# Using Geostatistical Gaussian Simulation for Designing and Interpreting Soil Surface Magnetic Susceptibility Measurements

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}was located in a forest in Upper Silesian Industrial Area, in southern Poland (Figure 1), WGS84 coordinates 50.323N, 19.450E. The majority of the natural forest where measurements were made was overgrown by coniferous trees. The major types of geological substrata in the study area were sands and gravels, eolian sands, and partially loess, mostly in the south-western part of the area [21].

#### 2.2. Measurements of Soil Magnetic Susceptibility

^{−5}SI] units. At the selected location, 10 to 15 single readings were made on the soil surface in a circle with a radius of 2 m. These values of soil magnetic susceptibility measured at every single reading will be furthermore labeled as κ

_{non-avg}. The total number of κ

_{non-avg}values was equal to 460. Within each of the measuring points, values of κ

_{non-avg}were averaged, and the calculated average value of soil magnetic susceptibility was furthermore labeled as κ

_{avg}. The total number of κ

_{avg}values was equal to 46. Each of these MS2D readings was assigned with a geographical coordinate.

_{non-avg}were analyzed individually at each of 46 sample points in order to determine outlier values, which were defined as lower than Q

_{25%}− 1.5 × IQR or higher than Q

_{75%}+ 1.5 × IQR. After the outlier values of soil magnetic susceptibility were removed, the average magnetic susceptibility was calculated for each of 46 sample points. These averages will be referred to below as κ

_{avg}. In the final results, two data sets were obtained:

- the set of 450 κ
_{non-avg}, non-averaged readings of the MS2D meter performed in each of the 46 measured points, - the set of 46 κ
_{avg}susceptibility values created after averaging the readings of the MS2D meter which were performed in each of the 46 measured points.

_{avg}and κ

_{non-avg}were later used as input data in the geostatistical analyses of spatial correlations and simulations.

#### 2.3. Spatial Simulations

_{non-avg}and κ

_{avg}.

_{non-avg}or κ

_{avg}. The average values of all realizations of SGS at sample point locations were approximately equal to measured values of κ

_{non-avg}or κ

_{avg}depending on which data were used as an input for SGS. The small differences between measured and simulated values could be observed because values were simulated at a grid cell that might not be located exactly in the same place as the sample points. Only one value of κ per location was used in the SGE. In case of the κ

_{avg}set, ten values of κ measured per sample point, as well as the XY coordinates of these measurements, were averaged. Then, the average values of κ and XY coordinates were used in simulations. In the case of the simulations based on the κ

_{non-avg}set, all 10 values of κ measured per sample point (and their coordinates) were used separately. As a consequence, when ten κ

_{non-avg}set was used, many more pairs of κ values existed for each variogram lag. (The number of pairs between measurements is proportional to the square of the number of measurements). A hundred simulations were made separately for two cases when the input data were κ

_{non-avg}and κ

_{avg}values. These SGS realizations were later used to calculate spatial distributions of:

- κ
_{sim-avg}—the average of all simulated realizations, - κ
_{sim-min}—the minimum of all simulated realizations, - κ
_{sim-max}—the maximum of all simulated realizations, - κ
_{sim-std}—the standard deviations of all simulated realizations.

#### 2.4. Analyses of Spatial Correlations

**x**

_{i}is a location,

**h**is a lag vector, Z(

**x**

_{i}) is the measured value at location

**x**

_{i}, and N is the number of pairs spaced by

**h**vector.

_{non-avg}and κ

_{avg}measured at the sample points. Next, variograms were also calculated using simulated values of κ

_{sim-avg}separately for two simulation cases where the input was κ

_{non-avg}or κ

_{avg}values. These simulated values were available not at the measuring points, but in each cell of the simulated grid. The cell size of this grid was equal to 10 m. All experimental variograms were later modeled using the spherical model with a nugget effect. Nugget effect describes the variability between values for distances shorter than a sampling distance and is also related to the variability resulting from measurement and instrumental errors.

## 3. Results and Discussion

_{non-avg}and κ

_{avg}, were tested for normality of their distributions. A Shapiro–Wilk test was performed with a significance level of 0.05. The results of this test suggested that for both sets of κ

_{non-avg}and κ

_{avg}, it can be assumed that their distributions are close to normal distribution. On the basis of these results, it was concluded that the κ

_{non-avg}and κ

_{avg}values could be used in further analyses, variogram calculation, and SGS simulation without data transformation.

_{non-avg}, was much more frequent than the set of κ

_{avg}. The average susceptibility values were similar for both sets of data and approximately equal to 65 × 10

^{−5}SI. More pronounced differences were observed in the case of quartiles and standard deviation values, where the averaging data resulted in a visible reduction in the spread of susceptibility values. For the κ

_{avg}, the minimum and maximum susceptibility values were 31 × 10

^{−5}SI and 108 × 10

^{−5}SI, respectively, and for the set of κ

_{non-avg}, 14 × 10

^{−5}SI and 149 × 10

^{−5}SI, respectively.

