# Evaluating Ecohydrological Model Sensitivity to Input Variability with an Information-Theory-Based Approach

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Multi-Layer Canopy Model and Forcing Data

- Air temperature (Ta) affects stomatal conductance of plant and evaporation from both soil and canopy. The Campbell 083E instrument located at the flux tower has a precision for measuring Ta of $\pm 0.1{\phantom{\rule{0.277778em}{0ex}}}^{\circ}$C.
- Wind speed (U) affects canopy evaporation and associated heat flux partitioning. The precision of the CSAT3 measurement instrument at the tower for U is $\pm 0.1$ m/s.
- Vapor pressure deficit (VPD) is the difference between saturation water vapor pressure (es) and actual water vapor pressure (ea). It is the measure of atmospheric desiccation strength [46] and indicates the atmospheric demand for water vapor. The ea is determined by a large number of drivers and is also related to Ta in that it is upper bounded by es.
- Shortwave radiation (Rg) is a principle driver of $C{O}_{2}$, vapor, and heat exchange available in the system which controls photosynthesis and leaf energy balance.

- Precipitation (PPT) is often zero at the 15-min resolution considered here, such that changing the precision of PPT would not have any effect for most time steps.
- Carbon dioxide concentration (Ca) is a model parameter and is constant.
- Air pressure (Pa) varies within a very small range, and the model is relatively unresponsive to fluctuations in Pa. In other words, quantizing Pa would not lead to measurable differences in model behavior relative to other inputs.
- Canopy structure is described by leaf area density (LAD) profiles and the total leaf area index (LAI). In the model, the LAI is independent of all other forcing conditions and is based on surveys of maize or soybean plants throughout each year. It typically increases monotonically through the growing season and does not vary on a sub-daily to daily timescale.

#### 2.2. Information Theory to Manipulate Forcing Precision

#### 2.2.1. Quantization

#### 2.2.2. Illustrative Example of Quantization: Air Temperature

#### 2.2.3. Quantized Model Cases

#### 2.3. Sensitivity Analysis

_{t}) to input variability by comparing the model response to changing input precisions individually and jointly. In this regard, we defined Equation (6) to calculate the diurnal root mean square errors (RMSE) between quantized and full model cases. Since this definition is the difference between the full and quantized model with an RMSE metric, it does not indicate an “error” but a “sensitivity”. Equation (6) defines SM

_{t}for a given time of day and variable over the 100-day study window:

## 3. Results

#### 3.1. Effects of Quantizing Individual Forcing Variables

#### 3.1.1. Entropy

#### 3.1.2. Quantizing Rg Only: Changing Energy Balance Precision

_{Q}= 100 W/m${}^{2}$ for a given time point, this effect is not equally split between LE and SH. This difference in partitioning of energy is due in part to the effect of saturating net leaf carbon dioxide uptake (An) on stomatal conductance (gs) (Figure 6a) [42], which would impose an upper limit to LE and cause SH to increase to maintain the energy balance. We previously noted that due to the small amount of observed $Rg$ in the forcing data during the night, Rg is not quantized at night.

_{Q}is negative, we see larger changes in Fc, as Fc-Fc

_{Q}. This larger change means that when quantization leads to overestimating radiation input during the day (Rg

_{Q}> Rg), the resulting carbon flux is lower (positive Fc-Fc

_{Q}). However, since daytime carbon fluxes are generally negative, this indicates that Fc

_{Q}is actually larger in magnitude, and more photosynthesis is occurring. Similarly, when quantization leads to underestimating Rg input (Rg > Rg

_{Q}), we see that daytime Fc-Fc

_{Q}is negative, indicating that Fc

_{Q}decreases in magnitude relative to the regular model but to a lesser extent. This behavior relates to soil carbon respiration and vegetation photosynthesis in the model. A change in radiation alters both soil respiration and photosynthesis, but this effect is nonlinear and depends on whether radiation is increased or decreased.

