# State-space prediction of soil respiration time series in temperate, semi-arid grassland in northern China.

Introduction

Soil is a major biospheric reservoir for carbon (C), globally containing twice as much C as the atmosphere and three times as much as vegetation (Raich and Potter 1995). Small changes in the size of the C pool in soils across large areas could therefore produce a large effect on the concentrations of atmospheric C[O.sub.2] and ultimately on global climate change (Grace and Rayment 2000; Granier et al. 2000; Schlesinger and Andraws 2000; Rodeghiero and Cescatti 2005). The accurate evaluation and prediction of levels of soil respiration are thus of great concern for research on global change.

Soil respiration relies on a series of processes with complex interactions and feedback among its components and influencing factors. The rate of soil respiration is controlled primarily by the rate of C[O.sub.2] production by biota within the soil, but is modified by factors influencing the movement of C[O.sub.2] out of the soil (Raich and Schlesinger 1992; Tufekcioglu et al. 2001). To understand the regulation of soil respiration by environmental factors and to quantify soil respiration in the global C budget, researchers have developed many models of soil respiration (Cook et al. 1998). Temperature (air temperature and soil temperature at different depths) and soil moisture are usually considered the most influential environmental factors controlling soil respiration, and some types of equations can be fitted to represent the relationship (e.g. linear, quadratic, power, and exponential models, etc.) (Kucera and Kirkham 1971; Peterson and Billings 1975; Schlentner and Van Cleve 1985; Raich and Potter 1995; Chen et al. 2003; Jia et al. 2006). Some of these models developed for soil temperature and moisture are good predictors of the temporal variation of soil respiration (Table 1). However, these models, based on traditional statistical tools (ANOVA and regression), do not consider the temporal/spatial coordinates of the observations and assume that observations are temporally/spatially independent of each other and have no correlation structure. The prediction of temporal patterns of soil respiration is highly affected by soil properties (i.e. soil moisture, temperature, etc.) and also by the state of soil C[O.sub.2] itself. Developing better predictive models of soil respiration through new statistical methods has become one of the important issues in the study of the global C cycle.

Autoregressive state-space methods have been shown to provide opportunities for suitable identification of temporal/ spatial relationships between soils and their vegetation, taking into account their temporal/spatial associations (Nielsen et al. 1994; Wendroth et al. 1999, 2003; Timm et al. 2003a, 2003b; Jia et al. 2011). State-space modelling is a technique that can filter noise underlying vegetative and soil processes at various scales if the density of observation supports the identification of the correlation length (Timm et al. 2003a). State-space modelling has also been demonstrated to be an effective research tool for quantifying localised variation (Wendroth et al. 2003; Timm et al. 2003a, 2003b). However, no reports have addressed the performance of state-space methods applied to the study of the soil respiration time series in a semi-arid grassland ecosystem. We have thus applied a state-space analysis to identify the underlying variables involved in soil respiration in semi-arid grassland in northern China. The emphasis for applying state-space modelling in this study, as in many current spatial and temporal data series, is to identify the underlying processes of one-dimensional series of observations (Nielsen et al. 1994; Wendroth et al. 2003) and to estimate model parameters based on maximum likelihood (Shumway 1988) for predicting future emissions of soil C[O.sub.2] at the study site.

We therefore investigated the effects of soil moisture, temperature (i.e. air temperature and soil temperature at the surface and at a depth of 5 cm), air pressure, and relative humidity on the rate of soil respiration in temperate, semi-arid grassland from July to October 2010 and analysed the relationships between soil respiration and environmental factors. Our specific objectives were to: (1) analyse the primary factors influencing soil respiration, (2) establish models of soil respiration with state-space and classical regression methods, and (3) compare the predictions obtained by the state-space methods with those from the classical regression methods.

Materials and methods

Study area

The study was conducted at the Shenmu Erosion and Environment Research Station of the Institute of Soil and Water Conservation, in the Liudaogou catchment in Shenmu County, Shaanxi Province (38[degrees]46'-38[degrees]51'N, 110[degrees]21'-110[degrees]23'E; elevation 1094-1274 m above sea level). This region belongs to a moderate-temperate and semi-arid zone with an average annual precipitation of 430 mm, which occurs mostly in July and August. The mean annual temperature is 8.4[degrees]C, and mean annual pan evaporation reaches 785 mm [year.sup.-1]. The study area is representative of the transitional belt subject to severe wind and water erosion. There are two main soil types in this catchment, a loessial and an aeolian sandy soil. The study area is under three main land-use types that cover >86% of the total area, i.e. farmland (16%), grassland (44%), and shrubland (26%). The remaining area is covered by wasteland, gully channels, and manmade structures.

Experimental design and field measurements

Experimental design

Two typical grasslands were selected for the study. The selected grasslands are in a level area with good drainage and a relatively low percentage of surface stones. The dominant plant species in the grasslands is purple alfalfa (Medicago sativa), which represented >95% of the sum peak above-ground biomass of this ecosystem during the experimental period.

Three plots 2 m by 2 m were established in each selected grassland. We collected two sets of data consisting of the rate of soil respiration, air temperature, soil surface temperature (0 cm), soil temperature at a depth of 5 cm, soil moisture in the top layer (6 cm in depth), air pressure, and relative humidity from the two grasslands. One group of data was used to evaluate the temporal effects of factors influencing soil respiration during the growing season and to establish models for predicting soil respiration with autoregressive state-space methods and classical statistical methods. The other group was used to compare the quality of prediction of the statespace and classical statistical models for assessing the validity and robustness of the state-space approach in studying the temporal relationships between soil respiration and influencing factors.

Soil respiration and influencing factors

Soil respiration (Rs) was measured using the CI-340 Photosynthesis System fitted with a CI-301SR Soil Respiration Chamber (CID Bio-Science, Inc., Camas, WA, USA). Three polyvinyl chloride (PVC) collars (11 cm inside diameter, 8 cm in height) were carefully pressed into the soil to a depth of 3 cm at three randomly selected positions in each plot. The mean value for each plot (three PVC collars) was considered as a replicate. Measurements were performed by placing the soil respiration chamber on the PVC collars in each plot.

Measurements of soil respiration were performed every 1-3 days (avoiding rainy days) between 1 July and 21 October 2010. Measurements were made during the day, from ~09:00 to 11 : 00 (local time), for 1-2 min on each collar before moving to the next collar; midday values of the rate of soil respiration are representative of daily averages (Davidson et al. 1998). During the growing season from July to October 2010, we measured soil respiration a total of 53 times, and the average of the measurements for each month was defined as the monthly mean soil respiration.

To evaluate the effect of net photosynthetic rate on soil respiration, we measured the photosynthesis of leaves of Medicago sativa three times per month using the CI-340 Ultra-Light Portable Photosynthesis System (CID BioScience, Inc., Camas, WA, USA) with a 6.5-[cm.sup.2], clamp-on leaf cuvette in each plot throughout the growing season. The measurements were taken between 09:00 and 10:00 (local time) on clear days. At each time of measurement, five healthy, fully expanded, representative leaves in each plot were randomly selected for measurements of photosynthesis, and the five values were averaged as one replicate. The average of three replicates in each month was defined as the monthly mean photosynthetic rate.

