# Implementation of the Kalman Filter for a Geostatistical Bivariate Spatiotemporal Estimation of Hydraulic Conductivity in Aquifers

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{2}/day

^{2}value for K for all the cases, however, the smallest number of cells with values above 2 m

^{2}/day

^{2}correspond to the bivariate spatiotemporal case) and the best agreement between the estimated errors and the selected model variance (SMSE values of 0.574 and 0.469) were found for the bivariate cases, which suggests that the implemented methodology could be used for reducing calibration efforts, particularly when the hydraulic parameters data are scarce.

## 1. Introduction

## 2. Materials and Methods

- Univariate estimation (based on the spatial correlation of hydraulic conductivity data only).
- Bivariate or cross estimation (hydraulic conductivity as the primary variable and hydraulic head for a single time as the secondary).
- Multivariate spatiotemporal estimation (based on the correlation between the hydraulic conductivity data and the hydraulic head spatiotemporal data).

#### 2.1. Geostatistical Theory

#### 2.1.1. The Spatial Variogram

_{i}) and Z(x

_{i}+ h) are the tail and head values, respectively. i is the position of coordinates (x

_{i}, y

_{i}). The separation vector h is specified with some direction and distance (lag) tolerance [36].

#### 2.1.2. The Cross Variogram

#### 2.1.3. The Spatiotemporal Variogram

_{s}and h

_{t}, respectively) is based on the following expression:

_{s},h

_{t}) is the number of pairs Z(x

_{i},t

_{k}) and Z(x

_{i}+ h

_{s},t

_{k}+ h

_{t}) separated by increments h

_{s}and h

_{t}, T is the total number of times with data. h

_{t}is the temporal distance between two spatiotemporal positions.

#### 2.2. Multivariate Spatiotemporal Methodology

- From geostatistical analyses, obtain the spatial variogram model for the hydraulic conductivity with the available data (in the case study, it is assumed that only 60 of the total 400 hydraulic conductivity data of the model are known), the spatiotemporal variogram model for hydraulic head with the values simulated each 4 months for a 2-year period (2400 values in total), and the cross variograms for each simulation time of the model between the hydraulic conductivity (60 data) and the corresponding hydraulic head data (400 values).
- Derive separately the covariance matrices for the hydraulic conductivity, the spatiotemporal hydraulic head data and the cross covariances between the hydraulic conductivity and the hydraulic head for each time. The covariance values are obtained for all the estimation and sampling nodes.
- Integrate a multivariable covariance matrix that includes the spatial, spatiotemporal and cross covariances (Table 4).
- Estimate using the static Kalman filter, the hydraulic conductivity in all the nodes where these data were not available for the geostatistical analyses.

#### 2.3. The Static Kalman Filter

_{t}. The transition between states X

_{t}⟶X

_{t+1}is characterized by the transition matrix A and the addition of a Gaussian white noise vector w

_{t}with covariance matrix Q

_{t}. The vectors and matrices are composed by N and N×N elements, respectively; the same applies for the subsequent cases.

_{t}to the state of the system X

_{t}through the measurement matrix H and the addition of a Gaussian white noise vector v

_{t}with covariance matrix R

_{t}. Both random variables w

_{t}and v

_{r}are assumed to be independent of each other. The measurement model is as follows:

_{t}and the measurement error R

_{t}in the most general form of the filter may change over time; however, for simplicity they can be assumed as constants.

#### 2.4. Case Study

^{3}/

^{3}, and the hydraulic conductivity ($\mathrm{K}$) values are in the range of 0.2–10 m/day corresponding to clays and sands. The domain was discretized in a block-centered finite difference grid composed by 20 columns and 20 rows (the size of the mesh cells is 250 m × 250 m), so information of hydraulic (hydraulic conductivity and storage coefficient) and hydrogeological (initial conditions) parameters were needed for 400 nodes. The boundary conditions were constant hydraulic head values of 900 m and 950 m at the north and south, respectively, and no-flow boundaries at east and west. Three fully-penetrating pumping wells were included with flow rates of 1000 m

^{3}/day each (Figure 1). The initial head was of 1000 m for all the modelled area.

