Integration of CO₂ flux and remotely-sensed data for primary production and ecosystem respiration analyses in the Northern Great Plains: potential for quantitative spatial extrapolation

Modelling

Temporal dynamics of CO₂ in the air between the soil and the CO₂ sensor (in non-forest ecosystems usually located between 1.5 and 4 m above the ground) are the result of the interactions of processes summarized in Fig. 2. Let AirCO₂(t) be the total amount of CO₂ between the soil surface and the sensor; P_g the rate of gross photosynthesis; R_e rate of total ecosystem respiration; and F, the CO₂ flux from atmosphere to ecosystem (when F > 0), or from ecosystem to atmosphere (when F < 0) at the sensor level. Total ecosystem respiration (R_e) is the sum of autotrophic and heterotrophic respiration (R_a and R_h), which in turn include above- and below-ground components. The conservation equation for the amount of CO₂ between the soil surface and the sensor (AirCO₂) becomes:

(1)

From this it immediately follows that the flux of CO₂ from the atmosphere to ecosystem, F, measured by the sensor, is equal to the difference between gross productivity and ecosystem respiration plus the storage term:

(2)

In BREB towers, where CO₂ flux is measured at approximately 1.5 m height, determination of the storage term is not included in the measurement protocol, and the flux F is approximated as:

(3)

In some eddy covariance systems, measurements of the storage term are included (Flanagan et al., 2002), while in other cases the storage term is neglected, and approximation (3) is used similarly to BREB systems (Meyers, 2001). In the following consideration, the term F is assumed to include the storage correction, when available, otherwise approximation (3) will be used.

Because processes controlling CO₂ exchange during daytime (when photon flux density Q > 0) and night-time (Q = 0) are different, for modelling purposes it is convenient to introduce two additional variables, daytime flux, P:

(4)

and daytime ecosystem respiration, R_d. The major equations for subsequent analysis may be formulated as:

(5)

Radiation use efficiencies

In the context of ecosystem-scale analysis, it is important to distinguish between physiological and ecological light-use efficiencies. Physiological coefficient of light-use efficiency of the gross primary productivity, ɛ_phys, is defined as the ratio of gross photosynthetic assimilation, P_g, to absorbed photosynthetically active radiation, Q_a (Larcher, 1995):

(6)

It is the most direct indicator of radiation use efficiency at the level of individual plants. Nevertheless, determination of ɛ_phys at the ecosystem scale is rather complicated because it is difficult to measure all necessary radiation components in the plant canopy. Although approaches to estimate Q_a through both direct and indirect measurements are currently under development (Monteith, 1994; Asner et al., 1998; Sinclair & Muchow, 1999; Asner et al., 2003), serious difficulties of precise estimation of absorbed radiation and, consequently, of ɛ_phys remain (e.g. Demetriades-Shah et al., 1992, 1994).

As an alternative to ɛ_phys, ecological radiation use efficiency, ɛ_ecol, is defined as a ratio of gross productivity to total incoming photosynthetically active radiation (Odum, 1959; Cooper, 1970; Austin et al., 1978; Colinvaux, 1993; Wofsy et al., 1993):

(7)

In contrast to ɛ_phys, ɛ_ecol depends not only on physiological plant properties but also on such ecosystem-scale characteristics as above-ground green biomass, leaf area index, etc. For example, a sparse stand with certain physiological light-use efficiency ɛ_phys will have lower ecological efficiency ɛ_ecol than a more dense stand with the same value of ɛ_phys. Following Wofsy et al. (1993) who described ɛ_ecol as a ‘well defined property of the ecosystem’, in this paper, we will use ecological efficiency to compare light-use efficiency in different ecosystems (later in this paper ecological efficiency will be denoted by ɛ without a subscript).

Light-response functions and their parameterization

The daytime CO₂ flux, P, may be influenced by a variety of factors, however, photosynthetically active radiation, Q, is usually the dominant driver. Our experience, based on the analysis of thousands of daily data sets for different ecosystem types shows that only in special circumstances (drought, extreme heat or cold, nutrient deficiency, pests and disease, etc.) do other factors assume the leading role in determining daytime CO₂ exchange. To describe the general relationship of the daytime flux, P, to photosynthetically active radiation, Q, and, if necessary, its other factors-predictors, {X_i}, we use the concept of the ecosystem-scale light-response function:

(8)

where {a_i, i = 1, … , p} are numerical parameters. We have found that of the great variety of light-response relationships P(Q, …) available in the literature, the nonrectangular hyperbolic function (Prioul & Chartier, 1977):

