errors in cam-ra angle. To increase separation between foliage and red soil, thelue brightness of the green color channel was increased by 100%.mages were then reduced to 250 × 250 pixels, the coarsest reso-ution that permitted detection of fine scale vegetation features byhe naked eye. To further simplify computation and data manage-

ent images were thresholded (i.e. transformed) at a brightnessalue of 80 to produce black and white (or binary) images. Dataere then imported into an R software environment (version 2.7; Revelopment Core Team, 2008), coupled to the Geographical Infor-ation System GRASS (version 6.0.4 GRASS Development Team,

006; Neteler and Mitasova, 2004; Bivand et al., 2008).The final processing step applied a smoothing filter to remove

oise from the images. All patches of nine pixels size or smallerere removed to produce an image series (Fig. 3). Table 1 summa-

izes the image metrics calculated for each image in the GRASS-Romputing environment. Our choice of metrics depended on aualitative review of their mathematical properties (such as scale,otation and translation invariance; see for example Frohn, 1998).

.4. Statistical methodologies

A two-dimensional (2D) state space was defined by the first twoxes produced from a principal component analysis of the imageetrics, with data points corresponding to individual images. A key

oncern of the analysis is how to classify the state space into dif-erent ecological phases. A classification could be elicited through aombination of expert opinion, on-ground evidence and knowledgef site history. However, such a classification should be comparedo an unsupervised classification that makes few assumptions ariori regarding the possible structure of the STM. Here we applyodel-based clustering to the image metric scores, grouping the

our replicates for each treatment × site × capture date combina-ion into a single multivariate observation through concatenation.his grouping of observations assumes that all replicates are situ-ted within the same ecological phase given they have the same

istory of fire and rainfall. The model-based clustering identifiedlusters of observations based on estimating finite mixture mod-ls that vary in both their location and spread, while choosing anptimal number of clusters through applying a Bayesian Informa-ion Criterion (BIC; implemented in the ‘Mclust’ package in R; Fraley

ig. 4. Density estimation of hypothesized phases. The data were allocated to one of fiverincipal components. The contour for each hypothesized phase represents the 85th percoints are plotted on the first two principal component axes.

lling 221 (2010) 433–444 437

and Raftery, 2002). The original data were then classified into one ofthe inferred clusters by their treatment × site × capture date infor-mation, with each identified cluster representing a hypothesizedecological phase. In addition to the initial principal componentsordination and model-based clustering, our approach included afurther three statistical procedures: (i) kernel density estimation;(ii) classification trees; (iii) penalized spline geoadditive regres-sion.

A ‘porous’ phase boundary for each cluster may be drawn inthe principal component state space through kernel density esti-mation. Kernel density estimation superimposes a density curve(or surface), known as the kernel, over each data point and sumsthe resulting collection of density curves, and may be interpretedsimply as a smoothed histogram (Bowman and Azzalini, 1997). Aphase boundary can therefore be represented by choosing a spe-cific contour (e.g. the countour given by the 85th percentile definesall points on the density surface of equal height that contains 85%of the volume under the density surface). A Gaussian kernel wasemployed, with the smoothing parameter controlling the kernel’sdispersion chosen automatically as the asymptotic normal smooth-ing parameter estimate (Wand and Jones, 1995). Kernel densityestimation was implemented using the ‘sm’ package of Bowmanand Azzalini (1997) in R.

A classification tree analysis identified which covariates, suchas time since fire and rainfall, were associated with transitionsbetween different hypothesized phases. Some hypothesized phasesmay be different in terms of structural patterning as captured by theimagery, but are not ecologically different in terms of what driversor covariates explain them. The operation of classification treesis relatively straightforward, choosing splits in a single variablethrough some optimality criterion (in this case a deviance measurebased on the conditional likelihood) before proceeding to furthersplits (Breiman et al., 1984). A split occurs at a ‘node’, and terminalnodes are known as ‘leaves’. Classification trees have been used fre-quently in the ecological literature (e.g. De’ath and Fabricius, 2000;Jackson and Bartolome, 2002) and were implemented using the‘rpart’ package in R (Venables and Ripley, 2002), with confusionmatrix statistics estimated with the ‘caret’ package.