## 3. Results and Discussion

#### 3.1. Univariate Estimation

#### 3.2. Bivariate Estimation

#### 3.3. Bivariate Spatiotemporal Estimation

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Hydraulic conductivity spatial configuration and location of pumping wells for the proposed synthetic model.

$\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{x}}_{\mathrm{i}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{x}}_{\mathrm{i}+1}\right)$ | $\dots $ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{x}}_{\mathrm{N}}\right)$ |

$\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1},{\mathrm{x}}_{\mathrm{i}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1},{\mathrm{x}}_{\mathrm{i}+1}\right)$ | $\dots $ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1},{\mathrm{x}}_{\mathrm{N}}\right)$ |

⋮ | ⋮ | ⋮ | ⋮ |

$\mathrm{C}\left({\mathrm{x}}_{\mathrm{N}},{\mathrm{x}}_{\mathrm{i}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{N},}{\mathrm{x}}_{\mathrm{i}+1}\right)$ | $\dots $ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{N}},{\mathrm{x}}_{\mathrm{N}}\right)$ |

${\mathrm{C}}_{\mathrm{ZY}}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{x}}_{\mathrm{i}}\right)$ | ${\mathrm{C}}_{\mathrm{ZY}}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{x}}_{\mathrm{i}+1}\right)$ | $\dots $ | ${\mathrm{C}}_{\mathrm{ZY}}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{x}}_{\mathrm{N}}\right)$ |

${\mathrm{C}}_{\mathrm{ZY}}\left({\mathrm{x}}_{\mathrm{i}+1},{\mathrm{x}}_{\mathrm{i}}\right)$ | ${\mathrm{C}}_{\mathrm{ZY}}\left({\mathrm{x}}_{\mathrm{i}+1},{\mathrm{x}}_{\mathrm{i}+1}\right)$ | $\dots $ | ${\mathrm{C}}_{\mathrm{ZY}}\left({\mathrm{x}}_{\mathrm{i}+1},{\mathrm{x}}_{\mathrm{N}}\right)$ |

⋮ | ⋮ | ⋮ | ⋮ |

${\mathrm{C}}_{\mathrm{ZY}}\left({\mathrm{x}}_{\mathrm{N}},{\mathrm{x}}_{\mathrm{i}}\right)$ | ${\mathrm{C}}_{\mathrm{ZY}}\left({\mathrm{x}}_{\mathrm{N},}{\mathrm{x}}_{\mathrm{i}+1}\right)$ | $\dots $ | ${\mathrm{C}}_{\mathrm{ZY}}\left({\mathrm{x}}_{\mathrm{N}},{\mathrm{x}}_{\mathrm{N}}\right)$ |

$\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}},{\mathrm{x}}_{\mathrm{i},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}},{\mathrm{x}}_{\mathrm{i},\mathrm{t}+1}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}},{\mathrm{x}}_{\mathrm{i},\mathrm{NT}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}},{\mathrm{x}}_{\mathrm{NP},\mathrm{NT}}\right)$ |

$\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}},{\mathrm{x}}_{\mathrm{i},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}},{\mathrm{x}}_{\mathrm{i},\mathrm{t}+1}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}},{\mathrm{x}}_{\mathrm{i},\mathrm{NT}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}},{\mathrm{x}}_{\mathrm{NP},\mathrm{NT}}\right)$ |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

$\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}},{\mathrm{x}}_{\mathrm{i},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}},{\mathrm{x}}_{\mathrm{i},\mathrm{t}+1}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}},{\mathrm{x}}_{\mathrm{i},\mathrm{NT}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}},{\mathrm{x}}_{\mathrm{NP},\mathrm{NT}}\right)$ |

$\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i},\mathrm{t}+1}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i},\mathrm{NT}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{NP},\mathrm{NT}}\right)$ |

$\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i},\mathrm{t}+1}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i},\mathrm{NT}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1},{\mathrm{x}}_{\mathrm{NP},\mathrm{NT}}\right)$ |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

$\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i},\mathrm{t}+1}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i},\mathrm{NT}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1},{\mathrm{x}}_{\mathrm{NP},\mathrm{NT}}\right)$ |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

