# Toward a Priori Evaluation of Relative Worth of Head and Conductivity Data as Functions of Data Densities in Inverse Groundwater Modeling

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Experimental Design

#### 2.1.1. Synthetic Domain

^{−1}to 200 m day

^{−1}. It needs to be noted that the absolute value of this range is irrelevant if non-dimensionalization of the system is valid, which means the conclusions are only dependent on the dimensionless numbers. Then, we fitted a spherical model to the extracted empirical semi-variogram. The formula for the spherical model is:

^{®}[30] to find the c and $\lambda $ that minimized the sum of squared differences between the theoretical model in Equation (3) and the empirical variogram. We cut off the empirical semi-variogram at 8000 m before fitting the variogram model.

#### 2.1.2. Synthetic Observations of H and K

#### 2.2. Inverse Modeling

**Q**is a weight matrix used to define greater contributions of certain pairs of observations. In the present simulation, a uniform weight is assigned to all K observations. $\mathit{M}$ is the model that predicts the system responses, given the parameter set $\mathit{K}$, and ${\mathit{K}}^{\mathit{c}}$ is the calibrated K field that minimizes ${\mathit{\Phi}}_{\mathit{g}}$.

**Q**is a diagonal matrix consisting of squared weights assigned to each observation,

_{r}**R**is a regularization operator that expresses a certain geostatistical constraint, e.g., the difference between a trial parameter value and the parameter value at a site, given neighbors’ values (and a variogram model), and $\mathit{d}$ is a ‘system-preferred’ state, which is 0 in this case.

#### 2.3. Non-Dimensionalization of Data Density and Errors

#### 2.4. Recharge and Boundary Conditions

#### 2.5. Experimental Design and Multivariate Polynomial Curve Fitting

**P**is a vector of coefficients in polynomial fitting, $\stackrel{\rightharpoonup}{\mathsf{\mu}}=\left(1,{\mathsf{\mu}}_{H},{\mathsf{\mu}}_{H}^{2},\dots ,{\mathsf{\mu}}_{K},{\mathsf{\mu}}_{K}^{2},{\mathsf{\mu}}_{H}{\mathsf{\mu}}_{K}\right)$ is the vector of predictors, and $e\left({\mathsf{\mu}}_{H},{\mathsf{\mu}}_{K}\right)$ is the calibration error. Our experiments were constrained within the range of $\mathsf{\mu}$ values.

**P**was fitted using Matlab

^{®}curvefitting Toolbox. The term ${\mathsf{\mu}}_{H}{\mathsf{\mu}}_{K}$ in Equation (9) represents an interaction term. A probability value (p-value) was calculated for the null hypothesis that the coefficient is equation to zero based on t-tests for each of the curve fitting coefficients. Furthermore, the goodness of fit was evaluated using the coefficient of determination (R

^{2}) and the root mean of squared error between the calibrated K/H errors and the values predicted by the polynomial function.

#### 2.6. The Influence of Measurement Noise

#### 2.7. Relative Data Worth

## 3. Results and Discussion

#### 3.1. Verification of the Effectiveness of Non-Dimensionalization

#### 3.2. Impact of Recharge and Boundary Condition on Model Calibration Errors

#### 3.3. Errors as a Function of Normalized Data Densities

#### 3.4. Multivariate Polynomial Curve Fitting

**P**of Equation (9). Meanwhile, the quadratic term, ${\mu}_{K}^{2}$, is statistically significant (p-value = 0.001) for ${e}_{H}$. However, the R

^{2}without the quadratic term is adequately high, and adding the term does not increase it notably. In the interest of parsimony, we chose not to include ${\mu}_{K}^{2}$ in the fitted formula for ${e}_{H}$ either. As there are no effective dimensionless numbers that characterize ${e}_{H}$, we focused on ${e}_{K}$.