_{sim-avg}simulated on the basis of two sets of data, κ

_{non-avg}and κ

_{avg}, did not differ significantly. Subareas with high and low values of κ

_{sim-avg}were located in similar parts of the study area; in the Western and Eastern parts, respectively. Further analysis of the differences between these distributions, using differential map (Figure 3) between κ

_{sim-avg}simulated using κ

_{non-avg}and κ

_{sim-avg}simulated using κ

_{avg}showed that maximum variation between these distributions was in the range of 10 × 10

^{−5}SI. This was rather low values in comparison to the range of measured κ

_{non-avg}and κ

_{avg}values, which was over 100 × 10

^{−5}SI. Based on this observation, it might be assumed that the differences between simulated distributions of κ

_{sim-avg}, which were simulated using data sets κ

_{non-avg}and κ

_{avg}, could be attributed to the specific sampling methodology.

_{non-avg}and κ

_{avg}data sets. Using these realizations, it was possible to assess the variability of the simulated values by calculating κ

_{sim-avg}, κ

_{sim-min}, κ

_{sim-max}, and κ

_{sim-std}. Firstly, standard deviations of simulated values κ

_{sim-std}were calculated, and subsequently, the coefficients of variations, i.e., κ

_{sim-std}divided by κ

_{sim-avg}. As can be observed in Figure 4 and Figure 5, values of the coefficient of variation of κ were under 20% for the majority of the study area. It was also observed that slightly lower values of the coefficient of variation were observed for values simulated on the basis of the κ

_{avg}data set. Such observations were due to the fact that the κ

_{avg}data set was characterized by lower variability, which was reduced during the averaging of κ

_{non-avg}values. As it was analyzed, distributions of measured susceptibility values, κ

_{non-avg}, at sample points were characterized, in average, by standard deviation, and coefficient of variation equal to 18 × 10

^{−5}SI and 27%, respectively. Therefore, it was evident that the variability of simulated values of soil magnetic susceptibility at individual points was at a similar level to that of measured values.

_{avg}and κ

_{non-avg}data sets, as well as simulated values of κ

_{sim-avg}, κ

_{sim-min}, κ

_{sim-max}, and κ

_{sim-std}. For this purpose, experimental variograms (Figure 6) were calculated with 12 lags, and a lag distance equal to 200 m. Before the calculation of experimental variograms, input data were transformed using the normal-score transformation. After the experimental variograms of measured and simulated soil magnetic susceptibility were calculated, they were modeled using a spherical model with the nugget effect. The goal of this part of analysis was to investigate if the parameters of spatial correlations of simulated κ

_{sim-avg}, κ

_{sim-min}, κ

_{sim-max}, and κ

_{sim-std}were similar to those of measured κ

_{non-avg}values.

_{sim-avg}had a similar shape to the variograms of measured κ

_{non-avg}and κ

_{avg}values, especially where they achieved sill. In order to investigate the similarities between spatial correlations of κ

_{sim-avg}, κ

_{non-avg}, and κ

_{avg}more precisely, all variograms were modeled, and the parameters of these models were placed in Table 2.

_{avg}variogram was noticeable but not much longer than the correlation range of κ

_{non-avg}variogram. As for the modeled nugget effect, it was lower for κ

_{avg}variograms in comparison with κ

_{non-avg}ones. Such observation could be explained by the fact that during the averaging of values of soil magnetic susceptibility, the impact of outliers was reduced. As can be observed in Table 2, the sill of the spherical model of κ

_{avg}was slightly higher as the sill of κ

_{non-avg}. Referring these observed differences to the calculated experimental variograms, it can be noted that the differences in the sill values of spherical models of κ

_{non-avg}and κ

_{avg}concerned practically only distances above 1700 m. This distance was longer than the ranges of correlation of both κ

_{non-avg}and κ

_{avg}values. Differences in sill values could result mainly from a very large difference in the number of values of κ

_{non-avg}and κ

_{avg}, which were equal to 450 and 46, respectively. As a result, according to the Formula (1), on the basis of which the semivariance values were calculated, in the case of a variogram of κ

_{non-avg}values, it was possible to find many more pairs of sample points. Due to the fact that the semivariance values of κ

_{non-avg}were calculated on the basis of a much larger number of pairs of points than in the case κ

_{avg}, the sill value of the spherical model of κ

_{non-avg}was lower. However, it should be stated that the spatial characteristics of measured κ

_{non-avg}and κ

_{avg}values were rather similar, and the observed differences resulted from the sampling method.