(9)

and its soil temperature (T_s) dependent modification describing the hysteresis of the radiation-flux relationship:

(10)

are especially convenient tools to describe patterns of light-response of ecosystem-scale fluxes at the 20–30 min time step during non-rainy days (Gilmanov, 2001; Gilmanov et al., 2003a, 2003b, 2004). Interpretation of parameters in equations 9 and 10 are well known: α is the initial slope of the light response curve, β is its plateau parameter, equal to the maximum rate of gross photosynthesis, γ is the respiration term, and θ is the curvature parameter (0 ≤ θ ≤ 1), modifying the shape of light-response curve from hyperbolic at θ = 0 to linear at θ = 1. The coefficient κ in the exponential term of equation 10 describes the strength of the hysteresis of the light-response curve.

Parameters α, β, γ, θ, and κ of equations 9 and 10 that provide the best fit for daytime fluxes were identified using the ‘Global Optimization’ package of the Mathematica® system (Loehle Enterprises, 2001). Examination of seasonal dynamics showed that parameters α, β, γ, and, to a lesser extent, θ, have distinct seasonal patterns, α(t), β(t), γ(t), and θ(t), which can be used for gap-filling purposes.

Estimating R_d from daytime flux measurements

Following Marshall & Biscoe (1980), we estimated daytime ecosystem respiration, R_d, using the γ parameter of the light-response function (9) or (10). Comparing daytime respiration estimates thus obtained with directly measured night-time fluxes (Gilmanov et al., 2003a, 2003b, 2004) demonstrated that light-curve derived estimates of γ are in fair agreement with values measured by tower during night (R_n). Combining R_d estimates with measured R_n values results in 24 h ecosystem respiration values R_e = R_d + R_n. Integration of P(t) from sunrise to sunset for every calendar day j gives daytime production, P_d(j), which combined with daytime respiration integral, R_d(j) leads to estimation of daily gross primary production, P_g(j):

(11)

While the classical rectangular light-response model P(Q) = αQβ/(αQ + β) − γ typically overestimates daytime respiration, R_d, due to its inability to describe the curvature of the P(Q) relationship, estimates from non-rectangular hyperbolic eqn. (10) may also lead to overestimation of R_d because (especially under drought stress) a decrease of P corresponding to the lower branch of the hysteresis loop may in fact be caused not only by an increase in respiration rate due to higher temperature (described by the term ) but also to a decrease in the rate of photosynthetic accumulation, e.g. as the result of stomatal regulation.

Annual CO₂ budgets

Finally, plotting the curves of daily integrals of P_g, P_d, and F on the same graph provides complete characterization of the seasonal dynamics of ecosystem CO₂ exchange (see Fig. 7a–d below). Note that the area between the P_g and F curves is equal to total ecosystem respiration, R_e, while areas between P_g and P_d and P_d and F correspond to daytime and night-time respiration totals, respectively.

Flux–NDVI relationship

During the three-decade long history of studies, considerable progress has been made in understanding the relationships among the extensive (e.g. biomass, leaf area, moisture content) and intensive (e.g. photosynthesis, evapotranspiration, respiration, net primary productivity, net CO₂ exchange) characteristics of terrestrial ecosystems, on the one side, and various remotely sensed variables (indices), on the other side. Compared to early studies relating satellite spectral variables to biomass of various ecosystem types (Tucker et al., 1983; Prince & Tucker, 1986), contemporary researchers are using remotely sensed indices to differentiate a wide array of biophysical properties of vegetation surfaces in terms of quality and quantity of biomass, its phenological status, moisture content, and productivity characteristics. Correlative and functional relationships were established between vegetation indices and fluxes of energy, water, and carbon dioxide in green canopies of different communities (Bartlett et al., 1990; Gamon et al., 1995; Veroustraete et al., 1996; Frank & Karn, 2003; Xiao et al., 2004). Normalized difference vegetation index (NDVI) is defined as (Rouse et al., 1973):