#### 3.5. Relative Data Value

#### 3.6. The Influence of Noise with K Observations

- When ${\mu}_{H}$ is fixed, $\beta $, in general, grows as a function of increasing ${\mu}_{K}$ (Figure 9), but when ${\mu}_{K}<3$, the error amplification is almost close to 1 and $\beta $ is not very sensitive. The largest impact in this category is with the case (${\mu}_{K}=1.45$, ${\mu}_{H}=1.3$, ${\mathsf{\sigma}}_{n}=1$). Recall in this case, 33% of the K data points have been perturbed more than an order of magnitude, but the impact on calibrated K nonetheless appears limited ($\beta =1.29$). However, when ${\mu}_{K}$ is larger than 3, the errors grow significantly. At (${\mu}_{K}=6.48$, ${\mu}_{H}=7.24$, ${\mathsf{\sigma}}_{n}=0.5$), $\beta =11.6$ (the upper-rightmost point in Figure 9), which means the inversion essentially failed. Another such case is (${\mu}_{K}=5.12$, ${\mu}_{H}=7.24$, ${\mathsf{\sigma}}_{n}=1$) and $\beta =4.57$ (visible in Figure 9 blue line). At (${\mu}_{K}=6.48$, ${\mu}_{H}=7.24$, ${\mathsf{\sigma}}_{n}=0.2$), even if the perturbation is moderate, it still causes a significant error amplification ($\beta =1.47$, the rightmost circle on the lower solid green line).
- Larger ${\mu}_{H}$ can help inhibit error amplification. When ${\mu}_{K}$ and ${\mathsf{\sigma}}_{n}$ are kept the same and ${\mu}_{H}$ is increased, $\beta $ always decreases. However, this effect is small when ${\mu}_{K}<3.5$ because the amplification is already small.
- Even though larger ${\mu}_{K}$ increases the error amplification from a noise-free baseline (Figure 9), incorporating K data points nonetheless reduces error compared to corresponding cases with ${\mu}_{K}=0$ (note ${\beta}_{0}$ in Figure 9), as long as ${\mu}_{K}$ is not too large. For example, under a sparse-data scenario (${\mu}_{K}=0.45$ or ${\mu}_{K}=1.3$), even if ${\mathsf{\sigma}}_{n}=1$, which means significant noise, the error is still less than the case without incorporating K data.
- Under the combined conditions of high ${\mu}_{K}$ ($>3.5$) (again, this means there are on average 3.5
^{2}= 12.25 conductivity data points in an area of ${\lambda}^{2}$) and high noise (${\mathsf{\sigma}}_{n}\ge 0.5$), the error amplification skyrockets and dominates over the information content of K. For example, (${\mu}_{K}=5.1$, ${\mu}_{H}=7.2$, ${\mathsf{\sigma}}_{n}=1$), $\beta =4.57$, and ${e}_{K}=0.21$ which is greater than the case with the same ${\mu}_{H}$ but without K data (${\mu}_{K}=0$). At ${\mu}_{K}=6.5$ and ${\mu}_{H}=7.2$, even a ${\mathsf{\sigma}}_{n}$ of 0.5 is sufficient to bring the error amplification factor to 11.6. For these cases, the errors are too large, and the inversion failed.

## 4. Conclusions

## 5. Limitations

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Experimental configurations and errors for the verifications of dimensionless numbers are shown in Figure 3.

Cluster | ${\mathsf{\mu}}_{\mathit{H}}$ | ${\mathsf{\mu}}_{\mathit{K}}$ | $\mathbf{\lambda}$ | ${\mathit{n}}_{\mathit{H}}$ | ${\mathit{n}}_{\mathit{K}}$ | ${\mathit{e}}_{\mathit{H}}$ | ${\mathit{e}}_{\mathit{K}}$ |
---|---|---|---|---|---|---|---|