The volumetric soil moisture in the upper 6cm was monitored concurrently with the measurements of soil respiration next to the collars but outside the cores to avoid disturbance with the conductivity probe (theta probe ML2x, Delta-T Devices Ltd, Cambridge, UK). Soil temperature at a depth of 5 cm ([T.sub.5cm]) was monitored using a geothermometer inserted into the soil in the vicinity of the soil collars. The temperature of the soil surface ([T.sub.0cm]) and air temperature were automatically recorded with a CI-310TS IR Temperature Sensor and an air-temperature sensor (thermocouple) connected to the CI-340, respectively. The CI-340 uses a humidity-sensitive capacitor to measure relative humidity. Soil moisture, soil and air temperatures, air pressure, and relative humidity were measured continuously in the same area as the measurements of soil respiration (53 observations).

Theory

The state-space model of a stochastic process involving observed datasets, all collected at the same locations, is based on the property of Markovian systems that establishes the independence of the future of the process in relation to its past, once given the present state. In these systems, the state of the process condenses all information of the past needed to forecast the future (Timm et al. 2003b).

State-space models generally consist of two systems of equations (state equation and observation equation) to describe the relationship between the input and output of a dynamic system (Shumway 1988; Wendroth et al. 1997). In this case, the state equation is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

where [X.sub.i] is the non-observed state vector, [PHI] is a p x p matrix of state coefficients, and [[omega].sub.i] is the model error vector. The state equation describes how the state vector [X.sub.i] at time (or space) i is related to that at time (or space) i - 1 through the state coefficient matrix [PHI] (transition matrix) and an error associated with the state [[omega].sub.i] with the structure of a first-order autoregressive model. The model error term COl is a zero-mean, uncorrelated, and normally distributed noise with an r x r covariance matrix Q, the latter being variance per unit time (or space), and depending on the interval between observations. In the case of the state-space model, however, the state-equation is solved simultaneously with the observation equation:

[Y.sub.i] = M[X.sub.i] + [v.sub.i] (2)

where the observation vector [Y.sub.i] is related to the state vector [X.sub.i] through the observation matrix M (usually known as an identity matrix) and by the observation noise vector [v.sub.i], also considered to be zero-mean, uncorrelated, and normally distributed. The noises [[omega].sub.i] and [v.sub.i] are assumed to be independent of each other. For further details, see Shumway (1988), Nielsen et al. (1994), and Nielsen and Wendroth (2003).

Before applying a state-space analysis, data were normalised with respect to their mean m and standard deviation s by the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

where the scaled value [z.sub.i] of the measured observation [Z.sub.i] is calculated based on the mean m and the standard deviation s (Wendroth et al. 1999). This scaling procedure allows state coefficients of having magnitudes directly proportional to their contribution to each state variable included in the statespace analysis (Hui et al. 1998).

Statistical analyses

The data for soil respiration (53 observations) were analysed with both autoregressive state-space and classical regression methods. The data for soil respiration and influencing factors from consecutive measurements were analysed as if they were equally distributed and belonging to the same interval class (1-3 days). The direction of statistical analyses followed the sampling pattern from July to October.

Temporal relationships among variables had to be examined before estimating Rs with the state-space method. Therefore, we calculated autocorrelation functions (ACF) and cross-correlation functions (CCF) (Timm et al. 2003a, 2003b). If sets of transect data were autocorrelated and cross-correlated, the state-space models could be carried out (Nielsen and Wendroth 2003). The estimated values of soil respiration and the [PHI] coefficients of the state equation (Eqn 1) were obtained with the software Applied Statistical Time Series Analysis (ASTSA), developed by Shumway (1988). [Z.sub.i] data were scaled using Eqn 3. The classical methods were performed with SPSS v 16.0 (SPSS for Windows, Chicago, IL, USA). For evaluating the quality of estimation, the coefficients of determination ([R.sup.2]) from linear regressions between estimated and measured values of Rs were used in the study (all observations have been scaled using Eqn 3).

Model validation

To assess the quality of prediction of state-space models, the transition coefficients were applied in simple autoregressive predictions, where only the first value of soil respiration in the series is known, and all following values are calculated from the one previous and those from the underlying variables included in the respective state-vector. As a criterion for the quality of prediction, the root mean-squared error (RMSE) is determined from the measured soil respiration ([Rs.sub.meas]) and predicted soil respiration ([Rs.sub.prcd]). The RMSE is an accuracy measure of the prediction and should be as small as possible for accurate prediction. The RMSE was calculated by the following equation:

Results and discussion

Temporal patterns of soil respiration and influencing factors

Soil respiration exhibited a pronounced temporal trend from July to October 2010 (Fig. la), ranging from 0.29 to 2.40 [micro]mol[m.sup.-2][s.sup.-1], with a mean rate of 1.12[+ or -]0.45 [micro]mol [m.sup.-2][s.sup.-1] (mean [+ or -] s.d.). Monthly Rs increased through July and peaked in August, followed by a gradual decrease during September and October (data not shown). Our results are in agreement with Qi et al. (2008) showing that the seasonal patterns of soil respiration in semi-arid grassland were characterised by single-peak curves, which had the highest values in August. This finding may be ascribed to plant growth and/or soil microclimates. Hogberg et al. (2001) and Tang et al. (2005) showed that plant photosynthetic rate modulated soil respiration. Average plant net photosynthetic rate was significantly higher in August (38.7 [+ or -] 6.2 [micro]mol [m.sup.-2][s.sup.1]) than in the other three months and decreased through September (22.1 [+ or -] 3.8 [micro]mol[m.sup.-2][s.sup.-1]) and October (8.4 [+ or -] 1.2 [micro]mol [m.sup.-2][s.sup.-1]). However, no significant differences in photosynthetic rates or soil respiration were observed between July (25.2 [+ or -] 4.8 [micro]mol[m.sup.2][s.sup.1]) and September. Furthermore, soil microclimates (e.g. soil moisture and soil temperature) in August (Fig. lb, c) were more suitable for the activities of soil microbes and roots than in other months, thus resulting in higher emissions of C[O.sub.2] (Howard and Howard 1993; Qi et al. 2008).

Soil respiration originates mainly from root and microbial activity. Howard and Howard (1993) indicated that both components were strongly influenced by available soil moisture and temperature. This study focussed on the relationships among Rs, soil moisture (SM), air temperature (AT), [T.sub.0cm], [T.sub.5cm], air pressure (AP), and relative humidity (RH) to determine the crucial factors affecting temporal patterns of soil respiration at the study site. Exponential functions described the relationships between Rs and AT ([R.sup.2] = 0.38, P < 0.01), [T.sub.0cm] ([R.sup.2] = 0.44, P < 0.01), and [T.sub.5cm] ([R.sup.2] = 0.40, P < 0.01) to examine the effects of temperature on soil respiration during the experimental period. [T.sub.0cm] was able to explain 44% of the variation in Rs, more than did AT and [T.sub.5cm], indicating that soil surface temperature was a major factor influencing soil respiration. Fitted relationships of Rs with SM, AP, and RH had [R.sup.2] values of 0.13 (P<0.05), 0.10 (P<0.05), and 0.02 (P>0.05), respectively. Soil respiration was significantly positively correlated with temperature and soil moisture and negatively correlated with air pressure (Table 2), showing that soil respiration was inhibited by increasing air pressure (Jia et al. 2007). No significant relationship was observed between soil respiration and relative humidity. Soil respiration was therefore mainly affected by soil surface temperature, followed by soil moisture and air pressure, irrespective of relative humidity. Therefore, [T.sub.0cm], SM, and AP were used to predict Rs by the state-space models and classical statistical methods. The coefficient of variation (CV) values for Rs and other variables at the study site ranged from 0.6% to 41% during the experimental period. Variation in AP had the lowest CV and variation in AT and [T.sub.5cm] had the highest CV (Table 2). All of these variables exhibited point-to-point fluctuations due to temporal variation, which represents typical local characteristics, and may therefore be better represented by a local model (e.g. state-space model) (Timm et al. 2003a).