_{non-avg}, κ

_{avg}, and simulated κ

_{sim-avg}values were compared. In each case, the variogram determined from the measured κ

_{non-avg}or κ

_{avg}values was used as the reference point. As can be noticed in Table 2, the comparison was made separately for values simulated when the input data for SGS was set κ

_{non-avg}and κ

_{avg}.

_{sim-avg}were significantly lower than that of variograms of measured κ

_{non-avg}and κ

_{avg}. Simultaneously, the comparison of the parameters of spherical models showed that variograms of simulated values, κ

_{sim-avg}, were characterized by lower values of nugget effect than variograms of measured values of κ

_{non-avg}and κ

_{avg}data. Such observations might suggest that SGS was quite effective in recreating the local variability of soil magnetic susceptibility, especially for distances shorter than a distance between sample points. Measured values of soil magnetic susceptibility were not available for such small distances, though they were available for simulated data sets, κ

_{sim-avg}values were simulated for each simulation grid cell with a size of 10 m.

_{sim-avg}simulated using κ

_{non-avg}and κ

_{avg}data sets were comparable. It is important to underline here that a ratio of nugget effect to sill, which ranges from 0 to 1, is often recognized as a critical measure to define the spatial dependence of soil properties [23,24]. The closer this ratio is to zero, the stronger spatial correlations are observed. Precise assessment of this ratio is crucial to the quality of the results of geostatistical analyses in soil magnetometry. Usually, nugget effect is much more difficult to determine than sill. As it was observed, spherical models of simulated κ

_{sim-avg}were characterized by a shorter range of correlation in comparison with variograms of measured κ

_{non-avg}or κ

_{avg}. As can be noticed in Table 2, this difference was equal to about a few hundred meters. The explanation of such observations is related to the fact that values of simulated κ

_{sim-avg}might be characterized by greater spatial variability at shorter distances. It is related to the fact that simulated data were much more numerous than the measured data and were available at all 60 thousand grid cells, so small-scale spatial variability of magnetic susceptibility was well reproduced.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Spatial distributions of κ

_{sim-avg}simulated using κ

_{non-avg}(upper figure) and κ

_{avg}(bottom figure) data sets.

**Figure 3.**Spatial distribution of differences between κ

_{sim-avg}simulated using κ

_{non-avg}and κ

_{sim-avg}simulated using κ

_{avg}.

**Figure 4.**Spatial distributions of the coefficient of variation calculated as κ

_{sim-std}divided by κ

_{sim-avg}, simulated using κ

_{non-avg}(upper figure) and κ

_{avg}(bottom figure) data sets.

**Figure 5.**Spatial distribution of differences between coefficients of variation calculated using κ

_{non-avg}and κ

_{avg}data sets.

**Figure 6.**Experimental variograms of measured and simulated values of soil magnetic susceptibility using κ

_{non-avg}data (upper figure) and κ

_{avg}(bottom figure).

κ_{avg} | κ_{non-avg} | |
---|---|---|

(10^{−5} SI) | ||

Average | 65.2 | 65.7 |

Q_{25%} | 47 | 46 |

Q_{75%} | 82 | 87 |

Minimum | 31 | 14 |

Maximum | 108 | 149 |

Standard deviation | 20 | 27 |

Number | 46 | 450 |

Nugget Effect | Sill | Range of Correlation | |
---|---|---|---|

(10^{−10} SI) | (m) | ||

Measured Values | |||

κ_{non-avg} | 0.636 | 1.133 | 1580 |

κ_{avg} | 0.520 | 1.320 | 1700 |

Values Simulated Using κ_{non-avg} | |||

κ_{sim-avg} | 0 | 1.220 | 1100 |

Values Simulated Using κ_{avg} | |||

κ_{sim-avg} | 0 | 1.190 | 1150 |

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**MDPI and ACS Style**

Fabijańczyk, P.; Zawadzki, J. Using Geostatistical Gaussian Simulation for Designing and Interpreting Soil Surface Magnetic Susceptibility Measurements. *Int. J. Environ. Res. Public Health* **2019**, *16*, 3497.
https://doi.org/10.3390/ijerph16183497

**AMA Style**

Fabijańczyk P, Zawadzki J. Using Geostatistical Gaussian Simulation for Designing and Interpreting Soil Surface Magnetic Susceptibility Measurements. *International Journal of Environmental Research and Public Health*. 2019; 16(18):3497.
https://doi.org/10.3390/ijerph16183497

**Chicago/Turabian Style**

Fabijańczyk, Piotr, and Jarosław Zawadzki. 2019. "Using Geostatistical Gaussian Simulation for Designing and Interpreting Soil Surface Magnetic Susceptibility Measurements" *International Journal of Environmental Research and Public Health* 16, no. 18: 3497.
https://doi.org/10.3390/ijerph16183497