A geoadditive model was used to understand how biomass var-

ied across the state space after taking into account such variablesas time since fire and rainfall, and thereby give some ecologicalmeaning to the state space (in a manner similar to surface fitting inordinated spaces; Dixon, 2003). Geoadditive models combine bothsemiparametric regression (i.e. including linear and smooth func-

hypothesized phases identified through model-based clustering of the entire set ofentile of a 2D kernel density estimate of points assigned to the hypothesized phase.

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438 R.J. Sadler et al. / Ecological Modelling 221 (2010) 433–444

Table 2Classification of phases by events

Node 1 Node 2 Node 3 Class (leaf) Positive prediction rate a Number of observations

Rain over last 2months> 106 mm

HighNovember–Januaryrainfall > 215 mm

5 75% 16

LowNovember–Januaryrainfall < 106 mm

4 71% 59

Rain over last 2months< 106 mm

HighNovember–Januaryrainfall > 62 mm

Time since fire ≤ 6.5months

3 25% 16

Rain over last 6 months< 36 mm & time since lastfire between 20 and 85months

5 75% 16

Otherwise 1 and 2 100% 104LowNovember–Januaryrainfall < 62 mm

Time since fire > 5 years 1 and 2 83% 24

Rain in last 12 months< 99 mm and time sincefire < 5 years

4 83% 12

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a The positive prediction rate is the difference between the total number of predif positives for each leaf of the classification tree.

ions such as splines of regressor variables) and universal krigingi.e. covariate regression and autocorrelated error) within a shared

ixed model framework (Kammann and Wand, 2003), facilitat-ng model fitting and selection. Smoothing parameters for fittingrend lines were selected automatically using generalised cross-alidation (Ruppert et al., 2003), whilst the number of knots usedo define the splines was constrained to five knots for univariateegressors and 20 knots for bivariate regressors to highlight anyain trends in the data. Standard F-statistics were used in choosing

n optimal model through combined backward and forward selec-ion of both smooth and linear functions of covariates. Althoughhere is no explicit form for the distribution of these statisticsithin semiparametric regression, some guidance is provided by

omparison to a corresponding F distribution in the absence of com-utationally intensive simulation (Hastie and Tibshirani, 1990).hus a test with significance levels set to 0.05 in this context pro-ides only some evidence (as opposed to strong evidence) of a trendr effect in response to explanatory variables. Implementation ofeoadditive models used the ‘SemiPar’ package in R (Ruppert et al.,003).

. Results

.1. Defining the state space and ecological phases

A state space was constructed using the first two principal com-onents of the sphered image metric data (where sphering divideshe data by its correlation matrix), explaining 84.5% of the varia-

able 3onfusion matrix for predicting hypothesized phases

Predicted Hypothesized phase

Phase 1 and 2 Phase 3

Phases 1 and 2 124 0Phase 3 4 4Phase 4 0 4Phase 5 16 0

Sensitivity 86.1% 50.0%

a Specificity is the proportion of observations not in a given class that are falsely predichat are correctly identified as belonging to that class.

5 75% 48

ositives and number of falsely predicted positives, as a proportion of total number

tion in the image metric data (66% principal component 1; 18.5%principal component 2). Each point in the state space correspondsto an observed image with observations concentrated to the righthand side of the state space (Fig. 4a). A user defined mask coveredall points, illustrating that not all parts of the state space containobservations.

Five phases (clusters) were hypothesized by the model-basedalgorithm: phases 1 and 2 shared the same location, but with phase2 more widely dispersed across the state space than phase 1 (Fig. 4band c). In order of proximity to phases 1 and 2 the phases weredefined as phase 5, phase 4 and phase 3. Both phase 5 and phase 4were more broadly dispersed than phases 1 and 2, whereas phase3 was the least frequently occurring phase containing just eight ofthe 295 observations in total. Note that at the 85th percentile thereis significant overlap between the different hypothesized phases inthe state space.