A | 3.4 | 0 | 725 | 1444 | 0 | 0.0041 | 0.3 |

3.5 | 0 | 1752 | 256 | 0 | 0.008 | 0.29 | |

3.66 | 0 | 2088 | 196 | 0 | 0.017 | 0.27 | |

B | 4.8 | 0 | 2088 | 361 | 0 | 0.006 | 0.19 |

5 | 0 | 2401 | 256 | 0 | 0.007 | 0.2 | |

4.8 | 0 | 2966 | 169 | 0 | 0.011 | 0.17 | |

5 | 0 | 3621 | 121 | 0 | 0.014 | 0.08 | |

C | 7.2 | 0 | 3621 | 256 | 0 | 0.101 | 0.005 |

7.4 | 0 | 2966 | 169 | 0 | 0.980 | 0.004 | |

E | 1.5 | 0.45 | 725 | 256 | 25 | N/A | 0.36 |

1.4 | 0.64 | 725 | 256 | 49 | N/A | 0.35 | |

1.5 | 0.69 | 1751 | 49 | 16 | N/A | 0.33 | |

F | 3.1 | 2.95 | 1751 | 256 | 196 | N/A | 0.18 |

3.4 | 3.13 | 2088 | 169 | 144 | N/A | 0.18 | |

4.3 | 3.04 | 2088 | 256 | 121 | N/A | 0.16 | |

G | 4.8 | 4.3 | 2401 | 256 | 196 | N/A | 0.081 |

5.9 | 4.194 | 2966 | 256 | 121 | N/A | 0.08 | |

4.7 | 4.2 | 2088 | 324 | 256 | N/A | 0.1 |

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**Figure 1.**Random log(K) fields generated for our experiments. λ is the correlation length of the log(K) field. Smaller λ produces more heterogeneous fields and larger distances between data points.

**Figure 2.**Locations of observation points (black) and pilot points (red). Boundaries of cells constituting the domain grid are shown with black grid lines. The elevation of the ground surface is uniformly set at 150 m. North and south sides of the domain are no-flow boundary conditions while the eastern side is Dirichlet with a value of h

_{BC2}; h

_{BC2}is 130 m by default but varied during the tests for the effects of recharge and boundary conditions (Section 2.4). The Western side of the domain is no-flow by default but Dirichlet was also tested during in the experiments about boundary conditions.

**Figure 3.**Verifications of normalized data densities as effective dimensionless numbers for ${e}_{K}$ (

**a**) and ${e}_{H}$ (

**b**). A–G annotate clusters. When we tested the effectiveness of ${\mu}_{H}$ (blue circles), ${\mu}_{K}$ was kept constant. Similarly, when we tested ${\mu}_{K}$ (green squares), ${\mu}_{H}$ was kept constant. In the three blue circle clusters, water head is the only synthetic observational data in the model. Thus, their ${\mu}_{K}$ are zero. From the clustering pattern in the left panel, it is clear that ${e}_{K}$ is similar for similar (${\mu}_{H}$, ${\mu}_{K}$) pairs, even though the correlation lengths and the number of data points are different. This figure suggests (${\mu}_{H}$, ${\mu}_{K}$) are effective dimensionless parameters controlling ${e}_{K}$. However, it is not the case for ${e}_{H}$. The experimental configurations and results for the clusters on these figures are provided in Table A1.

**Figure 4.**Rescaled water head error (

**a**) and conductivity error (

**b**) as functions of recharge under two different H data densities. The errors are obviously impacted by ${\mathsf{\mu}}_{H}$, but at each ${\mu}_{H}$ level, recharge does not influence ${e}_{K}$ or ${e}_{H}$. This pattern allows us to remove recharge as a control variable from our experiments. (

**c**,

**d**) ${e}_{K}$ (

**c**) and ${e}_{H}$ (

**d**) as functions of normalized data densities. BC1 means domain with one Dirichlet boundary and BC2 stands for the domain with two Dirichlet boundaries. Since different boundary conditions generate the same curves, it indicates our analysis of ${e}_{K}$ can be valid for different boundary conditions.

**Figure 5.**Normalized K error (

**a**) and H error (

**b**), and dimensionalized K error (

**c**) and H error (

**d**), as functions of normalized data densities. Each point on the plot represents the mean of three calibrations, except when $x\%=0$. It is obvious that ${e}_{K}$ decreases as ${\mu}_{K}$ or ${\mu}_{H}$ increases and ${e}_{K}~{\mu}_{K}$ at each ${\mu}_{H}$ level is almost linear.