Auto- and cross-correlation functions

To understand how soil moisture, temperature, and air pressure affect soil respiration, we examined their temporal relationships. We calculated the ACF for each dataset (i.e. Rs, [T.sub.0cm], SM, and AP; Fig. 2a-d, respectively), with the objective of evaluating the temporal correlation of the observations, i.e. if they had been monitored for a time interval sufficient for identifying their temporal representativity. Using a t-test at P = 0.05, the ACF of the Rs and [T.sub.0cm] data (Fig. 2a, b) manifested significant temporal correlations up to eight lags. Temporal dependence between adjacent observations of AP (Fig. 2d) was significant up to seven lags; the ACF of SM (Fig. 2c) showed temporal correlation only up to two lags because it had no discernible trend. The ACF is a tool that reflects the local variation between samples for different distances of separation and is used to identify the range of the spatial/temporal correlations in the observations of a variable (Journel and Huijbregts 1991). The ACF indicated that the sampling-time intervals of these variables were sufficient to identify their temporal representativity.

For describing the spatial/temporal degree of linkage between two variables, the CCF can be used (Shumway 1988; Wendroth et al. 1997). In our study, CCF was calculated to analyse the temporal cross-correlation structures between Rs and [T.sub.0cm], SM, and AP (Fig. 3a-c, respectively). At a 95% confidence level, Rs was positively cross-correlated with [T.sub.0cm] and SM and negatively cross-correlated with AP. Using the t-test at P=0.05, the cross-correlogram between Rs and [T.sub.0cm] showed a strong temporal dependence (Fig. 3a). Similar results were found for the cross-correlogram between Rs and AP (Fig. 3c). The cross-correlogram between Rs and SM (Fig. 3b), however, showed a weak temporal dependence. The CCF of Rs and the significant influencing factors manifested the potential for describing their temporal distributions across the experimental period through state-space analysis, verifying that the use of the CCF leads to additional information on the temporal variability of soil respiration.

Classical regression models of soil respiration

With 53 observations throughout the entire experimental period, stepwise linear regression was used to determine the most significant factors affecting Rs, including Toxin, SM, and AP. The effects of [T.sub.0cm] and SM on Rs were the most important, and their coefficients were significant. Therefore, all classical models of Rs were established with [T.sub.0cm] SM, and their interactions (Table 3). This study was mainly focussed on exponential, stepwise linear regression, twice-linear regression, quadratic, power, and exponential-power models, because they had been widely used to estimate the efflux of soil C[O.sub.2] with soil temperature and soil moisture (Kucera and Kirkham 1971; Peterson and Billings 1975; Davidson et al. 1998; Davidson et al. 2000; Li et al. 2000; Wang et al. 2003; Jia et al. 2006, 2007; Qi et al. 2008). The classical equations were all significant (P < 0.05).

When [T.sub.0cm] was considered as an independent controlling factor, the quadratic model ([R.sup.2] = 0.54) performed best for estimating Rs, compared with the exponential ([R.sup.2] = 0.44) and power ([R.sup.2] = 0.39) models. Considering the relatively smaller contribution of SM to the temporal variation of Rs throughout the entire experimental period, SM was not analysed as an independent factor in predicting Rs. Notably, the [R.sup.2] values indicated that >60% of the variance of Rs was explained by the combination of [T.sub.0cm] and SM in the classical models, in particular the exponential-power model (~70%) (Table 3). This result supported the findings of Qi et al. (2008), who showed that the exponential-power model for interactions between soil temperature and soil moisture accounted for >70% of the variance in soil respiration in semi-arid grasslands. Jia et al. (2007), however, introduced the twice-linear regression model with soil moisture and air temperature as combined factors in modelling soil respiration in a semi-arid steppe. Nevertheless, we conclude that soil temperature and soil moisture were primary factors affecting soil respiration in semi-arid grassland, and the performance of estimation would improve greatly with the addition of more variables to the classical statistical models.

State-space models of soil respiration

The autoregressive state-space analysis quantifies how strongly Rs at time i was temporally based on Rs and other influencing factors at time i - 1. For comparing the quality of prediction by the state-space models with the classical statistical methods, autoregressive state-space equations were established using [T.sub.0cm], SM, and their interactions (Table 3). The state equations were all significant (P < 0.01).

Coefficients of determination ([R.sup.2]) indicated that the state-space model that included [T.sub.0cm] could represent ~95% of the variance in Rs, which was relatively better than the models that included SM ([R.sup.2] = 0.84) and the combination of [T.sub.0cm] and SM ([R.sup.2] = 0.94). This result suggested that the performance of estimating Rs may not improve with an increase in the number of variables in the state-space analysis, which differs from the case with classical statistical methods. Our result agreed with those of Timm et al. (2003a), who showed that estimating performance had no relation to the number of variables included in the state-space models used to evaluate sugarcane production with physical and chemical properties of soils. Jia et al. (2011) also showed that the performance of state-space models was not mediated by the number of variables. Therefore, the state equation established by SM was not used to predict Rs in this study due to its relatively poor performance in estimating Rs (Table 3).

The bivariate state-space equation with [T.sub.0cm] showed that [T.sub.0cm] at time i - 1 contributed 31% to the estimated Rs at time i, while [Rs.sub.i-1] contributed 69% (Fig. 4a), indicating that the contribution of the first neighbour was much larger in the case of Rs compared with that of [T.sub.0cm]. The trivariate statespace equation showed that SM at time i - 1 contributed only 17% to the estimated Rs at time i, smaller than that of [T.sub.i-1] (41%) and [Rs.sub.i-1] (44%) (Fig. 4b). The contribution of [T.sub.0cm] and the weight of the previous Rs observation in trivariate models changed significantly compared with the bivariate state-space equation. The contribution of [T.sub.0cm] increased from 31% to 41%, whereas Rs itself decreased from 69% to 44%. The sharp decrease in the contribution of Rs itself may explain why the performance of estimation is not improved by an increase in the number of variables in the state-space analysis. The 95% confidence interval represented the interval [+ or -] s.d. for each estimated value at time i. The closer the 95% confidence intervals, the better the model estimation (Wendroth et al. 2003). The magnitudes of the 95% confidence intervals, however, varied slightly between the two models (Fig. 4) due to the similar performance in estimating. By comparing the values of contribution and the [R.sup.2], we concluded that [T.sub.0cm] was more important than SM for describing the temporal process of Rs during the entire experimental period.

Prediction of soil respiration with the two approaches To understand how well the results can contribute in a more predictive scenario (i.e. in a case where the main variable of interest, Rs, is not known), transition coefficients from the state equations obtained above should be evaluated via application in ordinary autoregressive models, where only the initial Rs value at time i is given. Furthermore, the predicted results of the state-space models were compared with those from classical statistical methods (Figs 5, 6).

The results indicated that the traditional classical models might accurately predict soil respiration at relatively low temperatures but overestimated soil respiration at higher temperatures with low soil moisture (Fig. 6). The twice-linear regression and exponential-power models that included [T.sub.0cm] and SM accurately predicted Rs in August with moderate temperatures and sufficient soil moisture, indicating that the rates of soil respiration were highly dependent on the combined effects of soil moisture and temperature in August (Fig. 6c,f). In general, predicted Rs from all models, including state-space models and classical regression models, was relatively higher than the measured Rs in July (Figs 5, 6). This result may be attributed to the dry soils and relatively high temperatures (Fig. 1). For example, the trivariate state-space model that included [T.sub.0cm] and SM behaved poorly during days 12-35 when observed Rs decreased while predicted Rs was almost constant, even considering the soil moisture influence in the model (Fig. 5b). This could be explained by the much larger contribution of soil temperature (41%) than soil moisture (17%). The state-space model with a larger contribution of soil temperature may not predict soil respiration time series well during the dry period. Therefore, it is important to establish predictive models in different growing periods to more accurately study the C[O.sub.2] flux in a region with substantial differences between dry and wet seasons (e.g. the northern China).

