# Recent Advances in Geomathematics in Croatia: Examples from Subsurface Geological Mapping and Biostatistics

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## Abstract

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## 1. Introduction

- Full deterministic, models where volume is well known, without uncertainties, possible correlated, and settings are known and established. Such knowledge is rare, but many areas are approximated in such a way.
- Stochastic volumes, where uncertainties cannot be full described and permanently exists. However, the probability model allows predictions and estimations to be made with different geomathematical algorithms. This is mostly done with analysed (sub)surface volumes, but the stochastic approach asks for more experiences and, contradictorily, more data than the deterministic approach.
- Unpredictable volumes, where analysed variables could not be described by any algorithm or just the number of data elements is not high enough so that any observation is valid and general.

## 2. Mathematical Basics of Algorithms Applied in the Presented Case Studies

#### 2.1. Kriging Method

- Z
_{k} - value of the regionalised variable (variable that described some geological property in a selected space with clear structure and known statistics) calculated at location “k”;
- Z
_{i} - value of the regionalised variable measured at location “I”;
- λ
_{i} - weighting coefficient calculated by kriging matrices for location “I”.

_{i}values are characterised with normal distribution or, at least, that such property is assumed for that variable in the case of a large number of measurements. Compared with simpler estimation algorithms, kriging is a more time-consuming interpolation method, but also better tool for handling with highly clustered data. By contrast, the kriging results in very weak works with small datasets (n < 20), unable to give an origin to meaningful spatial models. The spatial (variogram, co-variance or madogram) tools are powerful when applied with enough data and background knowledge. The variogram is the most often applied among them. It is defined as squared difference between two pints at some distance.

_{1}…Z

_{n}—known measured values in spatial points (hard data), x—location where value is estimated from known values, λ—weighed coefficients for location 1…n.

#### 2.2. Inverse Distance Weighting (IDW) Interpolation Method

- Z
_{IU} - estimated value,
- d
_{1}…d_{n} - distance between locations (points) with measurements (1…n) and estimated location (IU),
- p
- power (distance) exponent,
- z
_{1}…z_{n} - known values at locations 1…n.

#### 2.3. Basics of the Nearest Neighbourhood (NN) Estimation Method

- d
- distance,
- n
- n-th pair of points,
- x and T
- unknown and measured point,
- X and T
- length of line segment in Euclidean space connecting the points X and T for pairs 1…n, in the Euclidian n-space.

#### 2.4. Basics of the Natural Neighbourhood (NaN) Estimation Method

- X(x,y)
- estimated value in point (x,y),
- A(X
_{i},Y_{i}) - known value in point (X
_{i},Y_{i}), - w
_{i} - proportion of polygon, i“ in total area.

#### 2.5. Modified Shepard’s Method (MSM)

- F
- MSM function,
- W
- relative weights,
- Q
_{k} - bivariate quadratic function,
- x, y
- data coordinates,
- n
- number of data elements.

- d
_{k} - Euclidean distance between points at locations (x, y) and (x
_{k}, y_{k}), - R
_{ω} - radius of influence around node (x
_{k}, y_{k}).

#### 2.6. Cross-Validation as Numerical Estimation of Mapping Error

- MSE
- mean square error value,
- n
- number of known values,
- SV
- measured value in point “i”,
- P
- estimated value in point “i”,
- i
- i-th point.

#### 2.7. Shannon–Wiener Index or Shannon Diversity Index (H)

- p
_{i} - a proportion of individuals belonging to the i
^{th}species in the sample, - ln
- a natural logarithm,
- R
- total number of species in the community (richness).

_{i}(proportion of individuals, and if all species in the sample are equally represented, H is at its maximum [42].

## 3. Recent Advances in Geomathematical Mapping in Small Datasets and Case Studies from Croatia

_{i}) using Equation (12):

- ${\overline{\mathrm{x}}}_{\mathrm{i}}$
- mean of sub-dataset, where i-th data is skipped,
- n
- number of data points,
- j
- data currently analysed.