(12)

where R_nir and R_red are reflectances in the near-infrared and red spectral bands, respectively. Yoder & Waring (1994) and Gamon et al. (1995) identified NDVI as quantifying potential photosynthetic activity or an indicator or physiological change at the canopy level. NDVI became the vegetation index most widely used in the context of ecosystem studies because it was shown to be closely related to biomass (Wylie et al., 2002; Boelman et al., 2003), biomass moisture (Chladil & Numez, 1995), leaf area index (Seen et al., 1995; Gower et al., 1999), absorbtion of photosynthetically active radiation (Hall et al., 1995; Gower et al., 1999), trends of photosynthesis and transpiration (Running & Nemani, 1988; Slayback et al., 2003), respiration (Boelman et al., 2003; Wylie et al., 2003) and CO₂ uptake (Frank & Karn, 2003; Wylie et al., 2004). On the other hand, NDVI has been shown to be sensitive to view angle effects (Epiphanio & Huete, 1995), standing dead or litter biomass (Huete & Jackson, 1987), saturation at high LAI (Gao et al., 2000), and soil and atmospheric effects. The Soil Adjusted Vegetation Indices (SAVI) and other SAVI related indices attempt to minimize soil background effects while the Enhanced Vegetation index (EVI) attempts to minimize both atmospheric and soil effects (Huete et al., 1997). However, SAVI was found to be more sensitive to view angle than NDVI (Epiphanio & Huete, 1995). Broge & Leblanc (2001) found NDVI to outperform SAVI on low or moderate levels of LAI, which are more typical for arid and semiarid rangelands. Purevdorj et al. (1998) found both a SAVI index and NDVI to track rangeland vegetation cover the best over a wide range of grass densities and Seen et al. (1995) found NDVI to track LAI well when atmospheric effects were minimized. Similarly, according to unpublished data by Gallo et al. (2004), comparison of MODIS EVI and NDVI at 4347 random US grassland locations, shows a strong linear relationship between the two indices with no increased scatter in NDVI at low index values where soil background effects should be important, and a larger dynamic range of NDVI over EVI. Given the historical NDVI data sets on record and because each index has its own weaknesses, we used NDVI in this study.

Until recently attempts to model the dynamics of the principal components of the ecosystem CO₂ exchange on geographical scales (gross primary production, ecosystem respiration, soil respiration, net primary production, net ecosystem production, etc.) were based largely on net primary production estimates (mostly represented by above-ground production, ANPP), soil respiration, and, since the mid-1990s, net ecosystem CO₂ exchange, F, provided by flux tower networks. Because F is the difference of two large and oppositely directed components, gross photosynthetic assimilation and ecosystem respiration (F = P_g – R_e), it was found difficult to establish relationships between F and remotely-sensed factors and to use net flux data sets to differentiate among various models. Based on the analysis of AVHRR–NDVI data, Box et al. (1989) found little consistent relationship across different vegetation types, between NDVI and net biospheric CO₂ flux. Contemporary literature is abundant with empirical models representing gross primary productivity using the radiation-use efficiency approach when P_g is expressed as a product of P_g,max and factor modifiers representing the ‘effects’ of radiation f₁(Q) or absorbed radiation f_1a(Q_a), air (or soil) temperature f₂(T), relative air humidity or other moisture factor f₃(M) and, possibly, other factors: P_g = P_g,max * f₁(Q) * f₂(T) * f₃(M) * … (e.g. Gilmanov, 1977; Prince & Goward, 1995; Ruimy et al., 1996; Running et al., 1999; Tian et al., 1999; Ciais et al., 2001; Medlyn et al., 2003; Veroustraete et al., 2004). Such models may have limited robustness given high correlations among radiation input, temperature, and humidity; and Choudhury (2001) has reasonably cautioned about possible regional biases. Their apparent ability to generate ‘reasonable’P_g values and maps may be explained by the fact that typically (with few exceptions: Aber et al., 1996; Reich et al., 1999) they were validated only against ANPP data (with monthly or biweekly time step) or at best against net flux {F = P_g – R_e} data sets leaving wide room for calibration of model parameters. Nevertheless, even most recent models of this type occasionally demonstrate errors as high as several hundreds of per cent when compared with the tower flux measurements (Veroustraerte et al., 2004). In contrast, in those rare cases when P_g data were available for validation, multiplicative radiation-use efficiency models show poor agreement with tower flux-based P_g curves (e.g. Ruimy et al., 1996; Turner et al., 2003b).