The predicted results of the state-space models were compared with those from the classical statistical methods by calculating RMSE values for examining the overall application of this approach. Autoregressive Rs predictions were given for [T.sub.0cm] and the combination of [T.sub.0cm] and the SM state-vector. The RMSE between prediction and observation was 0.192 (Fig. 5a) if transition coefficients were used from the scenario using only [T.sub.0cm], and was 0.199 (Fig. 5b) if based on the scenario using the combination of [T.sub.0cm] and SM. The predicted results obtained by the classical models showed that an exponential-power model including [T.sub.0cm] and SM had a relatively higher quality of prediction (RMSE 0.372) (Fig. 6f), followed by stepwise regression (RMSE 0.405) (Fig. 6b), power (RMSE 0.451) (Fig. 6e), quadratic (RMSE 0.452) (Fig. 6d), twice-linear regression (RMSE 0.458) (Fig. 6c), and exponential (RMSE 0.478) (Fig. 6a) models. In general, the RMSE values indicated that the state-space prediction of soil respiration time series using any combination of soil temperature and/or soil moisture was much better than the equivalent linear- and multiple-regression equations. The state-space models performed better because the classical linear- and multiple-regression methods ignored the local behavioural tendencies of observations (i.e. assuming independence of the data) (Nielsen and Alemi 1989; Wendroth et al. 1999). In addition, soil surface temperature was identified in our study as being an important state variable for predicting soil respiration time series during the experimental period, because it performed better than the state-space model that did not include soil temperature (Table 3). We therefore concluded that the state-space method was consistently more effective than the classical statistical methods for predicting soil respiration time series. We also concluded that soil surface temperature made effective contributions to describing the temporal distribution of soil respiration in temperate, semi-arid grassland in northern China.

Our results indicate that autoregressive state-space models provide a quantitatively meaningful means of predicting soil respiration time series. The state-space approach presented here was based on an empirical analysis of temporal processes underlying variation in soil respiration, which had not previously included the dynamics of plant phenology or the mechanisms of how environmental factors influence the temporal process of soil respiration. When the temporal factor is scaled up, the roles of other biotic factors (e.g. vegetative biomass, plant photosynthesis, leaf area index, etc.) may become more important than abiotic factors (e.g. soil temperature, soil moisture, etc.) in affecting soil respiration and should be included in future predictive models.

Conclusions

We present an initial study on the feasibility of applying state-space models to the prediction of soil respiration in semi-arid grassland. State-space analyses and classical multiple regressions resulted in different predictions of soil respiration time series by soil temperature and soil moisture. All established state-space models predicted soil respiration better than the classical statistics, indicating that the state-space models were helpful in understanding the association between soil respiration and influencing factors. As revealed by the state-space analysis, soil surface temperature was a strong predictor of soil respiration during the experimental period. The simple autoregressive state-space model, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], explained ~95% of the variation in soil respiration and may be used to predict future emissions of soil C[O.sub.2] at the study site.

Our study demonstrated the regulation of the plant photosynthetic rate on soil respiration; the question thus remains open of how to integrate biogeochemical equations into state-space models for clarifying the principles of the impact on the temporal process of soil respiration. The temporal scale of soil respiration and soil-based variables was the focus of this investigation. Future work should address different temporal and spatial scales in the study of the complex relationships among soil respiration, various biotic factors, and the physical and chemical properties of soil, employing state-space techniques for the accurate prediction of soil respiration.

doi.org/ 10.1071/SR12068

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 40801111), the Science and Technology Development Research Program in Shaanxi Province (2011kjxx25), and the Program for Youthful Talents in North-west A & F University. The authors are indebted to the editors of the journal and the reviewers for their constructive comments and suggestions that greatly improved the earlier version of this manuscript.

Received 15 March 2012, accepted 24 May 2012, published online 3 July 2012

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Xiaoxu Jia (A), Ming'an Shao (B,C), and Xiaorong Wei (A)

(A) State Key Laboratory of Soil Erosion and Dryland Farming on the Loess Plateau, Northwest A & F University, Yangling 712100, China.

(B) Key Laboratory of Ecosystem Network Observation and Modeling, Institute of Geographical Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China.

(C) Corresponding author. Email: mashao@ms.iswc.ac.cn

Soil is a major biospheric reservoir for carbon (C), globally containing twice as much C as the atmosphere and three times as much as vegetation (Raich and Potter 1995). Small changes in the size of the C pool in soils across large areas could therefore produce a large effect on the concentrations of atmospheric C[O.sub.2] and ultimately on global climate change (Grace and Rayment 2000; Granier et al. 2000; Schlesinger and Andraws 2000; Rodeghiero and Cescatti 2005). The accurate evaluation and prediction of levels of soil respiration are thus of great concern for research on global change.

Soil respiration relies on a series of processes with complex interactions and feedback among its components and influencing factors. The rate of soil respiration is controlled primarily by the rate of C[O.sub.2] production by biota within the soil, but is modified by factors influencing the movement of C[O.sub.2] out of the soil (Raich and Schlesinger 1992; Tufekcioglu et al. 2001). To understand the regulation of soil respiration by environmental factors and to quantify soil respiration in the global C budget, researchers have developed many models of soil respiration (Cook et al. 1998). Temperature (air temperature and soil temperature at different depths) and soil moisture are usually considered the most influential environmental factors controlling soil respiration, and some types of equations can be fitted to represent the relationship (e.g. linear, quadratic, power, and exponential models, etc.) (Kucera and Kirkham 1971; Peterson and Billings 1975; Schlentner and Van Cleve 1985; Raich and Potter 1995; Chen et al. 2003; Jia et al. 2006). Some of these models developed for soil temperature and moisture are good predictors of the temporal variation of soil respiration (Table 1). However, these models, based on traditional statistical tools (ANOVA and regression), do not consider the temporal/spatial coordinates of the observations and assume that observations are temporally/spatially independent of each other and have no correlation structure. The prediction of temporal patterns of soil respiration is highly affected by soil properties (i.e. soil moisture, temperature, etc.) and also by the state of soil C[O.sub.2] itself. Developing better predictive models of soil respiration through new statistical methods has become one of the important issues in the study of the global C cycle.

Autoregressive state-space methods have been shown to provide opportunities for suitable identification of temporal/ spatial relationships between soils and their vegetation, taking into account their temporal/spatial associations (Nielsen et al. 1994; Wendroth et al. 1999, 2003; Timm et al. 2003a, 2003b; Jia et al. 2011). State-space modelling is a technique that can filter noise underlying vegetative and soil processes at various scales if the density of observation supports the identification of the correlation length (Timm et al. 2003a). State-space modelling has also been demonstrated to be an effective research tool for quantifying localised variation (Wendroth et al. 2003; Timm et al. 2003a, 2003b). However, no reports have addressed the performance of state-space methods applied to the study of the soil respiration time series in a semi-arid grassland ecosystem. We have thus applied a state-space analysis to identify the underlying variables involved in soil respiration in semi-arid grassland in northern China. The emphasis for applying state-space modelling in this study, as in many current spatial and temporal data series, is to identify the underlying processes of one-dimensional series of observations (Nielsen et al. 1994; Wendroth et al. 2003) and to estimate model parameters based on maximum likelihood (Shumway 1988) for predicting future emissions of soil C[O.sub.2] at the study site.