## 4. Recent Advances in Biostatistics Applied in Palaeontology and Case Studies from Croatia and the Wider Region

## 5. Discussion and Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Matheron, G. Traité de géostatistique appliquée; Editions Technip: Paris, France, 1962; Volume 1, p. 334. [Google Scholar]
- Matheron, G. Principles of geostatistics. Econ. Geol.
**1963**, 58, 1246–1266. [Google Scholar] [CrossRef] - Matheron, G. Les Variables Régionalisées et leur Estimation; Masson & Cie: Paris, France, 1965; p. 306. [Google Scholar]
- Krige, D.G. A Statistical Approach to Some Basic Mine Valuation Problems on the Witwatersrand. J. Chem. Metall. Min. Soc. S. Afr.
**1951**, 52, 119–139. [Google Scholar] - Journel, A.G.; Huijbregts, C.J. Mining Geostatistics; Academic Press: London, UK, 1978. [Google Scholar]
- Ripley, B.D. Spatial Statistics; Wiley & Sons Ltd.: New York, NY, USA, 1981; p. 272. [Google Scholar]
- Davis, J.C.; Sampson, R.J. Statistics and Data Analysis in Geology; John Wiley & Sons Inc.: New York, NY, USA, 1973. [Google Scholar]
- Cressie, N. Statistics for Spatial Data; Wiley & Sons Ltd.: New York, NY, USA, 1991. [Google Scholar]
- Agterberg, F.P. An undulation of the rate of sedimentation in southern Gotland. Geol. Mijnb.
**1958**, 20, 253–260. [Google Scholar] - Agterberg, F.P. On the measuring of strongly dispersed minor folds. Geol. Mijnb.
**1959**, 21, 133–137. [Google Scholar] - Agterberg, F.P. The skew frequency distribution of some ore minerals. Geol. Mijnb.
**1961**, 23, 149–162. [Google Scholar] - Krumbein, W.C. Regional and local components in facies maps. AAPG Bull.
**1956**, 40, 2163–2194. [Google Scholar] - Krumbein, W.C. Trend-surface analysis of contour type maps with irregular control-point spacing. J. Geophys. Res.
**1959**, 64, 823–834. [Google Scholar] [CrossRef] - Krumbein, W.C.; Graybill, F.A. An Introduction to Statistical Models in Geology; McGraw-Hill: New York, NY, USA, 1965. [Google Scholar]
- Vistelius, A.B. Studies in Mathematical Geology (Translated from Russian by George V. Keller); Consultants Bureau: New York, NY, USA, 1967. [Google Scholar]
- Merriam, D.F. Geologic use of the computer. In Symposium on Recently Developed Geologic Principles and Sedimentation of the Permo-Pennsylvanian of the Rocky Mountains—Wyoming Geologists’s Association 20th Annual Conference; Petroleum Inf.: Casper, WY, USA, 1967; pp. 109–112. [Google Scholar]
- Isaaks, E.; Srivastava, R. An Introduction to Applied Geostatistics; Oxford University Press Inc.: New York, NY, USA, 1989. [Google Scholar]
- Jensen, J.L.; Lake, L.W.; Corbett, P.W.M.; Goggin, D.J. Statistics for Petroleum Engineers and Geoscientists; Prentice Hall PTR: Upper Saddle River, NJ, USA, 2000. [Google Scholar]
- Dubrule, O. Geostatistics in Petroleum Geology; AAPG Education Course Note, Series #38; AAPG and Geological Society Publishing House: Tulsa, Oklahoma, 1998. [Google Scholar]
- Kelkar, M.; Perez, G. Applied Geostatistics for Reservoir Characterization; Society of Petroleum Engineers: Richardson, TX, USA, 2002. [Google Scholar]
- Sokal, R.R.; Rohlf, F.J. Introduction to Biostatistics, 2nd ed.; Dover Publications Inc.: Dover, DE, USA, 2009; pp. 1–363. [Google Scholar]