#### 3.1.3. Quantizing Ta Only: Temporally Varying Responses to Quantization

_{t}, Equation (6)) within each time window for LE. First, we note that the model response to levels of quantization is nonlinear. For example, during all time windows, ${N}_{T}=2$ levels of quantization leads to the most changes in the model, and the difference between ${N}_{T}=2$ and ${N}_{T}=3$ is usually greater than that between ${N}_{T}=4$ and ${N}_{T}=5$. This shows that quantization to a very few levels makes a larger difference relative to changing precision more slightly.

_{t}for the quantized cases are lower than for later times, which means the quantized and full models are more similar. The largest SM

_{t}occurs in the last window (DOY 231-250) for ${N}_{T}=2$ levels of quantization. This time window, late in the growing season, is during mid-August until early-September which is relatively dry and warm and more water-limited relative to other time windows. During this time window, vegetation is very mature (highest LAI) which results in a large potential to uptake and release water. This large water potential causes a larger temperature dependency, such that the same quantization of Ta leads to a larger response in LE relative to earlier in the growing season, as Ta influences atmospheric demand for water and the ability to photosynthesize. In other words, when water is limited and there is high demand, the precision of Ta can have a large effect on heat fluxes. The largest SM

_{t}occurs in the last window for ${N}_{T}=2$ levels of quantization. This largest difference between the quantized model and the full model occurs at midday (Figure 7e) when Ta is maximum. This is the time when the quantized Ta time series essentially switches from its “low” value at night to its “high” value during the day.

_{t}follows the diurnal cycle in that it increases with flux magnitudes, Figure 7 shows that nighttime values are also impacted by quantization, particularly for ${N}_{T}=2$ levels of quantization. For example, the ratios of nighttime to daytime SM

_{t}value of LE (about 1:2) are a lot larger than the ratio of nighttime to daytime actual value of LE (about 1:10), meaning that nighttime values are actually affected disproportionately. This variability based on times of day and over different time windows through the growing season indicates that model sensitivity varies along with different thresholds and system states.

#### 3.2. Model Sensitivity to Combinations of Forcing Precisions

_{t}is higher during the day for all fluxes, which is associated with higher flux magnitudes during the day. We previously noted that the effect on LE due to quantization of Ta is actually disproportionately higher at night when flux magnitude is taken into account. This effect of higher flux magnitude is also true for the effect of U on LE, but not for VPD on LE or for the effect of both (U and VPD) on SH (Figure 8a,b, Case 2 and Case 3). Otherwise, the effect of quantization varies over inputs and times of day for each model case. For LE and SH (Figure 8a,b), we see that the cases for which Rg and Ta are quantized (Case Rg, Case 1, and portions of Cases 4, 5, and 7) particularly lead to more sensitivity to quantization. Meanwhile, quantizing Ta influences SH more than LE (Figure 8a,b, Case 1). SH results show a less smooth pattern relative to LE over the diurnal cycle. LE is constrained by stomatal conductance and water availability in addition to temperature, and we see changes in Ta have an effect where the diurnal cycle dominates. However, since SH is more directly associated with Ta and Rg changes, we see quantization has a more varied effect with levels of quantization and times of day.

#### 3.3. Forcing Complexity and Model Behaviors

_{t}, Equation (6)) with forcing complexity (${C}_{m}$, Equation (7). We first consider ${C}_{m}$ and 3-h averaged modeled diurnal cycle sensitivity for all seven quantized cases between 12:00–15:00 (Figure 9). In general, decreasing forcing data complexity for all quantized cases leads to larger differences between the quantized and the full model. For example, from Figure 9 we see that a 50% change in ${C}_{m}$ leads to about a 20–30 W/m${}^{2}$ change in SM

_{t}. Quantization of any input variable to $N=5$ levels or fewer will result in at least 15 W/m${}^{2}$ of change in the modeled LE (blank area in lower right corner of Figure 9). We also detect general behaviors related to each individual input variable as follows:

- Quantized VPD (yellow asterisks in Figure 9) shows a horizontal pattern in the (${C}_{m}$, SM
_{t}) space, which means quantization decreases forcing data complexity, but different levels of quantization (between $N=$ 2 and 5) do not affect model output for LE. This indicates that the modeled LE is somewhat responsive to a change away from the instrument precision of VPD but does not vary further as the precision VPD is decreased even more. In general, this forcing variable can be greatly simplified without changes in model sensitivity. Moreover, quantizing VPD leads to the lowest SM_{t}indicating the lowest sensitivity at this time of day of Cases 1-3, for which a single variable is quantized. - Quantized Ta (blue asterisks in Figure 9) similarly decreases forcing data complexity but results in different model behavior based on each level of quantization. For example, the $N=2$ scenario of Case 1 has the largest LE sensitivity among other individually quantized forcing variables at this time of day. This also translates to other quantized cases where Ta is quantized to $N=2$ levels (Case 4, 5, 7) that lead to higher model sensitivity.
- Finally, quantizing U (red asterisks in Figure 9) also influences both model behavior and forcing complexity. Quantization decreases forcing data complexity similarly, but model sensitivity is less than when Ta is quantized.

_{t}in which Ta and U are quantized (purple circles in Figure 9). In Case 5, for which Ta and VPD are quantized (green circles in Figure 9), the tightly clustered subsets along the ${C}_{m}$ axis are associated with quantized VPD, and the differences in SM

_{t}are mainly attributable to different quantization levels for Ta. This follows the previous finding that quantizing VPD on its own (Case 3) does not lead to changes in model behavior. In Case 6, in which U and VPD are quantized (orange circles in Figure 9), while there is a similar horizontal pattern for changes in VPD quantization levels, smaller differences in SM

_{t}and ${C}_{m}$ between groupings are due to quantizing U.

_{t}and differences in ${C}_{m}$ in each cluster. This indicates that there are groupings of model inputs for which the forcing complexity can be greatly decreased without changing model sensitivity, but behavior depends on which forcing variable is altered. Namely, for LE the differences between clusters are associated with changing the precision of Ta as determined from the other cases.

_{t}during the day for the heat fluxes (Figure 9b,e), where the lowest precision cases lead to a large reduction in complexity and tend to have the most impact on modeled fluxes. In other words, there is an approximate line that relates ${C}_{m}$ to SM

_{t}during the day (middle panels) even though ${C}_{m}$ reflects a different combination of forcing variables that are manipulated. Meanwhile for early morning and evening time ranges, the relationship between forcing complexity and model behavior is more related to whether air temperature (Ta) is quantized or not.

_{t}during the nighttimes, indicating the model is particularly unresponsive to the precision of those variables. This can be explained in that U and VPD are typically low and relatively stable during the night, so quantizing has less effect on model results. Finally, for Fc, the gap from the model to the quantized cases is always the same (Figure 10g–i). This indicates that Fc has a response to quantization with a different threshold than the other fluxes. For example, Fc may show a more variable response for quantization to many higher levels which are not considered here.