We therefore investigated the effects of soil moisture, temperature (i.e. air temperature and soil temperature at the surface and at a depth of 5 cm), air pressure, and relative humidity on the rate of soil respiration in temperate, semi-arid grassland from July to October 2010 and analysed the relationships between soil respiration and environmental factors. Our specific objectives were to: (1) analyse the primary factors influencing soil respiration, (2) establish models of soil respiration with state-space and classical regression methods, and (3) compare the predictions obtained by the state-space methods with those from the classical regression methods.

Materials and methods

Study area

The study was conducted at the Shenmu Erosion and Environment Research Station of the Institute of Soil and Water Conservation, in the Liudaogou catchment in Shenmu County, Shaanxi Province (38[degrees]46'-38[degrees]51'N, 110[degrees]21'-110[degrees]23'E; elevation 1094-1274 m above sea level). This region belongs to a moderate-temperate and semi-arid zone with an average annual precipitation of 430 mm, which occurs mostly in July and August. The mean annual temperature is 8.4[degrees]C, and mean annual pan evaporation reaches 785 mm [year.sup.-1]. The study area is representative of the transitional belt subject to severe wind and water erosion. There are two main soil types in this catchment, a loessial and an aeolian sandy soil. The study area is under three main land-use types that cover >86% of the total area, i.e. farmland (16%), grassland (44%), and shrubland (26%). The remaining area is covered by wasteland, gully channels, and manmade structures.

Experimental design and field measurements

Experimental design

Two typical grasslands were selected for the study. The selected grasslands are in a level area with good drainage and a relatively low percentage of surface stones. The dominant plant species in the grasslands is purple alfalfa (Medicago sativa), which represented >95% of the sum peak above-ground biomass of this ecosystem during the experimental period.

Three plots 2 m by 2 m were established in each selected grassland. We collected two sets of data consisting of the rate of soil respiration, air temperature, soil surface temperature (0 cm), soil temperature at a depth of 5 cm, soil moisture in the top layer (6 cm in depth), air pressure, and relative humidity from the two grasslands. One group of data was used to evaluate the temporal effects of factors influencing soil respiration during the growing season and to establish models for predicting soil respiration with autoregressive state-space methods and classical statistical methods. The other group was used to compare the quality of prediction of the statespace and classical statistical models for assessing the validity and robustness of the state-space approach in studying the temporal relationships between soil respiration and influencing factors.

Soil respiration and influencing factors

Soil respiration (Rs) was measured using the CI-340 Photosynthesis System fitted with a CI-301SR Soil Respiration Chamber (CID Bio-Science, Inc., Camas, WA, USA). Three polyvinyl chloride (PVC) collars (11 cm inside diameter, 8 cm in height) were carefully pressed into the soil to a depth of 3 cm at three randomly selected positions in each plot. The mean value for each plot (three PVC collars) was considered as a replicate. Measurements were performed by placing the soil respiration chamber on the PVC collars in each plot.

Measurements of soil respiration were performed every 1-3 days (avoiding rainy days) between 1 July and 21 October 2010. Measurements were made during the day, from ~09:00 to 11 : 00 (local time), for 1-2 min on each collar before moving to the next collar; midday values of the rate of soil respiration are representative of daily averages (Davidson et al. 1998). During the growing season from July to October 2010, we measured soil respiration a total of 53 times, and the average of the measurements for each month was defined as the monthly mean soil respiration.

To evaluate the effect of net photosynthetic rate on soil respiration, we measured the photosynthesis of leaves of Medicago sativa three times per month using the CI-340 Ultra-Light Portable Photosynthesis System (CID BioScience, Inc., Camas, WA, USA) with a 6.5-[cm.sup.2], clamp-on leaf cuvette in each plot throughout the growing season. The measurements were taken between 09:00 and 10:00 (local time) on clear days. At each time of measurement, five healthy, fully expanded, representative leaves in each plot were randomly selected for measurements of photosynthesis, and the five values were averaged as one replicate. The average of three replicates in each month was defined as the monthly mean photosynthetic rate.

The volumetric soil moisture in the upper 6cm was monitored concurrently with the measurements of soil respiration next to the collars but outside the cores to avoid disturbance with the conductivity probe (theta probe ML2x, Delta-T Devices Ltd, Cambridge, UK). Soil temperature at a depth of 5 cm ([T.sub.5cm]) was monitored using a geothermometer inserted into the soil in the vicinity of the soil collars. The temperature of the soil surface ([T.sub.0cm]) and air temperature were automatically recorded with a CI-310TS IR Temperature Sensor and an air-temperature sensor (thermocouple) connected to the CI-340, respectively. The CI-340 uses a humidity-sensitive capacitor to measure relative humidity. Soil moisture, soil and air temperatures, air pressure, and relative humidity were measured continuously in the same area as the measurements of soil respiration (53 observations).

Theory

The state-space model of a stochastic process involving observed datasets, all collected at the same locations, is based on the property of Markovian systems that establishes the independence of the future of the process in relation to its past, once given the present state. In these systems, the state of the process condenses all information of the past needed to forecast the future (Timm et al. 2003b).

State-space models generally consist of two systems of equations (state equation and observation equation) to describe the relationship between the input and output of a dynamic system (Shumway 1988; Wendroth et al. 1997). In this case, the state equation is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

where [X.sub.i] is the non-observed state vector, [PHI] is a p x p matrix of state coefficients, and [[omega].sub.i] is the model error vector. The state equation describes how the state vector [X.sub.i] at time (or space) i is related to that at time (or space) i - 1 through the state coefficient matrix [PHI] (transition matrix) and an error associated with the state [[omega].sub.i] with the structure of a first-order autoregressive model. The model error term COl is a zero-mean, uncorrelated, and normally distributed noise with an r x r covariance matrix Q, the latter being variance per unit time (or space), and depending on the interval between observations. In the case of the state-space model, however, the state-equation is solved simultaneously with the observation equation:

[Y.sub.i] = M[X.sub.i] + [v.sub.i] (2)

where the observation vector [Y.sub.i] is related to the state vector [X.sub.i] through the observation matrix M (usually known as an identity matrix) and by the observation noise vector [v.sub.i], also considered to be zero-mean, uncorrelated, and normally distributed. The noises [[omega].sub.i] and [v.sub.i] are assumed to be independent of each other. For further details, see Shumway (1988), Nielsen et al. (1994), and Nielsen and Wendroth (2003).

Before applying a state-space analysis, data were normalised with respect to their mean m and standard deviation s by the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

where the scaled value [z.sub.i] of the measured observation [Z.sub.i] is calculated based on the mean m and the standard deviation s (Wendroth et al. 1999). This scaling procedure allows state coefficients of having magnitudes directly proportional to their contribution to each state variable included in the statespace analysis (Hui et al. 1998).

Statistical analyses

The data for soil respiration (53 observations) were analysed with both autoregressive state-space and classical regression methods. The data for soil respiration and influencing factors from consecutive measurements were analysed as if they were equally distributed and belonging to the same interval class (1-3 days). The direction of statistical analyses followed the sampling pattern from July to October.