3.2. Interpretation of transition pathways

A partition of the state space using a classification tree shows theintensities and types of event combinations most associated withthe hypothesized phases (Table 2). For example, phase 4 was asso-ciated with two differing sets of conditions defined by two separate

branches of the classification tree: (i) under dry November–Januaryconditions (combined seasonal rainfall < 215 mm), but when rain-fall in the previous 2 months was greater than 106 mm; or, (ii)under more extreme drought conditions (previous yearly rainfall< 99 mm), but within 5 years of the most recent fire. Phase 1 and

Phase 4 Phase 5 Specificity a

0 4 97.4%4 4 95.8%

52 15 91.9%4 60 90.6%

86.7% 72.3% n = 295

ted as that given class. Sensitivity is the proportion of observations in a given class

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R.J. Sadler et al. / Ecological Modelling 221 (2010) 433–444 439

Fig. 5. Cattlewell vegetation pattern dynamics, 1996–2005. Each plot maps the images defined as points on the first two principal components for individual sampling datesa embel in Apri of theb accoun

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t the Cattlewell site. Control (©) and treatment plots experimentally burnt in Novast fire. A large plotting symbol indicates a recent fire, with a wildfire occurrings represented by the height of the bar at the bottom of each image. Dark shadingiomass, after factors such as site, time since fire and rainfall have been taken into

hase 2 were largely congruous, but with 36% of Phase 2 observa-ions distinguished from Phase 1 observations by a time since firef greater than 7 years. Phases 1 and 2 were then combined to lifthe overall classification accuracy rate up from 73% to 81% (a statis-ically significant increase in accuracy at the ˛ = 0.05 significanceevel), suggesting that in terms of the ecological drivers examinedere the two phases differed little. Together, phases 1 and 2 wereredicted well by low recent rainfall (< 106 mm) but with a highombined rainfall in the most recent November–January period> 62 mm), or when time since fire was greater than 8 years. Phasewas the most poorly predicted (a sensitivity rate of 50% in Table 3),ut was associated with recent fire within the previous 6 months.ignificantly, the previous phase of the system was not found to ben important predictor of the current system phase.

Exploratory data analysis (EDA) may be employed to support orontradict the model derived from the model-based clustering andree classification. The event histories at the two sites in the tim-ng of fire in relation to a drought over 2001 and 2002. The fire at

attlewell in April 2001 preceded the drought and led to phases 4nd 5 type behaviour (Fig. 5). The fire at Ridge in December 2002as at the tail of the drought: images captured during the drought

re tightly clustered within the phases 1 and 2 region (Fig. 6). Pro-ounced behaviour such as this response to the interaction of fire

r 1996 (�) are plotted, with the size of each observation dependent on time sinceil 2001. Rainfall (mm) summed over the 3 months previous to the sampling datestate space corresponds to high biomass whilst light shading corresponds to lowt in a geoadditive regression.

and drought is encapsulated in the classification tree (Table 2).More transitory behaviour, such as a sheet flow event in early 2004that was observed to strip the T. triandra grasslands of much of itssenesced biomass, in part explains why phase 4 and phase 5 wereassociated with both drought and extreme wet events, both beingbiomass removing ecological events. Here is an example of simi-lar structural configurations, as captured by the imagery, resultingfrom different ecological stimuli. Further, where events or hypoth-esized phases were infrequently observed then the model-basedclustering and classification tree performs less reliably, as when theNovember 1996 images for Ridge being classed as phase 4 whereasideally they would be classed as phase 3 as they were capturedwithin 2 months of an experimental burn. The low positive predic-tion rate for phase 3 (Table 3) arises from the allocation of June 1998images from Cattlewell into phase 3 through the location/spreadform of the mixture models estimated in the model-based cluster-ing. This result supports an assertion that distances between imagesin the state space should not be automatically assumed to represent

‘ecological distance’, and effort should be invested in EDA to querythe underlying drivers of state space movements and transitions.

Not all state space mappings will have the advantage of thesedata in having available auxillary state variables such as biomassto verify or ground-truth the dynamics captured by the image

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Fig. 6. Ridge vegetation pattern dynamics, 1996–2005. Each plot maps the images defined as points on the first two principal components for individual sampling datesat the Ridge site. Control (©) and treatment plots experimentally burnt in November 1996 (�) are plotted, with the size of each observation dependent on time since lastfi cembi of theb accoun

ss1rtwrtbb(aTrpp

3

pCe

re. A large plotting symbol indicates a recent fire, with a wildfire occurring in Des represented by the height of the bar at the bottom of each image. Dark shadingiomass, after factors such as site, time since fire and rainfall have been taken into

eries. Biomass data for the T. triandra grassland, harvested at theame time as when the imagery were captured, ranged from 0.2 to0 t ha−1. In general, high scores on the first principal componentepresented a 2 t ha−1 higher biomass than low scores, once site,ime since fire and rainfall (summed over the previous 6 months)ere taken into account in a geoadditive semi-parametric (spline)