## 4. Discussion and Conclusions

**Energy balance constraints:**When incoming solar radiation is quantized, sensible and latent heat fluxes shift to meet the energy balance with some gaps that can be attributed to other energy fluxes that are not analyzed here, specifically ground heat flux and longwave radiation. Canopy carbon fluxes respond more nonlinearly and depend on whether the quantization increases or decreases the apparent radiation.**Temporal variability and dependence on system states:**We next explored model effects of air temperature quantization for different growing season time windows. This highlighted the connection between model sensitivity to forcing precision and system states, such as wetness or vegetation conditions. For example, quantized air temperature most influences latent heat fluxes during a relatively hot and dry period at the end of the growing season. This connects to previous findings based on observed data that causal interactions in ecohydrologic systems vary seasonally and with wetness conditions [55].**Joint effects of multiple forcing precisions:**Finally, we quantized combinations of air temperature, vapor pressure deficit, and wind speed to different extents, and plotted model responses along a forcing complexity axis, where forcing complexity indicated the extent to which the joint entropy of the forcing variables was reduced. From this, we can determine which combinations of variables are “most compressible” in terms of their ability to reduce model forcing complexity and which lead to large changes in model behavior. For example, a quantization that leads to vary forcing complexity with a very small change in model behavior would indicate that the model is not sensitive to a particular input or combinations of inputs, and the model forcing or use of that input could be simplified in some way. On the other hand, this could also indicate that the model is not utilizing the full extent of available information in the forcing and could motivate other changes to the model structure. We saw a general trend in which simulated heat fluxes diverge more from the original model case as multiple variables were increasingly quantized, but there was a wide range of forcing complexities that led to the same model sensitivity. This illustrates that the effect of forcing precision varies depending on the degree of quantization (N) and the combination of variables involved. In other words, it is possible to determine a set of forcing variables that can be maximally simplified with minimal change in model behavior.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

IT | Information Theory |

SA | Sensitivity Analysis |

SM | Sensitivity Metric |

MLCan | Multi-Layer Canopy-soil-root |

CINet | Critical Zone Interface Network |

IMLCZO | Managed Landscape Critical Zone Observatory |

RMSE | Root Mean Square Error |

GSA | Global Sensitivity Analysis |

SUMMA | Structure for Unifying Multiple Modeling Alternatives |

JULES | Joint UK Land Environment Simulator |

MMSE | Minimum Mean Square Error |

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**Figure 2.**(

**a**–

**d**) Illustration of rate-distortion theory applied to air temperature (Ta) to quantize the data to different precisions. Red lines indicate the representation points and dashed lines indicate final decision thresholds for ${N}_{T}=2,\cdots ,5$ levels of quantization for Ta in the growing season (DOY 150-250). The blue bars indicate the underlying probability mass function of Ta. Representation points alternately calculated with the fixed binning method are shown with red dots.

**Figure 3.**The original Ta forcing data and quantized forcing data for the growing season based on the Lloyd algorithm: (

**a**) original Ta time-series data, where red lines indicate start and end of the peak growing season (DOY 150-250); (

**b**) simplified Ta time- series data with ${N}_{T}=2$ levels of quantization, (

**c**), ${N}_{T}=3$ levels of quantization, (

**d**) ${N}_{T}=4$ levels of quantization, and (

**e**) ${N}_{T}=5$ levels of quantization.

**Figure 4.**Model cases of simplified forcing data formed on different individual or joint quantization of the input variables (indicated by $\widehat{}$ symbol) and levels of quantization ($N\in [2,3,4,5]$). $\widehat{Ta}=$ quantized air temperature to ${N}_{T}$ levels, $\widehat{U}=$ quantized wind speed to ${N}_{U}$ levels, $\widehat{VPD}=$ quantized vapor pressure deficit to ${N}_{VPD}$ levels and $\widehat{Rg}=$ quantized shortwave radiation to ${N}_{Rg}$ levels.

**Figure 5.**Differences between the entropy of the quantized forcing data of Ta (red dots), U (blue dots), VPD (green dots), and Rg (magenta dots) into N bins using fixed binning (lower points) and the Lloyd algorithm (higher points). A longer vertical line indicates that the Lloyd algorithm retains significantly more information relative to fixed binning. Black dots indicates maximum possible entropy (${H}_{max}$) for each level of quantization based on a uniform distribution.

**Figure 6.**Quantized diurnal flux variation: (

**a**) LE (blue circles), SH (red stars), and LE+SH (green squares); and (

**b**) Fc (black diamonds) with quantized shortwave radiation Rg for $N=2$ level of quantization. X-axis indicates the difference between quantized shortwave radiation Rg and original forcing variable Rg. Y-axis indicates the difference between quantized model results and the full model results. The dashed line shows that the change in Rg is typically larger than LE+SH.

