*2.3. Simulation Decomposition*

The simulation decomposition (SD) [12] approach is an extension to the Monte Carlo simulation [21] that enhances the explanatory capability of the simulation results by exploiting the inherent cause-and-effect relationship between the input and output parameters [13].

SD has recently been developed and successfully used on problems involved in different domains such as geology, business, and environmental science [22]. It has been shown to provide a deeper understanding of the interaction between different sources of uncertainties and its impact on output uncertainty and its distribution to stakeholders. The current section provides a brief description of SD from an application point of view. A detailed description of SD can be found in [12].

In this section, the fundamental steps of implementing SD are described using an analytical model problem. Consider a simple analytical function given as

$$y = v\_1 + v\_{2\prime}^2 \tag{7}$$

where *v*<sup>1</sup> and *v*<sup>2</sup> are the real numbered input parameters and *y* the real number output parameter. The SD process has the following steps [12]:


**Figure 3.** Probability distribution of simulation output for example problem: (**a**) output full distribution, and (**b**) decomposed distribution based on scenarios. **Figure 3.** Probability distribution of simulation output for example problem: (**a**) output full distribution, and (**b**) decomposed distribution based on scenarios.

**Parameter Distribution/Range State Boundary**  ଵ U [0, 10] Low [0–5)

ଶ U [0, 10] Low [0–5)

**Scenario Combination of States**  *Sc*<sup>1</sup> ଵ: Low, ଶ: Low *Sc*<sup>2</sup> ଵ: Low, ଶ: High *Sc*<sup>3</sup> ଵ: High, ଶ: Low *Sc*<sup>4</sup> ଵ: High, ଶ: Low

High (5–10]

High (5–10]

**3. Results Table 2.** Input parameter details.

**Table 2.** Input parameter details.

**Table 3.** Generating scenarios from parameter states.


ature is assumed to be normally distributed with a mean of 74 °F and a standard deviation of 2 °F, whereas the July precipitation is assumed to have a lognormal distribution with a **Table 3.** Generating scenarios from parameter states.


#### stant and set as 185 kg/ha and 17 kg/ha based on the Iowa State University extension **3. Results**

This section presents the results of applying SD to the proposed extended nitrogen export model which includes a weather model. In particular, the current work focuses on understanding the effects of weather parameters on the nitrogen surplus in different scenarios.

For this study, the weather parameters temperature (T) and precipitation (PPT) for July are taken as input parameters, whereas soil nitrogen surplus is considered as an output parameter computed from the IFEW simulation model. Furthermore, the July temperature is assumed to be normally distributed with a mean of 74 ◦F and a standard deviation of 2 ◦F, whereas the July precipitation is assumed to have a lognormal distribution with a standard deviation of 0.4 in., a shape parameter of 0, and median at 4 as shown in Table 4. All other parameters considered in the IFEW simulation model are kept constant.


**Table 4.** Input parameter details for performing simulation decomposition with IFEW simulation model.

In the crop-weather model, May plantation progress and June precipitation is assumed to be 80% and 5.5 in., respectively. The parameters used in the animal agriculture model (*x*5–8) are based on the 2012 Iowa animal population data [19]. The commercial nitrogen application rate for corn (*x*3) and soybean (*x*4) agriculture are considered to be constant and set as 185 kg/ha and 17 kg/ha based on the Iowa State University extension guidelines for the nitrogen application rate for corn [23] and on the fertilizer use and price data [24].

After setting up the IFEW model, Monte Carlo simulations are performed using Latin hypercube sampling (LHS) [25]. The LHS sampling method ensures that the input parameter ranges are represented appropriately. The input parameter states and boundary details are presented in Table 4. For July temperature, any temperature above 76 ◦F is considered to be under state high where all other temperature values are considered to be under state regular. Similarly, for July precipitation, any precipitation value below 2.5 in. is labeled under state low precipitation and all other values are under state regular. Table 5 presents the scenarios based on a combination of states. The parameter states are selected to produce some of the extreme condition scenarios (e.g., Table 5 dry condition).

**Table 5.** Scenarios for simulation decomposition approach with IFEW model.


A total 10<sup>5</sup> samples of input weather parameters (*w*<sup>1</sup> and *w*2) are generated using LHS and SD approach is implemented using the IFEW simulation model. Figure 4 shows the distribution of sampled weather parameters in two states and four scenarios as mentioned in Tables 4 and 5. Most of the generated samples are observed under regular condition (*Sc*2) whereas the least number of samples are observed in dry condition (*Sc*3).