- Lincoln Edwards, C. Biometry as a Method in Taxonomy. Am. Nat.
**1908**, 42, 537–540. [Google Scholar] [CrossRef] - Billard, L. Sir Ronald A. Fisher and The International Biometric Society. Biometrics
**2014**, 70, 259–265. [Google Scholar] [CrossRef] - Hohn, M.E. Geostatistics and Petroleum Geology; Van Nostrand Reinhold: New York, NY, USA, 1988. [Google Scholar]
- Liebhold, A.M.; Rossi, R.E.; Kemp, W.P. Geostatistics and Geographic Information System in Applied Insect Ecology. Annu. Rev. Entomol.
**1993**, 38, 303–327. [Google Scholar] [CrossRef] - Balić, D.; Velić, J.; Malvić, T. Selection of the most appropriate interpolation method for sandstone reservoirs in the Kloštar oil and gas field. Geol. Croat.
**2008**, 61, 27–35. [Google Scholar] - Medved, I.; Pribičević, B.; Medak, D.; Kuzmanić. Usporedba metoda interpolacije batimetrijskih mjerenja za praćenje promjena volumena jezera (Comparison of Interpolation Methods of Bathymetry Data Used for Monitoring of Lake Volume Change—In Croatian). Geod. List
**2010**, 2, 71–86. [Google Scholar] - Ly, S.; Charles, C.; Degré, A. Geostatistical interpolation of daily rainfall at catchment scale: The use of several variogram models in the Ourthe and Ambleve catchments, Belgium. Hydrol. Earth Syst. Sci.
**2011**, 15, 2259–2274. [Google Scholar] [CrossRef] [Green Version] - Husanović, E.; Malvić, T. Review of deterministic geostatistical mapping methods in Croatian hydrocarbon reservoirs and advantages of such approach. Nafta
**2014**, 65, 57–63. [Google Scholar] - Ivšinović, J. Deep mapping of hydrocarbon reservoirs in the case of a small number of data on the example of the Lower Pontian reservoirs of the western part of Sava Depression. In Proceedings of the 2nd Croatian Congress on Geomathematics and Geological Terminology, Zagreb, Croatia, 6 October 2018; Malvić, T., Velić, J., Rajić, R., Eds.; University of Zagreb, Faculty of Mining, Geology and Petroleum Engineering: Zagreb, Croatia, 2018; pp. 59–65. [Google Scholar]
- Traversoni, L. Natural neighbour finite elements. Trans. Ecol. Environ.
**1994**, 8, 291–297. [Google Scholar] - Boissonnat, J.-D.; Cazals, F. Natural neighbor coordinates of points on a surface. Comput. Geom.
**2001**, 19, 155–173. [Google Scholar] [CrossRef] [Green Version] - Tsidaev, A. Parallel Algorithm for Natural Neighbor Interpolation. In Proceedings of the 2nd Ural Workshop on Parallel, Distributed, and Cloud Computing for Young Scientists, Yekaterinburg, Russia, 6 October 2016; Sozykin, A., Akimova, E., Ustalov, D., Eds.; Ural-PDC: Yekaterinburg, Russia, 2016; pp. 78–83. [Google Scholar]
- Shepard, D. A two-dimensional interpolation for irregularly spaced data function. In Proceedings of the 1968 ACM National Conference, New York, NY, USA, 27–29 August 1968; Blue, R.B., Rosenberg, A.M., Eds.; Association for Computing Machinery: New York, NY, USA, 1968; pp. 517–523. [Google Scholar]
- Franke, R.; Nielson, G. Smooth Interpolation of Large Sets of Scattered Data. Int. J. Numer. Methods Eng.