$\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{NT}},{\mathrm{x}}_{\mathrm{i},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{NT}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{NT}},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{NT}},{\mathrm{x}}_{\mathrm{i},\mathrm{t}+1}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{NT}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{NT}},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{NT}},{\mathrm{x}}_{\mathrm{i},\mathrm{NT}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{NT}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i},\mathrm{NT}},{\mathrm{x}}_{\mathrm{NP},\mathrm{NT}}\right)$ |

$\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}},{\mathrm{x}}_{\mathrm{i},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}},{\mathrm{x}}_{\mathrm{i},\mathrm{t}+1}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}},{\mathrm{x}}_{\mathrm{i},\mathrm{NT}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}},{\mathrm{x}}_{\mathrm{NP},\mathrm{NT}}\right)$ |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

$\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{NT}},{\mathrm{x}}_{\mathrm{i},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{NT}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{NT}},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{NT}},{\mathrm{x}}_{\mathrm{i},\mathrm{t}+1}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{NT}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{NT}},{\mathrm{x}}_{\mathrm{NP},\mathrm{t}+1}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{NT}},{\mathrm{x}}_{\mathrm{i},\mathrm{NT}}\right)$ | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{NT}},{\mathrm{x}}_{\mathrm{i}+1,\mathrm{NT}}\right)$ | … | $\mathrm{C}\left({\mathrm{x}}_{\mathrm{NP},\mathrm{NT}},{\mathrm{x}}_{\mathrm{NP},\mathrm{NT}}\right)$ |

_{i,t}= spatiotemporal position that corresponds to the spatial position i of coordinates (x

_{i},y

_{i}) and time t. NP = number of spatial positions. NT = number of times. C(x

_{i,t1},x

_{j,t2}) Covariance value between the spatiotemporal position that corresponds to the spatial position i of coordinates (x

_{i},y

_{i}) and time t

_{1}and the spatiotemporal position that corresponds to the spatial position j of coordinates (x

_{j},y

_{j}) and time t

_{2.}

Univariate spatial covariance matrix (hydraulic conductivity only) | Cross-covariance matrix (hydraulic conductivity–hydraulic head for time 1) | Cross-covariance matrix (hydraulic conductivity–hydraulic head for time 2) | … | Cross-covariance matrix (hydraulic conductivity–hydraulic head for time NT) |

Cross-covariance matrix (hydraulic head for time 1–hydraulic conductivity) | Space–time covariance matrix (hydraulic head from time 1 to time NT) | |||

Cross-covariance matrix (hydraulic head for time 2–hydraulic conductivity) | ||||

… | ||||

Cross-covariance matrix (hydraulic head for time NT–hydraulic conductivity) |

Case | Variable | Correlation | Nugget (m^{2}) | Sill(m^{2}) | Range (m) |
---|---|---|---|---|---|