**Figure 6.**3D visualization of adjusted multiple polynomial curve fitting of ${e}_{K}$ (

**a**) and ${e}_{H}$ (

**b**) as a function of normalized data densities. The data fall close to the surface. Some data points fall below the surface and are not visible at the shown angles.

**Figure 7.**Contour representation of the surface fitted to ${e}_{K}$ (

**a**) and ${e}_{H}$ (

**b**) as functions of data densities. Scattered points indicate the data points used to construct the contours.

**Figure 8.**${R}_{\mu}$ of K error (

**a**) and H error (

**b**) as functions of normalized data densities ${\mu}_{\mathrm{H}}$ and ${\mu}_{\mathrm{K}}$. Bold lines highlight the less-than-one values among the contours. (

**c**–

**f**) ${R}_{n}$ and ${R}_{n}^{\prime}$ as functions of ${\mathrm{n}}_{\mathrm{H}}$, ${\mathrm{n}}_{\mathrm{K}}$, and λ. ${R}_{n}$ is calculated assuming each new K data point entails a new H data point. ${R}_{n}^{\prime}$ is calculated assuming new K and H data points are independent of each other.

**Figure 9.**Error amplification factor $\beta $, the ratio of errors between “with noisy K data,” and “with noise-free data” as functions of ${\mu}_{K}$, ${\mu}_{H}$, and ${\sigma}_{n}$. As explained before, since this experiment is very expensive, we could only afford to explore a few lines. For comparison, dashed lines indicate ${\beta}^{0}$.

**Table 1.**Multivariate polynomial curve fitting for the following equation: $e\left({\mathsf{\mu}}_{H},{\mathsf{\mu}}_{K}\right)={p}_{0}+{p}_{1}{\mathsf{\mu}}_{H}+{p}_{2}{\mathsf{\mu}}_{K}+{p}_{3}{\mathsf{\mu}}_{H}^{2}+{p}_{4}{\mathsf{\mu}}_{K}^{2}+{p}_{5}{\mu}_{H}{\mu}_{K}$. p-value is the probability of the null hypothesis that the corresponding coefficient is equal to 0 according to the t-statistic.

Calibration Errors | Coefficients in Fitting Equation | Fitting Goodness | |||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{p}}_{0}$ | ${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | ${\mathit{p}}_{3}$ | ${\mathit{p}}_{4}$ | ${\mathit{p}}_{5}$ | R^{2} | ${\mathit{e}}_{\mathit{f}}$ | ||

${e}_{K}$ | p-Value | 0 | 0 | 0 | 0.946 | 0.697 | 0 | 0.975 | 0.0162 |

Value | 0.464 | −0.0533 | −0.061 | 0 | 0 | 0.0075 | |||

${e}_{H}$ | p-Value | 0 | 0 | 0 | 0.939 | 0.001 | 0 | 0.973 | 0.00043 |

Value | 0.012 | 0.0011 | −0.00156 | 0 | 0 | 0.00015 | −0.982 | −0.00035 |

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## Share and Cite

**MDPI and ACS Style**

Sun, N.; Fang, K.; Shen, C.
Toward a Priori Evaluation of Relative Worth of Head and Conductivity Data as Functions of Data Densities in Inverse Groundwater Modeling. *Water* **2019**, *11*, 1202.
https://doi.org/10.3390/w11061202

**AMA Style**

Sun N, Fang K, Shen C.
Toward a Priori Evaluation of Relative Worth of Head and Conductivity Data as Functions of Data Densities in Inverse Groundwater Modeling. *Water*. 2019; 11(6):1202.
https://doi.org/10.3390/w11061202

**Chicago/Turabian Style**

Sun, Nuan, Kuai Fang, and Chaopeng Shen.
2019. "Toward a Priori Evaluation of Relative Worth of Head and Conductivity Data as Functions of Data Densities in Inverse Groundwater Modeling" *Water* 11, no. 6: 1202.
https://doi.org/10.3390/w11061202