**1980**, 15, 1691–1704. [Google Scholar] [CrossRef] - Renka, R.J. Multivariate Interpolation of Large Sets of Scattered Data. ACM Trans. Math. Softw.
**1988**, 14, 139–148. [Google Scholar] [CrossRef] - Davis, B. Uses and Abuses of Cross Validation in Geostatistics. Math. Geol.
**1987**, 19, 241–248. (In Dordrecht) [Google Scholar] [CrossRef] - Rodrıguez, J.D.; Perez, A.; Lozano, J.A. Sensitivity Analysis of k-Fold Cross Validation in Prediction Error Estimation. IEEE Trans. Pattern Anal. Mach. Intell.
**2010**, 3, 569–575. [Google Scholar] [CrossRef] - Arlot, S.; Lerasle, M. Choice of V for V-Fold Cross-Validation in Least-Squares Density Estimation. J. Mach. Learn. Res.
**2016**, 17, 7256–7305. [Google Scholar] - Malvić, T.; Ivšinović, J.; Velić, J.; Rajić, R. Kriging with a Small Number of Data Points Supported by Jack- Knifing, a Case Study in the Sava Depression (Northern Croatia). Geosciences
**2019**, 9, 36. [Google Scholar] [CrossRef] [Green Version] - Malvić, T.; Ivšinović, J.; Velić, J.; Rajić, R. Interpolation of Small Datasets in the Sandstone Hydrocarbon Reservoirs, Case Study of the Sava Depression, Croatia. Geosciences
**2019**, 9, 201. [Google Scholar] [CrossRef] [Green Version] - Murray, J.W. Ecology and Applications of Benthic Foraminifera; Cambridge—University Press: Cambridge, UK, 2006. [Google Scholar] [CrossRef]
- Malvić, T.; Ivšinović, J.; Velić, J.; Sremac, J.; Barudžija, U. Application of the Modified Shepard’s Method (MSM): A Case Study with the Interpolation of Neogene Reservoir Variables in Northern Croatia. Stats
**2020**, 3, 68–83. [Google Scholar] [CrossRef] [Green Version] - Kochansky-Devidé, V. O fauni marinskog miocena i o tortonskom šliru Medvednice (Zagrebačke gore) [Ueber die Fauna des marinen Miozäns und über den Tortonischen “Schlier” von Medvednica (Zagreber Gebirge)]. Geološki Vjesnik
**1957**, 10, 39–50. [Google Scholar] - Sokač, A. Panonska fauna ostrakoda Donjeg Selišta jugozapadno od Gline (Pannonische Ostrakodenfauna von Donje Selište südwestlich von Glina). Geološki Vjesnik
**1963**, 15, 391–401. [Google Scholar] - Prlj Šimić, N.; Sremac, J.; Ćosović, V. Taxonomy and Biometry (Applied to the Eocene Corals from the Island of Krk –Croatia). 1st Croatian Geological Congress; Abstracts Book. 1995, pp. 495–498. Available online: http://geol.pmf.hr/~jsremac/radovi/znanstveni/1995_opat_koralji.pdf (accessed on 14 May 2020).
- Pezelj, Đ.; Sremac, J.; Sokač, A. Palaeoecology of the Late Badenian foraminifera and ostracoda from the SW Central Paratethys (Medvednica Mt., Croatia). Geol. Croat.
**2007**, 60, 139–150. [Google Scholar] - Pezelj, Đ.; Sremac, J.; Bermanec, V. Shallow-water benthic foraminiferal assemblages and their response to the palaeoenvironmental changes—example from the Middle Miocene of Medvednica Mt. (Croatia, Central Paratethys). Geol. Carpathica
**2016**, 67, 329–345. [Google Scholar] [CrossRef] [Green Version] - Pezelj, Đ.; Sremac, J. Badenian Marginal Marine Environment in the Medvednica Mt. (Croatia). Joannea Geol. Paläont.
**2007**, 9, 83–84. [Google Scholar] - Pezelj, Đ.; Drobnjak, L. Foraminifera-based estimation of water depth in epicontinental seas: Badenian deposits from Glavnica Gornja (Medvednica Mt., Croatia), Central Paratethys. Geol. Croatica
**2019**, 72, 93–100. [Google Scholar] [CrossRef] - Hohenegger, J. Growth-invariant Meristic Characters Tools to Reveal Phylogenetic Relationships in Nummulitidae (Foraminifera). Turk. J. Earth Sci.