Temporal relationships among variables had to be examined before estimating Rs with the state-space method. Therefore, we calculated autocorrelation functions (ACF) and cross-correlation functions (CCF) (Timm et al. 2003a, 2003b). If sets of transect data were autocorrelated and cross-correlated, the state-space models could be carried out (Nielsen and Wendroth 2003). The estimated values of soil respiration and the [PHI] coefficients of the state equation (Eqn 1) were obtained with the software Applied Statistical Time Series Analysis (ASTSA), developed by Shumway (1988). [Z.sub.i] data were scaled using Eqn 3. The classical methods were performed with SPSS v 16.0 (SPSS for Windows, Chicago, IL, USA). For evaluating the quality of estimation, the coefficients of determination ([R.sup.2]) from linear regressions between estimated and measured values of Rs were used in the study (all observations have been scaled using Eqn 3).

Model validation

To assess the quality of prediction of state-space models, the transition coefficients were applied in simple autoregressive predictions, where only the first value of soil respiration in the series is known, and all following values are calculated from the one previous and those from the underlying variables included in the respective state-vector. As a criterion for the quality of prediction, the root mean-squared error (RMSE) is determined from the measured soil respiration ([Rs.sub.meas]) and predicted soil respiration ([Rs.sub.prcd]). The RMSE is an accuracy measure of the prediction and should be as small as possible for accurate prediction. The RMSE was calculated by the following equation:

Results and discussion

Temporal patterns of soil respiration and influencing factors

Soil respiration exhibited a pronounced temporal trend from July to October 2010 (Fig. la), ranging from 0.29 to 2.40 [micro]mol[m.sup.-2][s.sup.-1], with a mean rate of 1.12[+ or -]0.45 [micro]mol [m.sup.-2][s.sup.-1] (mean [+ or -] s.d.). Monthly Rs increased through July and peaked in August, followed by a gradual decrease during September and October (data not shown). Our results are in agreement with Qi et al. (2008) showing that the seasonal patterns of soil respiration in semi-arid grassland were characterised by single-peak curves, which had the highest values in August. This finding may be ascribed to plant growth and/or soil microclimates. Hogberg et al. (2001) and Tang et al. (2005) showed that plant photosynthetic rate modulated soil respiration. Average plant net photosynthetic rate was significantly higher in August (38.7 [+ or -] 6.2 [micro]mol [m.sup.-2][s.sup.1]) than in the other three months and decreased through September (22.1 [+ or -] 3.8 [micro]mol[m.sup.-2][s.sup.-1]) and October (8.4 [+ or -] 1.2 [micro]mol [m.sup.-2][s.sup.-1]). However, no significant differences in photosynthetic rates or soil respiration were observed between July (25.2 [+ or -] 4.8 [micro]mol[m.sup.2][s.sup.1]) and September. Furthermore, soil microclimates (e.g. soil moisture and soil temperature) in August (Fig. lb, c) were more suitable for the activities of soil microbes and roots than in other months, thus resulting in higher emissions of C[O.sub.2] (Howard and Howard 1993; Qi et al. 2008).

Soil respiration originates mainly from root and microbial activity. Howard and Howard (1993) indicated that both components were strongly influenced by available soil moisture and temperature. This study focussed on the relationships among Rs, soil moisture (SM), air temperature (AT), [T.sub.0cm], [T.sub.5cm], air pressure (AP), and relative humidity (RH) to determine the crucial factors affecting temporal patterns of soil respiration at the study site. Exponential functions described the relationships between Rs and AT ([R.sup.2] = 0.38, P < 0.01), [T.sub.0cm] ([R.sup.2] = 0.44, P < 0.01), and [T.sub.5cm] ([R.sup.2] = 0.40, P < 0.01) to examine the effects of temperature on soil respiration during the experimental period. [T.sub.0cm] was able to explain 44% of the variation in Rs, more than did AT and [T.sub.5cm], indicating that soil surface temperature was a major factor influencing soil respiration. Fitted relationships of Rs with SM, AP, and RH had [R.sup.2] values of 0.13 (P<0.05), 0.10 (P<0.05), and 0.02 (P>0.05), respectively. Soil respiration was significantly positively correlated with temperature and soil moisture and negatively correlated with air pressure (Table 2), showing that soil respiration was inhibited by increasing air pressure (Jia et al. 2007). No significant relationship was observed between soil respiration and relative humidity. Soil respiration was therefore mainly affected by soil surface temperature, followed by soil moisture and air pressure, irrespective of relative humidity. Therefore, [T.sub.0cm], SM, and AP were used to predict Rs by the state-space models and classical statistical methods. The coefficient of variation (CV) values for Rs and other variables at the study site ranged from 0.6% to 41% during the experimental period. Variation in AP had the lowest CV and variation in AT and [T.sub.5cm] had the highest CV (Table 2). All of these variables exhibited point-to-point fluctuations due to temporal variation, which represents typical local characteristics, and may therefore be better represented by a local model (e.g. state-space model) (Timm et al. 2003a).

Auto- and cross-correlation functions

To understand how soil moisture, temperature, and air pressure affect soil respiration, we examined their temporal relationships. We calculated the ACF for each dataset (i.e. Rs, [T.sub.0cm], SM, and AP; Fig. 2a-d, respectively), with the objective of evaluating the temporal correlation of the observations, i.e. if they had been monitored for a time interval sufficient for identifying their temporal representativity. Using a t-test at P = 0.05, the ACF of the Rs and [T.sub.0cm] data (Fig. 2a, b) manifested significant temporal correlations up to eight lags. Temporal dependence between adjacent observations of AP (Fig. 2d) was significant up to seven lags; the ACF of SM (Fig. 2c) showed temporal correlation only up to two lags because it had no discernible trend. The ACF is a tool that reflects the local variation between samples for different distances of separation and is used to identify the range of the spatial/temporal correlations in the observations of a variable (Journel and Huijbregts 1991). The ACF indicated that the sampling-time intervals of these variables were sufficient to identify their temporal representativity.

For describing the spatial/temporal degree of linkage between two variables, the CCF can be used (Shumway 1988; Wendroth et al. 1997). In our study, CCF was calculated to analyse the temporal cross-correlation structures between Rs and [T.sub.0cm], SM, and AP (Fig. 3a-c, respectively). At a 95% confidence level, Rs was positively cross-correlated with [T.sub.0cm] and SM and negatively cross-correlated with AP. Using the t-test at P=0.05, the cross-correlogram between Rs and [T.sub.0cm] showed a strong temporal dependence (Fig. 3a). Similar results were found for the cross-correlogram between Rs and AP (Fig. 3c). The cross-correlogram between Rs and SM (Fig. 3b), however, showed a weak temporal dependence. The CCF of Rs and the significant influencing factors manifested the potential for describing their temporal distributions across the experimental period through state-space analysis, verifying that the use of the CCF leads to additional information on the temporal variability of soil respiration.

Classical regression models of soil respiration

With 53 observations throughout the entire experimental period, stepwise linear regression was used to determine the most significant factors affecting Rs, including Toxin, SM, and AP. The effects of [T.sub.0cm] and SM on Rs were the most important, and their coefficients were significant. Therefore, all classical models of Rs were established with [T.sub.0cm] SM, and their interactions (Table 3). This study was mainly focussed on exponential, stepwise linear regression, twice-linear regression, quadratic, power, and exponential-power models, because they had been widely used to estimate the efflux of soil C[O.sub.2] with soil temperature and soil moisture (Kucera and Kirkham 1971; Peterson and Billings 1975; Davidson et al. 1998; Davidson et al. 2000; Li et al. 2000; Wang et al. 2003; Jia et al. 2006, 2007; Qi et al. 2008). The classical equations were all significant (P < 0.05).