egression of sub-plot biomass (Fig. 7). The single regressors illus-rated the following trends: (i) biomass first increased with rainfallut showed a relatively constant response thereafter (Fig. 7a); (ii)iomass increased linearly with time since last fire (Fig. 7b); and,iii) the ‘Cattlewell’ site had 0.45 t ha−1 higher biomass than ‘Ridge’,fter taking into account the other regressors (p-value = 0.023).he inclusion of a bivariate smoothing interaction between fire andainfall did not lead to a strong improvement in model fit. Thushases 1 and 2 may be interpreted as a high biomass phase, andhase 3 as the low biomass phase in association with recent fire.

.3. Mapping the state-and-transition model

A STM was mapped onto the 2D state space derived from therincipal component ordination of image metrics applied to theRP imagery (Fig. 8). The STM was composed of the hypothesizedcological phases that partition the state space. The overlapping of

er 2002. Rainfall (mm) summed over the 3 months previous to the sampling datestate space corresponds to high biomass whilst light shading corresponds to lowt in a geoadditive regression.

phase boundaries was permitted as there was a priori no reasonto assume different phases should be entirely discrete (for thesedata the 60th percentile of the 2D kernel density estimate of thedata points in each ecological phase was considered an acceptableheuristic for a relatively discrete representation of phases in Fig. 8).The ecological phases were simply connected as a graph by transi-tions defined by specific sets of ecological events deduced from thetree classifier.

In summary, driving events associated with transitions of thesystem from one phase to another are learned by: (i) hypothe-sizing phases either through an unsupervised classifier such asmodel-based clustering, through expert opinion, or through anEDA (such as breaking down the system’s dynamics in the imagestate space into individual sampling dates and searching for sim-ilar clustering behaviour); (ii) by associating threshold intensitiesof different ecological events to each of the hypothesized phasesthrough a supervised classifier such as a classification tree; (iii)assigning the state space some ecological meaning through query-

ing the performance of the classifier through EDA techniques, orif available the geoadditive regression of auxillary state variablesover the state space. Ecological meaning could further be validatedby applying the classifier to test image series captured at novellocations.

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R.J. Sadler et al. / Ecological Modelling 221 (2010) 433–444 441

F ransfos e; andT atedu in (c)

4

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FemodtT

ig. 7. Semi-parametric regression of biomass. Total biomass (g m−2; square-root tum of the previous 6 months rainfall; (b) the natural logarithm of time since last firhe PCA component shows high biomass in red and low biomass in blue. Total estimsed either 5 knots (rainfall and fire) or 20 knots (bivariate PCA). The plotting mask

. Discussion

We have demonstrated the direct construction of a state-and-

ransition model from a time series of image data. Thus far,tate-and-transition models have been used to cogently summa-ize the existing state of knowledge of an ecosystem’s complexynamics, to identify the transitions that are most important for

ig. 8. State-and-transition model of T. triandra grassland dynamics. Boundaries ofcological phases are represented by the 60th percentile of 2D kernel density esti-ates of all observations placed into each hypothesized phase, and were collapsed

nto the one plot. Note that the different phases overlap but in this plot have beeniscretised to an extent by the choice of percentile. Together the phases and transi-ions form a phase dynamics that characterise an ecological state of the system (i.e.. triandra grassland).

rmed for variance stabilization) was regressed on: (a) the natural logarithm of the(c) the two PCA axes. Grey regions in plots (a) and (b) indicate ±2 standard errors.

biomass is the additive sum of the component smooth regressions. The regressionswas user defined to cover all observed points.

management, to direct research towards conditions that lead tothose transitions, or as qualitative tools to evaluate the relativebenefits and risks of different management actions (Westoby et al.,1989; Stringham et al., 2003). High spatial and temporal variabil-ity in driving event processes were previously thought to precludea quantitative approach (Bestelmeyer et al., 2004). Where quan-titative transition rules have been assigned to a STM, as in themodelling of long term carbon dynamics in Australian savannas,parameter values for these rules were sourced from other stud-ies (Hill et al., 2005). Our approach builds on that of Jacksonand Bartolome (2002) by employing an unsupervised classifier forhypothesizing different system phases and states, and a super-vised classifier to associate ecological covariates and events tothe hypothesized phases. The methods proposed here differ fromJackson and Bartolome (2002), however, in a number of ways:

• The classifiers are applied to patterning in imagery (as charac-terised by a set of image metrics) as opposed to communitycomposition data.