Our experience with analysis and modelling of the net CO₂ flux data sets from a number of CO₂ flux towers in grasslands and shrublands (Wylie et al., 2003; Gilmanov et al., 2004; Wylie et al., 2004) has also demonstrated rather weak relationships between F and spectral vegetation indices (e.g. NDVI), contrary to the earlier opinion by Bartlett et al. (1990). We found that daytime CO₂ flux integrals (P_d) are more closely correlated with NDVI, than 24-h F integrals (Wylie et al., 2003). With the emergence of the methods for estimation of daytime ecosystem respiration based on daytime flux measurements using light response function analysis allowing net CO₂ flux partitioning into P_g and R_e components (Gilmanov, 2001; Suyker & Verma, 2001; Falge et al., 2002; Gilmanov et al., 2004; Xu & Baldocchi, 2004), the first long-term data sets of P_g and R_e values became available. Analysis of these data sets in relation to vegetation indices revealed significantly higher correlations between P_g and NDVI than between P_d and NDVI (the latter, in its turn, is more highly correlated with NDVI than 24-h net CO₂ flux integral) (Wylie et al., 2003; Gilmanov et al., 2004).

To examine relationships between ecosystem-scale CO₂ exchange and remotely-sensed NDVI we used data from the SPOT VEGETATION dataset integrated over 10-day time steps. More specifically, SPOT VEGETATION NDVI data were represented by temporally smoothed (Swets et al., 1999) 10-day maximum value composite NDVI values (http://www.vgt.vito.be/). To match this level of aggregation, corresponding 10-day statistics for Q, P_g, P_d, R_e and other variables (temperature, precipitation, moisture, etc.) were calculated. Analysis of the relationships of gross primary productivity and ecosystem respiration to NDVI and other environmental factors indicates that combining NDVI with other factors may further improve predictive power of models for P_g and R_e compared to models based only on NDVI. Formally speaking, we are looking for models in the form:

(13)

(14)

where {X_i, i = 1, … , n} are additional predictors for photosynthesis, {Y_j, j = 1, … , m} denote additional predictors for ecosystem respiration, f_P(NDVI, X₁, X₂, … , X_n) and f_R(NDVI, Y₁, Y₂, … , Y_m) are multivariate functions to be identified, and e_P and e_R are random error terms with certain stochastic characteristics.

It should be emphasized that within the framework of a ‘black box’ approach when only external driving factors (e.g. meteorological variables) are allowed as members of the sets {X_i} and {Y_j}, the chances of finding models (13) and (14) with acceptable predictive power might not be very high. Nevertheless, these chances are improved by switching to the ‘grey box’ models, when the input data sets {X_i} and {Y_j} are allowed to include variables characterizing internal state of the ecosystem such as green biomass (leaf area index), soil water content, soil temperature, stage of phenological development, etc., as well as their remotely-sensed surrogates such as NDVI.

Light-response functions

Light response of daytime CO₂ fluxes can be fit by the non-rectangular hyperbolae (9), as illustrated by Fig. 3, for various sites characterizing days with favourable conditions (e.g. no moisture stress) and high photosynthesis. On days with substantial drought stress, a hysteresis loop is observed on the light-response curves, so that the morning branch of the P(Q) curve lies higher than the evening branch (Fig. 4a,b,c,d,e left). In such cases, daytime flux dynamics are better described by the light-response function P(Q, T_s) depending not only on photosynthetically active radiation (Q) but also on soil temperature (T_s).

The parameters of light-response functions exhibited seasonal patterns of variation at all sites (Fig. 5). These patterns are characterized by the maximum values of the apparent quantum yield, α; the maximum photosynthetic rates, β; and the daytime ecosystem respiration values, γ, occurring at the peak of the season (June–July). This pattern, however, is complicated by the fluctuations reflecting weather peculiarities during portions of some seasons. Maximum weekly average values for all three parameters were observed for the most productive Cheyenne site: α_max,wk = 1.50 g CO₂ (mol quanta)⁻¹ = 0.034 mol CO₂ (mol quanta)⁻¹; β_max,wk = 1.25 mg CO₂ m⁻² s⁻¹, and γ_max,wk = 0.275 mg CO₂ m⁻² s⁻¹. Parameters α, β and γ established in this study for Northern Great Plains grasslands are definitely lower than those determined by Luo et al. (2000) for the sunflower culture under mesocosm conditions. They do lie, however, between the values established for sagebrush steppe in the US Intermountain West (Gilmanov et al., 2003a) and true steppe in Kazakhstan (Gilmanov et al., 2004) at the lower end and the values for mixed prairie, pasture, tallgrass prairie and the winter wheat crop in Oklahoma (Gilmanov et al., 2003b) at the upper end.