When [T.sub.0cm] was considered as an independent controlling factor, the quadratic model ([R.sup.2] = 0.54) performed best for estimating Rs, compared with the exponential ([R.sup.2] = 0.44) and power ([R.sup.2] = 0.39) models. Considering the relatively smaller contribution of SM to the temporal variation of Rs throughout the entire experimental period, SM was not analysed as an independent factor in predicting Rs. Notably, the [R.sup.2] values indicated that >60% of the variance of Rs was explained by the combination of [T.sub.0cm] and SM in the classical models, in particular the exponential-power model (~70%) (Table 3). This result supported the findings of Qi et al. (2008), who showed that the exponential-power model for interactions between soil temperature and soil moisture accounted for >70% of the variance in soil respiration in semi-arid grasslands. Jia et al. (2007), however, introduced the twice-linear regression model with soil moisture and air temperature as combined factors in modelling soil respiration in a semi-arid steppe. Nevertheless, we conclude that soil temperature and soil moisture were primary factors affecting soil respiration in semi-arid grassland, and the performance of estimation would improve greatly with the addition of more variables to the classical statistical models.

State-space models of soil respiration

The autoregressive state-space analysis quantifies how strongly Rs at time i was temporally based on Rs and other influencing factors at time i - 1. For comparing the quality of prediction by the state-space models with the classical statistical methods, autoregressive state-space equations were established using [T.sub.0cm], SM, and their interactions (Table 3). The state equations were all significant (P < 0.01).

Coefficients of determination ([R.sup.2]) indicated that the state-space model that included [T.sub.0cm] could represent ~95% of the variance in Rs, which was relatively better than the models that included SM ([R.sup.2] = 0.84) and the combination of [T.sub.0cm] and SM ([R.sup.2] = 0.94). This result suggested that the performance of estimating Rs may not improve with an increase in the number of variables in the state-space analysis, which differs from the case with classical statistical methods. Our result agreed with those of Timm et al. (2003a), who showed that estimating performance had no relation to the number of variables included in the state-space models used to evaluate sugarcane production with physical and chemical properties of soils. Jia et al. (2011) also showed that the performance of state-space models was not mediated by the number of variables. Therefore, the state equation established by SM was not used to predict Rs in this study due to its relatively poor performance in estimating Rs (Table 3).

The bivariate state-space equation with [T.sub.0cm] showed that [T.sub.0cm] at time i - 1 contributed 31% to the estimated Rs at time i, while [Rs.sub.i-1] contributed 69% (Fig. 4a), indicating that the contribution of the first neighbour was much larger in the case of Rs compared with that of [T.sub.0cm]. The trivariate statespace equation showed that SM at time i - 1 contributed only 17% to the estimated Rs at time i, smaller than that of [T.sub.i-1] (41%) and [Rs.sub.i-1] (44%) (Fig. 4b). The contribution of [T.sub.0cm] and the weight of the previous Rs observation in trivariate models changed significantly compared with the bivariate state-space equation. The contribution of [T.sub.0cm] increased from 31% to 41%, whereas Rs itself decreased from 69% to 44%. The sharp decrease in the contribution of Rs itself may explain why the performance of estimation is not improved by an increase in the number of variables in the state-space analysis. The 95% confidence interval represented the interval [+ or -] s.d. for each estimated value at time i. The closer the 95% confidence intervals, the better the model estimation (Wendroth et al. 2003). The magnitudes of the 95% confidence intervals, however, varied slightly between the two models (Fig. 4) due to the similar performance in estimating. By comparing the values of contribution and the [R.sup.2], we concluded that [T.sub.0cm] was more important than SM for describing the temporal process of Rs during the entire experimental period.

Prediction of soil respiration with the two approaches To understand how well the results can contribute in a more predictive scenario (i.e. in a case where the main variable of interest, Rs, is not known), transition coefficients from the state equations obtained above should be evaluated via application in ordinary autoregressive models, where only the initial Rs value at time i is given. Furthermore, the predicted results of the state-space models were compared with those from classical statistical methods (Figs 5, 6).

The results indicated that the traditional classical models might accurately predict soil respiration at relatively low temperatures but overestimated soil respiration at higher temperatures with low soil moisture (Fig. 6). The twice-linear regression and exponential-power models that included [T.sub.0cm] and SM accurately predicted Rs in August with moderate temperatures and sufficient soil moisture, indicating that the rates of soil respiration were highly dependent on the combined effects of soil moisture and temperature in August (Fig. 6c,f). In general, predicted Rs from all models, including state-space models and classical regression models, was relatively higher than the measured Rs in July (Figs 5, 6). This result may be attributed to the dry soils and relatively high temperatures (Fig. 1). For example, the trivariate state-space model that included [T.sub.0cm] and SM behaved poorly during days 12-35 when observed Rs decreased while predicted Rs was almost constant, even considering the soil moisture influence in the model (Fig. 5b). This could be explained by the much larger contribution of soil temperature (41%) than soil moisture (17%). The state-space model with a larger contribution of soil temperature may not predict soil respiration time series well during the dry period. Therefore, it is important to establish predictive models in different growing periods to more accurately study the C[O.sub.2] flux in a region with substantial differences between dry and wet seasons (e.g. the northern China).

The predicted results of the state-space models were compared with those from the classical statistical methods by calculating RMSE values for examining the overall application of this approach. Autoregressive Rs predictions were given for [T.sub.0cm] and the combination of [T.sub.0cm] and the SM state-vector. The RMSE between prediction and observation was 0.192 (Fig. 5a) if transition coefficients were used from the scenario using only [T.sub.0cm], and was 0.199 (Fig. 5b) if based on the scenario using the combination of [T.sub.0cm] and SM. The predicted results obtained by the classical models showed that an exponential-power model including [T.sub.0cm] and SM had a relatively higher quality of prediction (RMSE 0.372) (Fig. 6f), followed by stepwise regression (RMSE 0.405) (Fig. 6b), power (RMSE 0.451) (Fig. 6e), quadratic (RMSE 0.452) (Fig. 6d), twice-linear regression (RMSE 0.458) (Fig. 6c), and exponential (RMSE 0.478) (Fig. 6a) models. In general, the RMSE values indicated that the state-space prediction of soil respiration time series using any combination of soil temperature and/or soil moisture was much better than the equivalent linear- and multiple-regression equations. The state-space models performed better because the classical linear- and multiple-regression methods ignored the local behavioural tendencies of observations (i.e. assuming independence of the data) (Nielsen and Alemi 1989; Wendroth et al. 1999). In addition, soil surface temperature was identified in our study as being an important state variable for predicting soil respiration time series during the experimental period, because it performed better than the state-space model that did not include soil temperature (Table 3). We therefore concluded that the state-space method was consistently more effective than the classical statistical methods for predicting soil respiration time series. We also concluded that soil surface temperature made effective contributions to describing the temporal distribution of soil respiration in temperate, semi-arid grassland in northern China.

Our results indicate that autoregressive state-space models provide a quantitatively meaningful means of predicting soil respiration time series. The state-space approach presented here was based on an empirical analysis of temporal processes underlying variation in soil respiration, which had not previously included the dynamics of plant phenology or the mechanisms of how environmental factors influence the temporal process of soil respiration. When the temporal factor is scaled up, the roles of other biotic factors (e.g. vegetative biomass, plant photosynthesis, leaf area index, etc.) may become more important than abiotic factors (e.g. soil temperature, soil moisture, etc.) in affecting soil respiration and should be included in future predictive models.

Conclusions

We present an initial study on the feasibility of applying state-space models to the prediction of soil respiration in semi-arid grassland. State-space analyses and classical multiple regressions resulted in different predictions of soil respiration time series by soil temperature and soil moisture. All established state-space models predicted soil respiration better than the classical statistics, indicating that the state-space models were helpful in understanding the association between soil respiration and influencing factors. As revealed by the state-space analysis, soil surface temperature was a strong predictor of soil respiration during the experimental period. The simple autoregressive state-space model, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], explained ~95% of the variation in soil respiration and may be used to predict future emissions of soil C[O.sub.2] at the study site.