• The phases/states are mapped directly onto a state space definedby an ordination of the data through kernel density percentiles,rather than deduced by the expert from the classification. Theplotting of dynamic trajectories onto an ordinated state spacefollows Friedel (1991) and Voigt and Perner (2004).

• The model-based clustering provides an objective method ofchoosing the number of hypothesized phases/clusters, thoughexploring alternative numbers of clusters should be an importantcomponent of the EDA. The supervised classifier should ‘whittle’down the number of hypothesized classes when hypothesizedclasses may be explained by similar event histories (as with

phases 1 and 2 in the T. triandra data set).

• We employed the tree classifier to predict the occurrence ofsystem phases and not observed transitions (as in Jackson andBartolome (2002)). In the T. triandra system, previous systemphase was not found to be competitive with other ecological

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42 R.J. Sadler et al. / Ecologica

covariates in predicting the current system phase. While the pre-vious system phase is likely to be an important predictor in otherecological systems, the evidence presented here suggests that thedynamics of T. triandra grasslands are highly event dependent:processes such as fire occurrence and drought mask any possi-ble relationship between the current and previous system states(supported by the high classification rate > 80% overall). Our for-mulation of including previous system phase as a predictor alsoallows for the possible identification of events that sustain thepersistence of a system over time in a given phase.

Our study identified four phases in the fine scaled vegeta-ion pattern dynamics of T. triandra grasslands in the Pilbara: ahigh’ biomass ecological phase (combined phases 1 and 2) asso-iated either with greater early season rain in November–Januaryr when there is an extended absence of fire from the system> 7 years); a transitory ‘low’ biomass phase occurring immedi-tely after infrequent fire (phase 3); and two intermediate biomasshases spanning the remainder of the state space (4 and 5). Thereas no evidence of irreversible transitions in the observed dynam-

cs of the system, and so what is reported here represents ‘phase’ynamics within the T. triandra grassland state as opposed to tran-itions between system states.

Identification of multiple phases can be interpreted as aesponse to a highly variable and unpredictable environment ofres and droughts. Transitions between meta-stable phases inegetation patterning brought about by shifts in standing plantiomass and plant numbers are considered to confer resilience

n arid ecosystems to drought and grazing (van de Koppel andietkerk, 2004), and concurs with previous observations of north-rn T. triandra grasslands as being amongst the most resilientrazing systems in Australia (Tongway and Ludwig, 1994). Thefficacy by which image metrics of fine-scale grassland pattern-ng were able to distinguish distinct ecological phases also lends

eight to the close coupling of vegetation pattern and ecologicalrocess in arid systems (Noy-Meir, 1973; Ludwig et al., 1997). Theole of pattern dynamics as a useful surrogate for the underlyingynamics of ecological processes is to an extent supported. Twoey caveats in utilizing pattern dynamics as a surrogate for pro-ess dynamics remain: (i) similar vegetation pattern configurationss captured in the imagery may result from different ecologicaltimuli (as in phases 4 and 5 that may occur after sheet flow ornteracting drought/fire events); and (ii) distance between pointsn the state space is not to be confused with ‘ecological’ distance,n that infrequently observed phases (e.g. phase 3) may have bothoor sensitivity and specificity in their prediction, and are therebyot well identified. Both these issues result from data constraints

n the number and length of image series that can be generatedhrough more extensive monitoring in both time and space. Somef the misclassification may result from the difficulty of monitoringppropriate or sufficiently many covariates. The practical conse-uence of these data constraints is that there is still a need forigorous querying by the ecological expert of outputs from thelustering and classification methods, and of how they are to benterpreted.

The derivation of STMs from image time series complements thedaptive management paradigm that underpins the SBLM frame-ork for arid and semiarid ecosystems. For example, the hypothesis

hat an early season rainfall of greater than 62 mm allows a ‘high’iomass phase to persist can be refined and updated by reap-lying the STM methodology when further monitoring data are

cquired. Thus, the STM methodology provides a facility to learnhe dynamics of a largely uncertain system – to date not all inten-ities, timing and interactions between ecological events possiblender the current event regime have been observed, with their con-equent impacts on vegetation dynamics largely unknown (Holling

lling 221 (2010) 433–444

and Allen, 2002). However, caution is needed in interpreting modeloutput as exact values (such as 62 mm) are an automatic productof the classification tree methodology. Further analyses, utilizingrandom forests (Breiman, 2004), may be used to provide a mea-sure of variability about such estimates. Threshold intensities inevents associated with transitions between phases should there-fore be viewed as provisional, but valid, starting points in predictingecological phase, and in providing an increasingly precise under-standing of system behaviour over time.