Univariate | K | 0 * | 6.69 * | 2177.608 | |

H1 | 0 | 0.075 | 1616.284 | ||

H2 | 0 | 0.209 | 4327.53 | ||

H3 | 0 | 0.413 | 4364.75 | ||

H4 | 0 | 0.491 | 4364.75 | ||

H5 | 0 | 0.541 | 4364.75 | ||

H6 | 0 | 0.571 | 4364.75 | ||

Bivariate | K-H1 | −0.282 | 0 ^{µ} | −0.200 ^{µ} | 1500 |

K-H2 | −0.135 | 0 ^{µ} | −0.160 ^{µ} | 1500 | |

K-H3 | −0.120 | 0 ^{µ} | −0.200 ^{µ} | 1800 | |

K-H4 | −0.127 | 0 ^{µ} | −0.230 ^{µ} | 1800 | |

K-H5 | −0.131 | 0 ^{µ} | −0.250 ^{µ} | 1800 | |

K-H6 | −0.133 | 0 ^{µ} | −0.260 ^{µ} | 1800 | |

H1-K | −0.173 | 0 ^{µ} | −0.200 ^{µ} | 1550 | |

H2-K | −0.251 | 0 ^{µ} | −0.260 ^{µ} | 2000 | |

H3-K | −0.294 | 0 ^{µ} | −0.340 ^{µ} | 1800 | |

H4-K | −0.306 | 0 ^{µ} | −0.380 ^{µ} | 1800 | |

H5-K | −0.309 | 0 ** | −0.400 ^{µ} | 1800 | |

H6-K | −0.326 | 0 ** | −0.430 ^{µ} | 1800 | |

Spatiotemporal | Space H | 0.09 | 0.41 | 3000 | |

Time H | 0.14 ^{Ω} | 0.09 ^{Ω} | 1.20 ^{β} | ||

Space-time H | 0.41 ^{α} |

^{µ}${\mathrm{m}}^{2}/\mathrm{day}$,

^{Ω}${\mathrm{day}}^{-2}$,

^{β}${\mathrm{day}}^{-1}$,

^{α}$\mathrm{m}\xb7\mathrm{day}$; K is the hydraulic conductivity; HN are data for time N; K-HN means cross variogram for K as primary and HN as the secondary variable; HN-K means cross variogram for HN as primary and K as the secondary variable.

Case | Variable | Minimum Error (m) | Maximum Error (m) | Mean Error (m) | MSE (m^{2}) | RMSE | SMSE |
---|---|---|---|---|---|---|---|

Univariate | K | −2.171 * | 3.063 * | 0.0031 * | 1.0343 ** | 1.017 * | 0.501 |

H1 | −0.056 | 0.256 | −0.0002 | 0.0005 | 0.022 | 0.035 | |

H2 | −0.098 | 0.254 | −0.0001 | 0.0006 | 0.024 | 0.028 | |

H3 | −0.149 | 0.254 | −0.0001 | 0.0007 | 0.026 | 0.022 | |

H4 | −0.183 | 0.264 | −0.0001 | 0.0007 | 0.027 | 0.020 | |

H5 | -0.204 | 0.253 | −0.0001 | 0.0008 | 0.028 | 0.019 | |

H6 | −0.217 | 0.263 | −0.0001 | 0.0008 | 0.028 | 0.019 | |

Bivariate | K-H1 | −2.146 * | 2.608 * | 0.0346 * | 0.9754 ** | 0.988 * | 0.283 |

K-H2 | −2.140 * | 2.612 * | 0.0361 * | 0.9738 ** | 0.987 * | 0.282 | |

K-H3 | −2.138 * | 2.613 * | 0.0366 * | 0.9732 ** | 0.986 * | 0.282 | |

K-H4 | −2.138 * | 2.615 * | 0.0368 * | 0.9729 ** | 0.986 * | 0.282 | |

K-H5 | −2.137 * | 2.615 * | 0.0370 * | 0.9728 ** | 0.986 * | 0.282 | |

K-H6 | −2.137 * | 2.616 * | 0.0370 * | 0.9729 ** | 0.986 * | 0.282 | |

H1-K | −0.056 | 0.260 | −0.0002 | 0.0005 | 0.023 | 0.167 | |

H2-K | −0.093 | 0.255 | −0.0003 | 0.0006 | 0.025 | 0.080 | |

H3-K | −0.142 | 0.256 | −0.0004 | 0.0007 | 0.027 | 0.063 | |

H4-K | −0.175 | 0.265 | −0.0005 | 0.0008 | 0.029 | 0.058 | |

H5-K | −0.195 | 0.255 | −0.0005 | 0.0009 | 0.029 | 0.054 | |

H6-K | −0.208 | 0.265 | −0.0006 | 0.0009 | 0.030 | 0.054 | |

Spatiotemporal | Space-time H | −0.366 | 0.351 | −0.0018 | 0.0047 | 0.069 | 0.023 |

Case | Data within the Confidence Interval | Data above the Confidence Interval | Data below the Confidence Interval | Data out of the Confidence Interval |
---|---|---|---|---|