We evaluated our light-response estimates of daytime respiration, γ_day, by comparing them with directly measured night-time respiration rates, γ_night (Fig. 6). In all five cases, close linear relationships between γ_day and γ_night were observed (Table 2) with R² coefficients from 0.64 (Mandan) to as high as 0.95 (Lethbridge). On all the five graphs, more points occur above the 1 : 1 line than below it, indicating that on average, estimated daytime respiration γ_day is higher than the measured night-time respiration γ_night. To test the hypothesis that deviation of the regression line from the 1 : 1 diagonal is due to random errors only, ellipses describing the 99% critical zones for this hypothesis in the parametric space {b₀, b₁} were constructed. At all sites, points (b₀, b₁) describing observed regressions γ_day = b₀ + b₁γ_night lie outside these ellipses, demonstrating that the regressions are different from the 1 : 1 relationships.

Our observation that γ_day > γ_night is in agreement with the results by Grahammer et al. (1991) who, using chamber measurements in tallgrass prairie in Kansas, estimated average rates of daytime and night-time soil respiration during summer as γ_soil,day = 0.12 g CO₂ m⁻² s⁻¹ and γ_soil,night = 0.10 g CO₂ m⁻² s⁻¹. We propose that γ_day > γ_night respiration even though some experimental data suggest that respiration of green leaves may be inhibited by light (Villar et al., 1994, 1995). Gifford's (2003) review demonstrated that leaf mitochondrial respiration was unaffected by light (Loreto et al., 1999, 2001a, 2001b). It is presently well established that respiration of below-ground plant parts, soil micro-organisms, and fauna increases with soil temperature, as described by the exponential term in eqn. (10) (e.g. Krogh, 1914; Rabinowich, 1956; Howard, 1971; Lloyd & Taylor, 1994; Kirschbaum, 1995; Tjoelker et al., 2001; Burton et al., 2002). In addition, a number of studies using radiocarbon pulse labelling have demonstrated rapid connection of production of photosynthates and root respiration and exudation (Megonigal et al., 1999; Kuzyakov, 2002; Lu et al., 2002). For example, Kuzyakov & Domanski (2002) have found the isotopic label in soil micro-organisms and exudates within the first hours after pulse labelling of ryegrass, with maxima of microbial and exudates ¹⁴C observed approximately 4 h after isotopic labelling. Our working hypothesis is that increased levels of below-ground plant and heterotrophic soil (especially, rhyzosphere) respiration during daytime, associated with increasing soil temperature and the production and translocation of photosynthates is greater than the inhibition of leaf respiration by light (if the latter is taking place at all), resulting in a higher rate of ecosystem respiration during the day than at night.

Seasonal/annual CO₂ budgets

Seasonal curves of P_g, P_d, and F for the five sites for selected years are shown in Fig. 7. Table 3 summarizes the estimates of seasonal (annual) flux integrals for all available sites and years. These estimates are in reasonable agreement with numbers for gross primary production and ecosystem respiration of non-forest ecosystems available in the literature (cf. Gilmanov et al., 2003a,b; Suyker et al., 2003; Xu & Baldocchi, 2004). Similar to southern prairies (Suyker et al., 2003) and Mediterranean grasslands (Xu & Baldocchi, 2004), the Northern Great Plains grasslands exhibit substantial variability of CO₂ exchange, and may switch from being significant carbon sinks during years with favourable precipitation to significant sources of carbon in water stress years (Table 3).

Radiation use efficiencies

As an integral characteristic of the plant production process, the daily coefficient of ecological radiation use efficiency, ɛ = P_g/Q, demonstrates significant seasonal and year-to-year variability. (Fig. 8). The moving weekly average of ɛ has more regular dynamics (solid line of Fig. 8), but, contrary to the anticipated unimodal pattern (Lethbridge 1998; Cheyenne 1998), in certain years it displays several distinct maxima (e.g. Lethbridge 2000; Fort Peck 2000; Miles City 2000–01; Mandan 1999–2001; Cheyenne 1998). The range of maximum daily light-use efficiencies in our study (0.014 to 0.032 mol CO₂ mol quanta⁻¹) is close to the same range from literature data (0.015 to 0.041 mol CO₂ mol quanta⁻¹) (Table 4). The same is true for the range of maximum weekly light use efficiencies: our interval between 0.008 and 0.019 mol CO₂ mol quanta⁻¹ practically lies within the interval 0.009 to 0.028 mol CO₂ mol quanta⁻¹ of published data, which also include more productive grasslands. Our intervals for the average seasonal radiation use efficiency ɛ_seas = P_g,seas/Q_seas (0.0033 to 0.0057 mol CO₂ mol quanta⁻¹) and the average annual radiation use efficiency ɛ_year = P_year/Q_year (0.0025 to 0.0053) are less than the available literature data (0.0057 to 0.0173 and 0.0038 to 0.0091 mol CO₂ mol quanta⁻¹, respectively), which is also in agreement with the position of the Northern Great Plains grasslands in the productivity gradient of the literature data (Table 4).