The input weather parameters are supplied to a crop-weather module which computes corn yield (*x*1) and soybean yield (*x*2). The computed crop yield values are then passed to an agriculture module where *CN*, *FN*, and *GN* values are computed as mentioned in Section 2.2. Here, the contribution of *CN* will be constant for every IFEW model evaluation due to the assumption of a constant commercial nitrogen application rate for corn (*x*3) and soybean (*x*4).

Figure 5 shows the decomposed distribution of corn and soybean yield along with the variation in *FN* and *GN* values. The effect of different scenarios due to combinations of weather parameters can be clearly seen in crop yield distribution. It is interesting to note that in dry condition (*Sc*3) corn yield drops compared to the yield in regular condition, whereas higher soybean yield is observed in dry condition compared to the regular condition. The computation of *GN* is influenced by both corn and soybean yield values (Figure 5c). The computation of *FN* is only influenced by soybean yield values (Section 2.2); thus, the *FN* distribution is observed to be similar to soybean yield distribution.

**Figure 4.** Decomposed distribution of input parameters from simulation decomposition: (**a**) July temperature (*w*1), and (**b**) July precipitation (*w*2). **Figure 4.** Decomposed distribution of input parameters from simulation decomposition: (**a**) July temperature (*w*<sup>1</sup> ), and (**b**) July precipitation (*w*<sup>2</sup> ).

guidelines for the nitrogen application rate for corn [23] and on the fertilizer use and price

to produce some of the extreme condition scenarios (e.g., Table 5 dry condition).

(*Sc*2) whereas the least number of samples are observed in dry condition (*Sc*3).

After setting up the IFEW model, Monte Carlo simulations are performed using Latin hypercube sampling (LHS) [25]. The LHS sampling method ensures that the input parameter ranges are represented appropriately. The input parameter states and boundary details are presented in Table 4. For July temperature, any temperature above 76 °F is considered to be under state high where all other temperature values are considered to be under state regular. Similarly, for July precipitation, any precipitation value below 2.5 in. is labeled under state low precipitation and all other values are under state regular. Table 5 presents the scenarios based on a combination of states. The parameter states are selected

A total 105 samples of input weather parameters (*w*1 and *w*2) are generated using LHS and SD approach is implemented using the IFEW simulation model. Figure 4 shows the distribution of sampled weather parameters in two states and four scenarios as mentioned in Tables 4 and 5. Most of the generated samples are observed under regular condition

**Table 4.** Input parameter details for performing simulation decomposition with IFEW simulation

July temperature (*w*1) N [2, 74] Regular ≤76 °F

July precipitation (*w*2) LogN [0.4, 0, 4] Regular ≥2.5 in

**Table 5.** Scenarios for simulation decomposition approach with IFEW model.

**Parameter Distribution/Range State Boundary** 

**Scenario Combination of States Description** 

*Sc*<sup>1</sup> *w*1: Regular, *w*2: Low Regular-T Low-PPT *Sc*<sup>2</sup> *w*1: Regular, *w*2: Regular Regular condition *Sc*<sup>3</sup> *w*1: High, *w*2: Low Dry condition *Sc*<sup>4</sup> *w*1: High, *w*2: Regular High-T Regular-PPT

High >76 °F

Low <2.5 in

data [24].

model.

Figure 6 shows the decomposed distribution of nitrogen surplus (*Ns*), the final output of the IFEW simulation model. The soil nitrogen surplus is usually affected by *CN*, *MN*, *GN*, and *FN* magnitudes. However, in this study, only *GN* and *FN* influence the variation in nitrogen surplus. This is mainly because the parameters affecting *CN* and *MN* are kept constant. The variation in nitrogen surplus shown in this work is purely due to uncertainty in weather parameters. From Figure 6, it is observed that most of the variation in nitrogen surplus lies in regular condition (*Sc*2), varying approximately between 0 and 20 kg/ha. The scenarios with high July temperatures (*Sc*<sup>3</sup> and *Sc*4) are observed to produce mid to high nitrogen surplus values. Similarly, scenario *Sc*1, with very low July precipitation and regular July temperature, tends to produce higher nitrogen surplus than in regular conditions. The dry condition with high July temperature and low July precipitation produces the highest soil nitrogen surplus, varying between 20 and 30 kg/ha. The accumulated nitrogen in the soil is highly water-soluble and could get exported at a high rate to the Mississippi River through melting snow or rainfall before the next growing season. Figure 6 provides the expected magnitude of nitrogen load from state of Iowa to the Mississippi River in different weather scenarios.