Univariate | 59.25% | 23.25% | 17.50% | 40.75% |

Bivariate K–H1 | 58.25% | 23.25% | 18.50% | 41.75% |

Bivariate K–H2 | 57.50% | 23.25% | 19.25% | 42.50% |

Bivariate K–H3 | 58.50% | 23.25% | 18.25% | 41.50% |

Bivariate K–H4 | 58.50% | 23.25% | 18.25% | 41.50% |

Bivariate K–H5 | 58.50% | 23.25% | 18.25% | 41.50% |

Bivariate K–H6 | 58.50% | 23.50% | 18.00% | 41.50% |

Bivariate spatiotemporal | 58.50% | 22.25% | 19.25% | 41.50% |

Case | Variables | Mean Error (m/day) | MSE (m^{2}/day^{2}) | RMSE (m/day) | SMSE |
---|---|---|---|---|---|

Univariate | K | 0.091 | 0.691 | 0.831 | 0.403 |

Bivariate | K and H1 | 0.096 | 0.909 | 0.953 | 0.574 |

Bivariate | K and H2 | 0.099 | 0.722 | 0.850 | 0.449 |

Bivariate | K and H3 | 0.098 | 0.720 | 0.848 | 0.438 |

Bivariate | K and H4 | 0.099 | 0.722 | 0.850 | 0.441 |

Bivariate | K and H5 | 0.100 | 0.724 | 0.851 | 0.444 |

Bivariate | K and H6 | 0.099 | 0.724 | 0.851 | 0.445 |

Bivariate spatiotemporal | K, H1, H2, H3, H4, H5 and H6 | 0.095 | 0.757 | 0.870 | 0.469 |

Time | Univariate | Bivariate | Bivariate Spatiotemporal | |||
---|---|---|---|---|---|---|

Mean Error (m) | MSE (m^{2}) | Mean Error (m) | MSE (m^{2}) | Mean Error (m) | MSE (m^{2}) | |

1 | −0.0133 | 0.0044 | −0.0146 | 0.0031 | −0.0192 | 0.0033 |

2 | −0.0220 | 0.0072 | −0.0248 | 0.0050 | −0.0340 | 0.0056 |

3 | −0.0290 | 0.0088 | −0.0329 | 0.0064 | −0.0460 | 0.0077 |

4 | −0.0339 | 0.0097 | −0.0389 | 0.0073 | −0.0554 | 0.0094 |

5 | −0.0375 | 0.0103 | −0.0434 | 0.0081 | −0.0623 | 0.0109 |

6 | −0.0398 | 0.0107 | −0.0461 | 0.0085 | −0.0670 | 0.0120 |

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**MDPI and ACS Style**

Júnez-Ferreira, H.E.; González-Trinidad, J.; Júnez-Ferreira, C.A.; Robles Rovelo, C.O.; Herrera, G.S.; Olmos-Trujillo, E.; Bautista-Capetillo, C.; Contreras Rodríguez, A.R.; Pacheco-Guerrero, A.I.
Implementation of the Kalman Filter for a Geostatistical Bivariate Spatiotemporal Estimation of Hydraulic Conductivity in Aquifers. *Water* **2020**, *12*, 3136.
https://doi.org/10.3390/w12113136

**AMA Style**

Júnez-Ferreira HE, González-Trinidad J, Júnez-Ferreira CA, Robles Rovelo CO, Herrera GS, Olmos-Trujillo E, Bautista-Capetillo C, Contreras Rodríguez AR, Pacheco-Guerrero AI.
Implementation of the Kalman Filter for a Geostatistical Bivariate Spatiotemporal Estimation of Hydraulic Conductivity in Aquifers. *Water*. 2020; 12(11):3136.
https://doi.org/10.3390/w12113136

**Chicago/Turabian Style**

Júnez-Ferreira, Hugo Enrique, Julián González-Trinidad, Carlos Alberto Júnez-Ferreira, Cruz Octavio Robles Rovelo, G.S. Herrera, Edith Olmos-Trujillo, Carlos Bautista-Capetillo, Ada Rebeca Contreras Rodríguez, and Anuard Isaac Pacheco-Guerrero.
2020. "Implementation of the Kalman Filter for a Geostatistical Bivariate Spatiotemporal Estimation of Hydraulic Conductivity in Aquifers" *Water* 12, no. 11: 3136.
https://doi.org/10.3390/w12113136