Flux–NDVI relationships

Statistical analysis of the relationships among the 10-day averages P_g, P_d, and R_e as dependent variables, and the 10-day Normalized Difference Vegetation Index (NDVI) composites and other factors reveals complex interrelationships. First, the correlation between the average 10-day gross primary productivity (P_g) and the 10-day composite NDVI is significantly higher than the correlation between the average 10-day daytime CO₂ flux totals (P_d) and the 10-day composite NDVI. As illustrated in the second and third columns of Table 5, correlation coefficients r(P_g, NDVI) vary from 0.68 to 0.88, while coefficients r(P_d, NDVI) lie in the range 0.21 to 0.73. In all cases r(P_g, NDVI) > r(P_d, NDVI). To test the significance of this inequality, we used the test for comparison of correlation coefficients suggested by Steiger (1980), which is also applicable for correlated variables (note high correlations r(P_g, P_d) between corresponding P_g and P_d values in column 4, Table 5). Results presented in Table 5 show that for all sites the observed t-value, t_obs, is considerably higher than the 5% value of the Student's criterion with (n− 3) degrees of freedom t_crit(0.95; n− 3). In fact, P-values for all sites (last column) are less than 0.1%, indicating significance of the observation that P_g is more highly correlated with NDVI than P_d.

The important practical implication for scaling-up algorithms is that gross primary productivity, P_g, derived from daytime flux measurements using light-response analysis is a more appropriate variable than total net daytime flux, P_d. These findings are in agreement with observations of higher P_g–NDVI correlations compared with P_d–NDVI correlations in a true steppe ecosystem in northern Kazakhstan (Gilmanov et al., 2004; Wylie et al., 2004).

Ten-day average total ecosystem respiration (R_e) was also closely correlated with NDVI composites, with r-values varying from 0.75 to 0.88 (Table 6). This is not surprising, because a substantial part of total ecosystem respiration is represented by plant biomass respiration which is directly related to green above-ground biomass and, consequently, to NDVI (cf. Gifford, 2003). However, R_e is also related to such environmental factors as precipitation, soil temperature, and soil water content. We have used available data sets to identify phenomenological models linking R_e to NDVI and other environmental drivers.

In spite of significant correlation, there is a scatter in both the P_g–NDVI and the R_e–NDVI relationships. Part of this scatter is probably due to totally random effects. On the other hand, there are, obviously, other factors, e.g. photosynthetically active radiation, precipitation, soil water content, and soil temperature, that affect production and respiration of the grassland ecosystem, in addition to NDVI. For instance, at the Mandan site, a linear function of NDVI explains 53% of the variability in 10-day gross primary productivity and has a standard error SE = 3.15 g CO₂ m⁻² d⁻¹. Including photosynthetically active radiation, Q, provides improvement of the model. Figure 9 shows the response surface of the gross photosynthesis of the mixed prairie at Mandan to photosynthetic radiation, Q, and NDVI, described by the model:

(15)

where e_P denotes the residual error term. With estimated parameters ϕ = 0.342 g CO₂ (mol quanta)⁻¹, λ = 3.954, and µ = 8.759, the model, obtained using nonlinear regression fitted to a 10-day aggregated Mandan (1999–2001) data set, is characterized by the corrected R-squared value  = 0.65 and the standard error SE = 2.67 g CO₂ m⁻² d⁻¹. In addition to visual evaluation of the model fit by comparing the data points with the surface on Fig. 9a, direct comparison is provided in Fig. 9b by the scatter plot of the observed and predicted values with the 1 : 1 diagonal.

Further improvement of the description of gross primary productivity is achieved by modification of the model (15) to include bell-shaped dependence on soil water content W_s, resulting in the model:

(16)

With parameters ϕ = 0.3844 g CO₂ (mol quanta)⁻¹, λ = 3.886 (dimensionless), and µ = 8.169 (NDVI units)⁻¹, ξ = 4.588 (dimensionless), and θ = 0.407 m³ m⁻³, the model is characterized by  = 0.68 and SE = 2.61 g CO₂ m⁻² d⁻¹. A scatter diagram of the observed vs. predicted P_g values for model (16) is presented in Fig. 10. Compared to model (15), this model has somewhat higher and lower SE, though for practical applications the simpler model (15) explains a sufficient part of the P_g variability to be acceptable.