A quantitative basis to STM construction also provides newpossibilities for understanding and thus managing vegetationdynamics, in addition to detecting potential deviations from thedomain of normally observed dynamics, including: (i) comprisingan ‘early warning’ system that detects potential deviations awayfrom the domain of normally observed dynamics, observed as a tra-jectory moving away from commonly occurring phases in the statespace; (ii) classifying landscapes and landscape change according toecological phases; (iii) determining how different ecological phasesunderpin other processes such as the likelihood of fire; and (iv) pre-diction of future transitions, although this last application will bedependent on the ability to predict the occurrence and intensity ofecosystem events.

Overlapping boundaries of the polygons defining different eco-logical phases in the constructed STM (Fig. 8) may be compared tohigher order phase transitions in physical systems. In physical sys-tems, a system parameter may be increased so one phase becomesdominant over another (i.e. that phase becomes the meta-stablephase) but that dominance is not complete, or not realized imme-diately. For example, shrub dominance is in a constant state of fluxwhen there is no sustained overgrazing in semi-arid wooded grass-lands in eastern Australia, with changes in shrub dominance drivenby fire interacting with drought events (Westoby et al., 1989). Thehigh variability in timing and intensity of fire and rainfall ensuresthat shrub or grass dominance in wooded grasslands is never com-plete at landscape scales. An awareness of the lack of completedominance (or lack of discreteness of phase boundaries in the statespace) of any single phase should be maintained when interpretingSTMs generated from image series, and multivariate data in general.

Our approach of deriving STMs of ecosystem dynamics from CRPimagery has the potential to reduce the labour required in exten-sive long term monitoring by ground based methods. However,non-trivial issues in advancing the implementation of the method-ology include calculating how much ground-truthing is required toattribute some form of ecological meaning to phases of the statespace (exemplified in this study by the linkage of the state spaceto labour intensive biomass data), and the initial cost of developingimage processing protocols. Without auxillary state variables inter-pretation will have to rely on judgements inferred from the EDA andknowledge of event histories at different sites. An auxillary statevariable or other interpretation is not strictly necessary for pur-poses of detecting novel system dynamics, but is likely essential incommunicating the event driven STM behaviour from an abstractstate space to on-ground managers.

When developing protocols for deriving STMs from image series,four criticisms of applying image metrics should be answered: (i)the image processing procedure has the potential to alter whatprocesses are actually being monitored, and the ecological inter-pretation of metrics applied to the imagery; (ii) the same metricsmay have potentially different meanings at different scales andin different ecosystems, therefore representing possibly unrelatedprocesses (Li and Wu, 2004); (iii) image metrics are subject to crit-

icism similar to that of on-ground ecological indicators in that notany one metric may fully represent a process; and (iv) image met-rics also possess mathematical properties that may impact on theutility of a metric (e.g. ‘energy’ based measures will in generalhave a higher sensitivity to deviations from landscape uniformity

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han from landscape heterogeneity in comparison to other texturaleasures; Baraldi and Parmiggiani, 1995). In consequence, several

mage metrics are typically used to represent a process, analogouso what occurs for on-ground monitoring in SBLM and Landscapeunctional Analysis. However, while on-ground indicators are oftenesigned for measuring specific ecological processes (e.g. soil elec-rical conductivity for salinity) image metrics measure only pattern,ith little a priori knowledge of which image metrics will usefully

apture ecological processes (Riitters et al., 1995; Fortin et al., 2003;i and Wu, 2004). For example, there is not an automatic correla-ion between textural metrics of image pattern and the scale athich plants may utilize resources in a landscape. A further task

n constructing STM state spaces from image data is therefore theelection of the subset of image metrics that will be appropriate tospecific application. However, once established image process-

ng would be at least semi-automatic for the repeated capture ofmages and of benefit to any long term monitoring program.