**Figure 7.**Quantized model sensitivity (SM

_{t}) for Case 1 ($\widehat{\mathit{Ta}},\mathit{U},\mathit{VPD}$) of LE (W/m${}^{2}$) within five 20-day time windows through the growing season: (

**a**) for DOY 150-170; (

**b**) for DOY 171-190; (

**c**) for DOY 191-210; (

**d**) for DOY 211-230; (

**e**) for DOY 231-250; (

**f**) the total PPT (mm) and the averaged Ta (${}^{\circ}$C) within given time window.

**Figure 8.**Diurnal quantized model sensitivity (SM

_{t}) for: (

**a**) LE (W/m${}^{2}$); (

**b**) SH (W/m${}^{2}$); and (

**c**) Fc ($\mathsf{\mu}$mol/m${}^{2}\mathrm{s}$) for all cases (Equation (6)). Red colors indicate high sensitivities to quantization.

**Figure 9.**Comparison between the model and all quantized cases for ($N\in [2,3,4,5]$) based on their forcing data complexity, ${C}_{m}$, and 3-h averaged modeled diurnal sensitivity, SM

_{t}, at 12:00–15:00 for LE. The cyan star indicates the full model case (${C}_{m}=1$ and SM

_{t}$=0$). Asterisks indicate individual quantized variables (Case 1–3, 4 model runs for each case), circles indicate jointly quantized variables (Cases 4–6, 16 model runs for each case), and black squares indicate Case 7 where all three variables are quantized and there are 64 individual model runs.

**Figure 10.**Comparison between full model and quantized cases (${N}_{T}=2$, ${N}_{U}=2$ and ${N}_{VPD}=2$ levels) based on their forcing data complexity and 3-h averaged model diurnal SM

_{t}for LE (

**a**–

**c**), SH (

**d**–

**f**), and Fc (

**g**–

**i**) for three times of day. Left panels (

**a**,

**d**,

**g**) indicate all 7 quantized model behaviors at 3:00–6:00 am. Middle panels (

**b**,

**e**,

**h**) indicate all 7 quantized model behaviors at 9:00–12:00. Right panels (

**c**,

**f**,

**i**) indicate model behaviors at 18:00–21:00.

**Table 1.**Specification of quantized forcing variable. The optimal bin width (D) is derived from $\frac{3.5\sigma}{{n}^{\frac{1}{3}}}$ [48], where $\sigma $ is the standard deviation of the data, and n is the total number of data values.

Input Variable | Unit | Stdev | Opt Bin Width |
---|---|---|---|

Ta | ${}^{\circ}$C | 3.7878 | 0.5 |

U | m s${}^{-1}$ | 1.7032 | 0.3 |

VPD | kPa | 0.6029 | 0.1 |

Rg | W/m${}^{2}$ | 349.02 | 50 |

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## Share and Cite

**MDPI and ACS Style**

Farahani, M.A.; Vahid, A.; Goodwell, A.E.
Evaluating Ecohydrological Model Sensitivity to Input Variability with an Information-Theory-Based Approach. *Entropy* **2022**, *24*, 994.
https://doi.org/10.3390/e24070994

**AMA Style**

Farahani MA, Vahid A, Goodwell AE.
Evaluating Ecohydrological Model Sensitivity to Input Variability with an Information-Theory-Based Approach. *Entropy*. 2022; 24(7):994.
https://doi.org/10.3390/e24070994

**Chicago/Turabian Style**

Farahani, Mozhgan A., Alireza Vahid, and Allison E. Goodwell.
2022. "Evaluating Ecohydrological Model Sensitivity to Input Variability with an Information-Theory-Based Approach" *Entropy* 24, no. 7: 994.
https://doi.org/10.3390/e24070994