Because the empirical models (15) and (16) combine information on the phenomenology of relationships of grassland P_g with photosynthetically active radiation, NDVI, and soil water (where available), we estimated their parameters for all Northern Great Prairie sites (Table 7). For all three sites for which soil water data were available (Lethbridge, Mandan, and Cheyenne), model (16) provided the best fit, with values above 60%. For the Fort Peck northern mixed prairie and the Miles City mixed prairie, where W_s data were not available, model (15) performed well, with combined effect of radiation and NDVI explaining 60% and 64% of the variation of 10-day average gross primary productivity, respectively. It should be noted that the top-soil water variable, W_s, used in this analysis, does not provide complete characterization of the water availability and its effect on ecosystem productivity. Apparently, data on water content in the whole rooting zone, W_r, when available, might further improve the predictive power of phenomenological models P_g = f(Q, NDVI, W_r, … ).

Table 7. Site-specific estimates of the parameters of equation 17 relating 10-day average gross primary productivity (P_g) to 10-day averages of photon flux density (Q), NDVI, and top-soil water content (W_s)

Site, ecosystem (n = number of data points)	ϕ	λ	µ	ξ	θ	R 2		SE
Lethbridge, northern mixed shortgrass prairie (n = 77)	0.577	6.78	29.08	38.89	0.364	0.92	0.85	1.79
Fort Peck, northern mixed prairie* (n = 31)	0.383	2.85	6.81	0	0	0.78	0.60	2.19
Miles City, northern mixed prairie (n = 44)	0.202	8.59	37.42	0	0	0.80	0.64	1.65
Mandan, mixed prairie (n = 64)	0.384	3.89	8.01	4.60	0.406	0.82	0.68	2.61
Cheyenne, mixed prairie (n = 36)	0.543	5.15	15.05	32.21	0.200	0.73	0.67	4.13

From previous studies of the phenomenology of ecosystem respiration, it is known that R_e is directly related to the amount of the autotrophic and heterotrophic biomass and, consequently, leaf area index (LAI) and NDVI (Monsi & Saeki, 1953; McCree, 1970; Bunce, 1989; Norman et al., 1992; Polley et al., 1992; Hanan et al., 1997; Amthor & Baldocchi, 2001; Gifford, 2003). On the other hand, the metabolic activity of plant, microbial, and animal biomass is controlled by environmental drivers such as temperature, moisture, etc. While detailed characterization of the interaction of these extensive and intensive factors in determining dynamics of ecosystem respiration may apparently be implemented only in rather comprehensive simulation models, analysis of our data revealed statistically significant phenomenological relationships between R_e and other factors at the 10-day time scale. As we have seen in Table 6, linear functions of NDVI alone explain more than 50% of the variability of ecosystem respiration of northern Great Plains rangelands at the 10-day scale. Multivariate linear regression analysis has indicated that atmospheric precipitation, PCPN, and radiation, Q, complement NDVI most effectively, so that linear functions of these variables explain 63 to 87% of the dispersion of 10-day R_e values with SE values from 1.17 to 3.60 g CO₂ m⁻² d⁻¹.

To investigate the effect of nonlinearity of the relationship between biomass and NDVI, and consequently, between R_e and NDVI (cf. Turner et al., 1999; Wylie et al., 2002), we used a nonlinear (with respect to NDVI) model:

(17)

where λ, µ, υ, ρ, and ψ are empirical parameters, ρ = R_e(0, 0, 0)(1 + e^λ). Parameters of this model, estimated for all the sites, are listed in Table 8. As an illustration, Fig. 11 shows the response surface R_e(NDVI, PCPN) and the scatter diagram of observed vs. predicted R_e values for the Fort Peck site. For all sites except Lethbridge, the model (17) provided better fit for ecosystem respiration data, demonstrating R² values from 0.70 to 0.89 and SE values from 1.22 to 3.36 g CO₂ m⁻² d⁻¹.