The need of land managers to distinguish alternative ecologi-al phases at broader scales (i.e. paddock to landscape) than them2 quadrats captured here by CRP can in part be addressed byultiple image capture over a study site. In this study, for exam-

le, eight quadrats were photographed at each of the two sites atach sampling date. Contours of kernel density estimates defin-ng boundaries of the hypothesized ecological phases were thenne way of representing the mean field behaviour of the T. triandrarasslands over the paddock scale sites where the imagery wereaptured. However, land managers in other contexts will likelymploy imagery of different resolution at larger spatial scales. Dif-erently scaled vegetation patterns will therefore be detected (e.g. arass–woodland matrix), dominated by differently scaled ecolog-cal processes (e.g. landscape flows of water and plant invasion).his diversity of applications may be furthered by managers choos-ng other metrics, processing methods and driver variables thatre directed more towards management needs, rather than thepproach presented here which aims at detecting differences inmage pattern with as few a priori assumptions as possible (e.g.

anagers may be more interested in shifts in basal area of the grass-ands for assessing stocking rates, as opposed to overall changes inatterning of above ground biomass). However, the deploymentf alternative imagery and metrics will likely affect the generationf STMs, potentially resulting in different, hypothesized ecologi-al phases correlated to somewhat different system events. Thispecificity of the derived STM in relation to the image source willonstrain the capacity of managers to “mix-and-match” STMs gen-rated from different image sources and at varying spatial scalesor the one vegetation system. Despite the limitations of mov-ng beyond a specific spatial scale and imagery source, the salientoint here is that STMs can be generated from an image series. Theethodology may therefore be applied more generally to larger

cale imagery in future.

.1. Alternative applications

Our study of T. triandra grasslands of north western Australiaepresents a relatively simple system: a monoculture that respondsapidly to driving events, with experimental exclusion of grazing.

here grazing is a factor, changes in grazing intensity may be incor-orated as a driver of a system’s dynamics by including grazing

ntensity as a further covariate in the tree classification, assuminghe data are available. In addition to semi-arid and arid systems,he methodology may be applied to the pattern dynamics of other

vent-driven, multi-phase systems such as seagrass-macroalgaommunities (Fourqurean and Rutten, 2004), where strong changesn patterning may be captured by aerial imagery over relativelymall time scales. More complex vegetation systems, such as conif-rous forests whose event-driven dynamics evolve over decadal

lling 221 (2010) 433–444 443

scales, will require monitoring of multiple sites in a space-for-timesubstitution, with the divergence in event histories between sites tobe maximised (Pickett, 1989). For example, ‘Ridge’ and ‘Cattlewell’were sufficiently distant spatially to be subject to separate wild-fires, permitting the interaction between fire and drought to bebetter identified in the T. triandra image series (Figs. 5 and 6). Animage derived dynamics has then a potentially wide application toa range of vegetation systems.

More generally, our approach consists simply of defining a statespace, hypothesizing phases through an unsupervised classifier,and then assigning thresholds in driving events to those phasesusing a supervised classifier. As demonstrated by (Jackson andBartolome, 2002), the methodology may be extended to other mul-tivariate, long-term monitoring data and not just to pattern metricsderived from image data. For example, restoration projects usingplant community data in comparing phase dynamics between dis-turbed and benchmark sites could be used to evaluate projectcompletion targets (Grant, 2006). Remote sensing technologiesquantifying aspects of the system elements other than vegeta-tion pattern (e.g. biomass and net primary productivity; ‘greeness’indices; water yield and heat fluxes) would simply extend the num-ber of indicators to be incorporated into the state space, or betreated as auxiliary state variables. Existing rangeland monitoringdata comprising SBLM type indices to detect changes in ecosystemprocesses may also be integrated within this framework.

Acknowledgements

This project was primarily funded by an Australian ResearchCouncil grant: ARC-LP0214150. We thank, in addition to the anony-mous reviewers, Mark Westoby for useful comments on the paper;Lauren Bennett and Mark Adams for the initial design and imple-mentation of the long-term monitoring project on HamersleyStation; Pilbara Iron Pty. Ltd. for funding and on-ground resources;Matt and Gill Herbert for providing fire and rainfall data for Hamer-sley Station; and all contributing members of the EcosystemsResearch Group (The University of Western Australia), particularlyKate Bowler, Louise Cullen and Kelly Whyte. The first author wassupported by an Australian Postgraduate Award, with further fund-ing from the Bushfire Cooperative Research Centre.

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