#### 2.1. Observation Data

[8] We used the N_{2}O and NO flux data obtained from an Andosol soil lysimeter to which a chemical fertilizer was applied; original data were published in *Akiyama et al.* [2000] and *Akiyama and Tsuruta* [2002, 2003]. These fluxes were measured by the automated closed chamber method in lysimeter fields at Tsukuba, Japan (361°01′N, 141°07′E) [*Akiyama and Tsuruta*, 2003]. The mean annual air temperature and precipitation over the last 30 years were 13.6°C and 1259 mm, respectively, [*Minamikawa et al.*, 2010]. The soil type was Andosol with pH (H_{2}O) 5.9, total C 3.13%, total N 0.26%, and bulk density 0.92 g cm^{−1} [*Akiyama et al.*, 2000]. Soil temperature was measured at a depth of 5 cm using a thermometer. The volumetric water content at a soil depth of 10 cm was monitored using time domain reflectometry moisture sensors (CS615, Campbell Scientific Instruments, Logan, UT). The water-filled pore space (WFPS) of the soil was calculated on the basis of the measured volumetric water content and porosity of the soil.

[9] Six lysimeters (numbered from ID 1 to ID 6) were used for the flux measurement. In 1996, a mixture of urea and ammonium sulfate in a 2:1 ratio was used as the N fertilizer. *Daucus carota* L. was planted with a basal fertilizer application of120 kg N ha^{−1} and an additional fertilizer application of 80 kg N ha^{−1} at ID 3 and ID 6. In 1998 (at ID 5 and ID 6) and in 1999 (at ID 2 and ID 4), *Brassica rapa* var. *chinensis* was planted with a urea application of 150 kg N ha^{−1}. In 2000 (at ID 4 and ID 6), *Spinacia oleracea* was planted with a urea application of 150 kg N ha^{−1}.

#### 2.2. Model Description

[11] First, we constructed the data model, which represents the likelihood of the HB model. We assumed that each observed N oxide flux (*N*_{obs}) consisted of the true flux “*N*_{true}” with the normal distribution error (*ϵ*) derived from observable issues.

Then, the true flux (*N*_{true}) was assumed to be lognormally distributed with the scale parameter and shape parameter, as shown in the following equation:

*μ*_{flux} is the scale parameter, which determines the flux strength. In addition, *σ*_{2} is the parameter related to the variance and shape of the variability in the flux.

[12] *ϵ* was assumed to follow a normal distribution with a mean of 0 and is given by the following equation:

By using the lognormal and normal distributions for the data model, we could distinguish between the spatial variability and observation error in the estimation as follows: lognormal: spatial variability:: normal: observation error.

[13] Next, we constructed the process model. The scale parameter “*μ*_{flux}” in the lognormal distribution was determined by the following equation:

where *α* is the parameter indicating the soil N mineralization rate under nonfertilized conditions, *f*_{fert} is the fertilization response function, and *n* is the number of fertilizer applications (i.e., *n* = 1 indicates basal fertilizer and *n* = 2 indicates additional fertilization). *f*_{T} and *f*_{W} are temperature and soil moisture limiting functions, respectively. indicates the random effect, which is considered for the chamber position (described in the data model).

[14] Each function can be defined as follows:

where *f*_{fert} is the N fertilization response function, which is a type of one-compartment model, and and are the explanatory variables of this function. The parameters of the fertilization function are *γ*, *k*_{a}, and *k*_{b}. The parameter 1/*γ* determines the conversion factor of fertilization to each N oxide gas flux; the parameters *k*_{a} and *k*_{b} determine the duration of the N fertilization effect on the N oxide gases.

where *f*_{T} is the temperature response function modified from the van't Hoff exponential function, which varies from 0 to 1. The explanatory variable of this function is and *T*_{ref} is a constant, which was set at 35°C in this study. The temperature sensitivity parameter is *Q*_{tem}.

where the soil moisture response function is *f*_{W} ranging from 0 to 1 and is the same as the temperature response function. is the explanatory variable of this function and *c* and *W*_{opt} are the parameters. We used the Gaussian equation as a soil moisture response function because certain studies showed that the response of N_{2}O flux to WFPS could be approximated as unimodal [e.g., *Sleutel et al.*, 2008; *Rafique et al.*, 2011].

[15] Finally, we constructed the parameter model. For each chamber, we incorporated random effects as follows:

where _{i} indicates the chamber ID.

[16] For all parameter priors, we used vague or broad uniform priors. For the parameter *Q*_{tem}, the restriction was set at greater than 1. For the parameters *α*, *γ*, *c*, *k*_{a}, and *k*_{b}, the restriction was set at greater than 0. For *W*_{opt}, we used beta distribution (beta(*shape*, *shape*)) as a prior distribution, which varies from 0 to 1. For the variance, we used uniform priors after logarithmic transformation. The definitions are as follows:

Then, the joint posterior probability was described as follows:

where *θ* is the parameter vector, i.e., *θ* = (*α*,*Q*_{tem},*c*,*W*_{opt},*γ*,*α*,*β*,*σ*), and *p*(*θ*) denotes priors. For this model, we used Markov chain Monte Carlo methods implemented with the Bayesian inference by using the Gibbs sampling software WinBUGS (D. Spiegelhalter et al., WinBUGS, version 1.4.3, 2007, available at http://www.mrc-bsu.cam.ac.uk/bugs/). In addition, we ran the Gibbs sampler for 100,000 iterations for three chains with a thinning interval of 10 iterations. We discarded the first 50,000 iterations as burn-in and used the rest as chains to calculate posterior estimations. For convergence diagnostics, we used the Gelman and Rubin convergence diagnostic as an index. The R [*R Development Core Team*, 2012] and R2WinBUGS package [*Sturtz et al.*, 2005] were used to call WinBUGS and to calculate statistics in R.

#### 2.3. Simulation to Evaluate Fertilizer-Induced Emission Factor

[17] We simulated the fertilizer-induced EFs for the N_{2}O and NO fluxes by using the posterior distribution of the aforementioned model.

[18] In *Akiyama et al.* [2000] and *Akiyama and Tsuruta* [2002, 2003], the N_{2}O and NO fluxes from nonfertilized control plots were not measured. Therefore, we eliminated background emissions, such as the fluxes from nonfertilized emissions, by using the following equation as a scale factor of the lognormal distribution instead of *μ*_{flux}:

This equation is obtained by subtracting *α* from equation (3).

[19] Each daily N oxide emission (*E*_{daily}) was calculated in the following manner:

The sum of *E*_{daily} for the duration of the simulation period was equal to the total emissions. Finally, EFs were calculated by dividing the integrated *E*_{daily} by the fertilizer input.

[20] In the simulation used to estimate the EFs, the soil temperature and WFPS were obtained from the observed dataset of 1998 and the duration of integration was 364 days. For N fertilization, we used 150 kg N ha^{−1} as the input for the simulation. Then, we obtained the posterior simulations by drawing 1000 times from the posterior distributions.