Table 8. Site-specific estimates of the parameters of equation 16 relating average total ecosystem respiration, R_e, to composite NDVI, precipitation, PCPN, and photosynthetically active radiation, Q, at the 10-day scale

Site, ecosystem (n = number of data points)	ρ	λ	µ	ν	ψ	R 2		SE
Lethbridge, northern mixed shortgrass prairie* (n = 125)	4.757	6.534	23.84	0.068	0.051	0.83	0.68	1.47
Fort Peck, northern mixed prairie (n = 34)	61.84	4.069	4.457	0.057	0	0.89	0.79	1.22
Miles City, northern mixed prairie (n = 36)	10.11	5.696	21.39	0.075	0	0.74	0.55	1.97
Mandan, mixed prairie (n = 60)	1049.5	6.982	3.198	0.017	0	0.72	0.51	1.89
Cheyenne, mixed prairie (n = 36)	13.85	7.01	21.51	0.022	0	0.70	0.49	3.36

• * At the Lethbridge site, linear regression R_e = −2.16 + 12.12NDVI + 0.058PCPN + 0.77Q is characterized by R² = 0.86, SE = 1.17 g CO₂ m⁻² d⁻¹, F_3,121 = 246.37 and P < 0.00001.

Previous studies of the phenomenology of ecosystem respiration have demonstrated bell-shaped dependences on temperature (which below 25 °C is reasonably approximated by the exponential or Q₁₀-type function) combined with saturated or bell-shaped relationships to soil water (Flanagan & Bunnell, 1976; Singh & Gupta, 1977; Davidson et al., 1998; Davidson et al., 2000; Mielnick & Dugas, 2000; Frank et al., 2002). We have found that the simple model combining exponential response to soil temperature, T_s, with bell-shaped response to soil water, W_s:

(18)

provides a satisfactory description of the R_e(T_s,W_s) relationships for all the sites when both T_s and W_s data available. Compared to the linear and logistic models, model (17) provides poorer fit to data (R² from 0.19 to 0.77 and SE from 1.58 to 4.56 g CO₂ m⁻² d⁻¹). Nevertheless, it allows us to estimate the exponential temperature coefficient, κ, which provides quantitative characterization of the partial response of ecosystem respiration to temperature. The average of κ-value for all five sites, 0.063 (°C)⁻¹ (range 0.037 to
0.082 (°C)⁻¹), is close to the value  = 0.069 (°C)⁻¹ correspond
ing to the standard physiological value of Q₁₀ = 2. It should be noted here that higher Q₁₀ values often reported in the literature, as a rule do not reflect the partial temperature response in the strict sense because of difficulties maintaining unchanged levels of other factors.

Overall, NDVI-based models provided considerably better fit to the respiration data than the temperature-based model (18). The fact that NDVI-based models did not include temperature as a significant additional predictor, and the temperature-based models in most cases did not include NDVI, may be explained by high positive correlation r(T_s, NDVI) between soil temperature and NDVI in the 10-day data (Table 9). Only at the Miles City and Cheyenne sites characterized by lower r(T_s, NDVI) values (0.50 and 0.65, respectively), did the inclusion of NDVI as an additional predictor increase the fit.

Recently, Körner (2003a, 2003b) expressed serious concerns regarding approaches to scaling-up of the results of the eddy covariance tower flux measurements in forests. One of his points — the need to generalize plot-based fluxes only to the areas with similar dynamic parameters of carbon cycling — is also applicable to non-forest ecosystems, including grasslands. Because NDVI characterizes structural and functional ecosystem properties, we think that the application of NDVI-based models (13)–(14) derived from tower sites has the potential for robust extrapolation. GIS layers of NDVI and other relevant predictors {X_i}, {Y_j} for the pixels with similar temporal patterns and magnitudes of NDVI as the flux towers and within ecologically similar ecoregions (Fig. 1) can spatially constrain the empirically based method of upscaling tower CO₂ fluxes. This approach may be considered as complementary to the dynamic simulation approach including running a SVAT or biogeochemical simulation models driven by meteorological, topographic, and remotely sensed data in every pixel (or subsets of pixels) of the map (e.g. Nouvellon et al., 2001; Liu et al., 2002).

An important distinction of our model relative to the constructions based on the radiation-use efficiency approach sensu Monteith (e.g. Running et al., 1999; Veroustraete et al., 2002) is that our functions P_g( … ) and R_e( … ) are directly derived from on-site and satellite measurements and do not include a priori assumptions about the values and seasonal patterns of radiation-use efficiency coefficients. According to Nouvellon et al. (2000) these assumptions may lead to large estimation errors.

The most straightforward application of the multivariate functions P_g( … ) and R_e( … ) is for scaling-up of CO₂ fluxes at ecologically similar locations within the ecoregion and the year of measurements. Portability of these phenomenological models across the years requires additional testing with larger data sets. However, the results for the Lethbridge data set, that includes 4 years of observations with R² values sufficiently high for both photosynthesis and respiration (Tables 7 and 8), provide encouragement and stimulate additional research.