Spatial dynamics of water and nitrogen management in irrigated agriculture.

where [n.sub.st]([beta]) = nitrogen supply [kg/ha], [w.sub.50] and [ns.sub.50] are scaling parameters, and the estimated parameters are [w.sub.50], [[PHI].sub.uw], [ns.sub.50], and [[PHI].sub.un] along with [n.sup.max.sub.u]. (A.2) has similar qualitative characteristics as the yield function (A.1). Nitrogen supply is defined by the accounting relation

(A.3) [n.sub.st]([beta]) = [n.sub.t]([beta]) + [n.sub.at]([beta]) - [n.sub.et]([beta])

where [n.sub.t]([beta]) = inorganic soil nitrogen at the beginning of the season [kg/ha], [n.sub.at]([beta]) = applied nitrogen [kg/ha], and [n.sub.et]([beta]) = nitrogen leaching from the soil [kg/ha].

Equation (A.4) specifies nitrogen leaching as a function of initial soil nitrogen, along with applied nitrogen and infiltrated water

(A.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[phi].sub.e], [w.sub.50], and [[phi].sub.ew] are parameters to be estimated. In this relation, nitrogen emissions are a fraction of soil nitrogen supply. This fraction is zero for low levels of infiltrated water, consistent with minimal transport below the rootzone due to low soil moisture levels, but increases in a convex-concave manner, eventually approaching a value of one as infiltrated water depths become large enough. Thus the maximum amount that can be leached is the measure of nitrogen supply consistent with mass balance. The parameter [w.sub.50] is a scaling parameter as above.

Inorganic soil nitrogen loss from denitrification, volatilization, and other factors, [n.sub.zt]([beta])[kg/ha], is defined as

(A.5)

[n.sub.zt]([beta]) = [[phi].sub.z0] + [[phi].sub.z1] [[n.sub.at]([beta]) + [n.sub.t]([beta])] + [[phi].sub.z2][[[n.sub.at],([beta]) + [n.sub.t]([beta])].sup.2] + [[phi].sub.zw] [w.sub.t]([beta])

where [[phi].sub.z0], [[phi].sub.z1], [[phi].sub.z2], and [[phi].sub.zw] are fitting parameters to be estimated. In general, inorganic nitrogen losses [n.sub.zt] depend on soil nitrogen supply but can also be influenced by water supply. Finally, with these definitions we can specify soil inorganic nitrogen dynamics as

(A.6) [n.sub.t]+l([beta]) = [n.sub.t]([beta]) + [[bar.n].sub.at] - [n.sub.ut]([beta]) - [n.sub.zt]([beta]) - [n.sub.et]([beta])

which is an accounting identity reflecting mass balance. In particular, ending soil inorganic nitrogen equals initial soil inorganic nitrogen plus applied nitrogen minus inorganic nitrogen losses to uptake, leaching, and denitrification and other factors.

These relations were estimated with data from Tanji et al. (1979) and Pang et al. (1997a), which provide values for all variables. One observation for [n.sub.zt] in (A.5) was theoretically implausible and inconsistent with other observed values. This observation was treated as an outlier and replaced with a value determined by extrapolation. Graphical analysis and a trial and error specification search identified functional forms with suitable data fit and global properties. Estimated coefficients and associated statistics are reported in table A1. The estimated regressions provide excellent fit with [R.sup.2] values ranging from 0.78 to 0.95, and all estimated parameter values are significant at the 95% level or higher. Graphical analysis of the regressions indicate functional fits lying within bands defined by data in alternate years, and all exhibit global properties consistent with the generalized conceptual model in Lanzer and Paris (1981). Table A.1. Estimated Parameter Values and Associated Statistics for the Plant-Level Production Function Relations

Independent Parameter and Estimated Value Equation Variable [95% Confidence Interval] A1 Yield [y] [y.sup.max]/ [W.sub.50]/

12.085 29.99

[11.28, 12.78] [23.67, 35.12] A2 Uptake [n.sup.max.sub.u]/ [w.sub.50]/

[[n.sub.u]] 351.87 22.42

[347.3, 356.4] [21.225, 23.63] A4 Emissions [[phi].sub.e]/ [w.sub.50]/

[[n.sub.e]] .1444 7141

[0.137, 0.152] [68.70, 77.74] A5 Denitrification [[phi].sub.z0]/ [[phi].sub.z1]/

and other losses -57.38 0.0711

[[n.sub.z]] [-60.0, -54.8] [0.064, 0.078]

Independent Parameter and Estimated Value Equation Variable [95% Confidence Interval] A1 Yield [y] [[phi].sub.yw]/ [nu.sub.75]/

3.3963 158.24

[2.31, 5.34] [136.5, 182.0] A2 Uptake [[phi].sub.uw]/ [ns.sub.50]/

[[n.sub.u]] 1.311 198.00

[1.244, 1.384] [193.7, 202.5] A4 Emissions [[phi].sub.ew]/

[[n.sub.e]] 0.238 --

[0.181, 0.3961 A5 Denitrification [[phi].sub.z2]/ [[phi].sub.zw]/

and other losses 1.56 [10.sup.-4] -0.430

[[n.sub.z]] [1.44 [10.sup.-4], [-0.468, -0.393]

1.67 [10.sup.-4]]

Parameter and

Estimated Value

Independent [95% Confidence Equation Variable Interval] [R.sup.2] A1 Yield [y] [[phi].sub.yu]/

1.812 0.91

[1.40, 2.28] A2 Uptake [[phi].sub.un]/

[[n.sub.u]] 2.034 0.95

[1.934, 2.139] A4 Emissions -- 0.88

[[n.sub.e]] A5 Denitrification -- 0.78

and other losses

[[n.sub.z]]

[Received January 2006; accepted September 2007.]

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