The SD in this work uses the Monte Carlo sampling approach which could be used to provide approximate probability of a scenario occurring in any given year considering the assumptions made earlier are true. Based on the data available in the current study, probabilities of scenarios *Sc*1, *Sc*2, *Sc*3, and *Sc*<sup>4</sup> occurring are 0.1, 0.74, 0.02, and 0.12, respectively. The probability of dry condition (*Sc*3) occurring is lowest whereas regular condition (*Sc*2) has the highest probability of occurring (Figure 6).

The SD approach implemented in the current study provides valuable results to gauge the impact of weather parameters on soil nitrogen surplus along with crop yields and nitrogen transfer in agriculture systems. However, the particular distributions used for the weather parameters are not data based, and the two input weather parameters are assumed to be independent of each during the Monte Carlo sampling process. Temperature and precipitation are correlated. Thus, there is a possibility that some combination of scenarios may not entirely occur. For example, high precipitation and high temperature may not occur at the same time because with high precipitation, the average temperature drops. Further, the probability distributions of the weather parameters are challenging to estimate as they typically do not have continuous distributions. Thus, it is advisable to use weather generators which have been trained on historical datasets to predict weather parameters

*Sustainability* **2022**, *14*, x FOR PEER REVIEW 8 of 11

corn (*x*3) and soybean (*x*4).

rather than using continuous probability distributions. In future studies, weather generators will be included in the IFEW simulation model to predict weather data for more realistic predictions of soil nitrogen surplus. condition. The computation of *GN* is influenced by both corn and soybean yield values (Figure 5c). The computation of *FN* is only influenced by soybean yield values (Section 2.2); thus, the *FN* distribution is observed to be similar to soybean yield distribution.

The input weather parameters are supplied to a crop-weather module which computes corn yield (*x*1) and soybean yield (*x*2). The computed crop yield values are then passed to an agriculture module where *CN*, *FN*, and *GN* values are computed as mentioned in Section 2.2. Here, the contribution of *CN* will be constant for every IFEW model evaluation due to the assumption of a constant commercial nitrogen application rate for

Figure 5 shows the decomposed distribution of corn and soybean yield along with the variation in *FN* and *GN* values. The effect of different scenarios due to combinations of weather parameters can be clearly seen in crop yield distribution. It is interesting to note that in dry condition (*Sc*3) corn yield drops compared to the yield in regular condition, whereas higher soybean yield is observed in dry condition compared to the regular

**Figure 5.** Decomposed distribution of intermittent IFEW model parameters from simulation decomposition: (**a**) corn yield (*x*1), (**b**) soybean yield (*x*2), (**c**) *GN*, and (**d**) *FN*. **Figure 5.** Decomposed distribution of intermittent IFEW model parameters from simulation decomposition: (**a**) corn yield (*x*<sup>1</sup> ), (**b**) soybean yield (*x*<sup>2</sup> ), (**c**) *GN*, and (**d**) *FN*.

Figure 6 shows the decomposed distribution of nitrogen surplus (*Ns*), the final output of the IFEW simulation model. The soil nitrogen surplus is usually affected by *CN*, *MN*, *GN*, and *FN* magnitudes. However, in this study, only *GN* and *FN* influence the variation in nitrogen surplus. This is mainly because the parameters affecting *CN* and *MN* are kept constant. The variation in nitrogen surplus shown in this work is purely due to uncertainty in weather parameters. From Figure 6, it is observed that most of the variation in nitrogen surplus lies in regular condition (*Sc*2), varying approximately between 0 and 20 kg/ha. The scenarios with high July temperatures (*Sc*3 and *Sc*4) are observed to produce mid to high nitrogen surplus values. Similarly, scenario *Sc*1, with very low July precipitation and regular July temperature, tends to produce higher nitrogen surplus than in regular conditions. The dry condition with high July temperature and low July precipitation produces the highest soil nitrogen surplus, varying between 20 and 30 kg/ha. The accumulated nitrogen in the soil is highly water-soluble and could get exported at a high rate to the Mississippi River through melting snow or rainfall before the next growing season. Figure 6 provides the expected magnitude of nitrogen load from state of Iowa to the Mis-

**Figure 6.** Distribution of IFEW simulation model output: nitrogen surplus (*Ns*). **Figure 6.** Distribution of IFEW simulation model output: nitrogen surplus (*Ns*).

sissippi River in different weather scenarios.