Our study demonstrated the regulation of the plant photosynthetic rate on soil respiration; the question thus remains open of how to integrate biogeochemical equations into state-space models for clarifying the principles of the impact on the temporal process of soil respiration. The temporal scale of soil respiration and soil-based variables was the focus of this investigation. Future work should address different temporal and spatial scales in the study of the complex relationships among soil respiration, various biotic factors, and the physical and chemical properties of soil, employing state-space techniques for the accurate prediction of soil respiration.

doi.org/ 10.1071/SR12068

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 40801111), the Science and Technology Development Research Program in Shaanxi Province (2011kjxx25), and the Program for Youthful Talents in North-west A & F University. The authors are indebted to the editors of the journal and the reviewers for their constructive comments and suggestions that greatly improved the earlier version of this manuscript.

Received 15 March 2012, accepted 24 May 2012, published online 3 July 2012

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Xiaoxu Jia (A), Ming'an Shao (B,C), and Xiaorong Wei (A)

(A) State Key Laboratory of Soil Erosion and Dryland Farming on the Loess Plateau, Northwest A & F University, Yangling 712100, China.

(B) Key Laboratory of Ecosystem Network Observation and Modeling, Institute of Geographical Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China.

(C) Corresponding author. Email: mashao@ms.iswc.ac.cn

Table 1. Selected classical statistical functions relating soil respiration rate to soil moisture, temperature, and temperature-moisture interaction Equation Example Vegetation Reference Exponential: Rs ([micro]mol C[O.sub.2] Peatlands, Chimner Rs = [ae.sup.bT] [m.sup.-2][s.sup.-1]) = Forest, (2004) 0.375[e.sup.0.066T], [R.sup.2] = 0.46 (T, soil temp.) Rs (g C [m.sup.-2] Mixed Liu and [year.sup.-1]) = hardwood Fang 349.66[e.sup.0.0449T], forest (1997) [R.sup.2] = 0.47 (T, air temp.) Rs (g C[O.sub.2] Leymus Kang et [m.sup.-2][h.sup.-1]) = chinensis al. 0.14[e.sup.0.113T], steppe (2003) [R.sup.2] = 0.75 (T, mean annual temp.) Linear regression: Rs (g C[O.sub.2] Leymus Wang et Rs = a + bT + cSM [m.sup.-2][day.sup.-1]) = chinensis al. 0.715 + 0.21T steppe (2003) + 0.285[P.sub.3-1] + 0.083[P.sub.7-4], [R.sup.2] = 0.64 (T, air temp.; [P.sub.3-1], [P.sub.7-4], precipitation at 3 days and 7-4 days before experiment, cm) Twice linear Rs (mg C[O.sub.2] Leymus Jia et regression: [m.sup.-2][h.sup.-1]) = chinensis al. Rs = a + bSM * T -109.09 + 1.68SM * T, steppe (2007) [R.sup.2] = 0.83 (T, air temp.; SM, soil moisture at 0-10 cm depth, %) Quadratic: RS (mg C[O.sub.2] Tundra Peterson Rs = a + bT + [m.sup.-2][h.sup.-1]) = and [cT.sup.2] 89.78 + 1.54T + Billings 5[T.sup.2], [R.sup.2] = (1975) 0.83 (T, mean daily temp.) Power: Rs = Rs (mg C [m.sup.-2] Leymus Li et [aT.sup.b] [day.sup.-1]) = chinensis al. Rs = a 256.75[T.sup.0.6557], steppe (2000) [(T + 10).sup.b] [R.sup.2] = 0.58 (T, air temp.) LnRs (mg C[O.sub.2] Tallgrass Kucera [m.sup.-2][h.sup.-1]) = prairie and -1.66 + 2.20 Ln(T + Kirkham 10), [R.sup.2] = 0.89 (1971) (T, soil temp.) Exponential-power: Rs (mg C [m.sup.-2] Degraded Chen et Rs = [ae.sup.bT] [day.sup.-1]) = steppe al. * [SM.sup.c] 5911.65[e.sup.0.04T] * (2003) [SM.sup.0.91], [R.sup.2] = 0.86 (T, air temp.; SM, gravimetric water content at 10-20 cm depth, %) Rs ([micro]mol Semi-arid Qi et C[O.sub.2][m.sup.-2] grassland al. [s.sup.-1]) = (2008) 0.1512[e.sup.0.053T] * [SM.sup.0.427], [R.sup.2] = 0.76 (T, soil temp. at 0-10 cm depth; SM, volumetric water content at 0-6 cm depth, %) Table 2. Correlation coefficients between soil respiration (Rs) and influencing factors AT, Air temperature; [T.sub.0cm] soil surface temperature; [T.sub.5cm] soil temperature at depth 5 cm; SM, soil moisture; AP, air pressure; RH, relative humidity. Mean, standard deviation, coefficient of variation (CV, (s.d./mean) * 100), and correlation coefficient (R) of each variable are presented. * P<0.05; ** P<0.01 Variables Mean s.d. CV (%) R RS([micro]mol 1.12 0.45 40 1 [m.sup.-2] [s.sup.-1]) AT ([degrees]C) 21.9 9.08 41 0.588 ** [T.sub.0cm] 21.8 7.9 36 0.618 ** ([degrees]C) [T.sub.5cm] 20.5 8.45 41 0.594 ** ([degrees]C) SM ([10.sup.-2] 14.5 4.38 30 0.355 * [m.sup.-3] [m.sup.-3]) AP (kPa) 87.9 0.54 0.6 -0.310 * RH (%) 65.5 22.1 34 -0.12 Table 3. State-space models and selected classical statistical regression models with surface soil temperature ([T.sub.0cm]), soil moisture (SM), and their combinations, respectively [R.sup.2], Coefficient of determination, from linear regression between estimated and measured values of soil respiration (Rs); [[omega]sub.i];, model noise; **P<0.01 Models [R.sup.2] P State-space models [Rs.sub.i] = 0.69 * [Rs.sub.i-1] + 0.31 * 0.95 ** [T.sub.0cmi-1] + [[omega].sub.i] [Rs.sub.i] = 0.44 * [Rs.sub.i-1] + 0.41 * 0.94 ** [T.sub.0cmi-1] + 0.17 * [SM.sub.i-1] + [[omega].sub.i] [Rs.sub.i] = 1.07 * [Rs.sub.i-1] + 0.10 * 0.84 ** [SM.sub.i-1] + [[omega].sub.i] Exponential model Rs = 0.446 * [e.sup.0.038T0cm] 0.44 ** Stepwise regression model Rs = 0.048 * [T.sub.0cm] + 0.056 * 0.63 ** SM - 0.756 Twice-linear regression model Rs = 0.003 * SM * [T.sub.0cm] + 0.320 0.62 ** Quadratic model Rs = 0.504 + 0.130 * [T.sub.0cm] - 0.54 ** 0.002 * [T.sub.0cm.sup.2] Power model Rs = 0.049 * [([T.sub.0cm] + 0.39 ** 10).sup.0.904] Exponential-power model Rs = 0.090 * [e.sup.0.047T0cm] * 0.69 ** [SM.sup.0.554]

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Author: | Jia, Xiaoxu; Shao, Ming'an; Wei, Xiaorong |
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Publication: | Soil Research |

Article Type: | Report |

Geographic Code: | 9CHIN |

Date: | Jul 1, 2012 |

Words: | 7172 |

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