**2011**, 20, 655–681. [Google Scholar] [CrossRef] - Hohenegger, J.; Torres-Silva, A.I. Growth-invariant and growth-independent characters in equatorial sections of Heterostegina shells relieve phylogenetic and paleobiogeographic interpretations. Palaios
**2017**, 32, 30–43. [Google Scholar] [CrossRef] - Eder, W.; Hohenegger, J.; Briguglio, A. Test flattening in the larger foraminifer Heterostegina depressa: Predicting bathymetry from axial sections. Paleobiology
**2018**, 44, 76–88. [Google Scholar] [CrossRef] [Green Version] - Harzhauser, M.; Landau, B. A revision of the Neogene Conidae and Conorbidae (Gastropoda) of the Paratethys Sea. Zootaxa
**2016**, 4210, 001–178. [Google Scholar] [CrossRef] - Bošnjak, M.; Sremac, J.; Vrsaljko, D.; Aščić, Š.; Bosak, L. The Miocene “Pteropod event” in the SW part of the Central Paratethys (Medvednica Mt., northern Croatia). Geol. Carpathica
**2017**, 68, 329–349. [Google Scholar] [CrossRef] [Green Version] - Derežić, I.; Bošnjak, M.; Sremac, J. Biostatistic analyses of newly found pteropods (Mollusca, Gastropoda) in the Middle Miocene (Badenian) deposits from the southeastern Medvednica Mt. (Northern Croatia). In Mathematical Methods and Terminology in Geology; Malvić, T., Velić, J., Rajić, R., Eds.; Faculty of Mining, Geology and Petroleum Engineering, University of Zagreb: Zagreb, Croatia, 2018; pp. 95–102. [Google Scholar]
- Šeparović, A. Miocene Deposits with Scaphopods South from Veternica Cave (Medvednica). Master’s Thesis, University of Zagreb, Faculty of Science, Zagreb, Croatia, 2019; pp. 1–66. [Google Scholar]
- Kowalewski, M. The Fossil Record of Predation: An Overview of Analytical Methods. In The Fossil Record of Predation; Kowalewski, M., Kelley, P.H., Eds.; Paleontological Special Papers, 8; Yale University: New Haven, CT, USA, 2002; pp. 3–42. [Google Scholar]
- Zell, P.; Beckmann, S.; Stinnesbeck, W. Liostrea roemeri (Ostreida, Bivalvia) attached to Upper Jurassic ammonites of northeastern Mexico. Palaeobio Palaeoenv
**2014**, 94, 439–451. [Google Scholar] [CrossRef] - Curman, D. Digital Model Analysis of Theropod Footprints from Solaris Tracksite (Istria). Master’s Thesis, University of Zagreb, Faculty of Science, Zagreb, Croatia, 2017; pp. 1–72. [Google Scholar]
- Dalla Vecchia, F.M. Remains of Sauropoda (Reptilia, Saurischia) in the Lower Cretaceous (Upper Hauterivian/Lower Barremian) limestones of SW Istria (Croatia). Geol. Croat.
**1998**, 51, 105–134. [Google Scholar] - Mezga, A.; Cvetko Tešović, B.; Bajraktarević, Z.; Bucković, D. A new dinosaur tracksite in the late Albian of Istria, Croatia. Riv. Ital. Paleontol. Stratigr.
**2007**, 113, 139–148. [Google Scholar] - Mezga, A.; Cvetko Tešović, B.; Pretković, V.; Jovanović, N.; Bajraktarević, Z. Dinosaur footprints in the Lower Hauterivian deposits of Palud Cove in Istria, Croatia. Geol. Croat.
**2015**, 68, 113–122. [Google Scholar] [CrossRef] - Costa-Pérez, M.; Joaquín Moratalla, J.; Marugán-Lobón, J. Studying bipedal dinosaur trackways using geometric morphometrics. Palaeontol. Electron.
**2019**, 22, 1–13. [Google Scholar] [CrossRef]

**Figure 3.**Results of inverse distance weighting (IDW), nearest neighbourhood (NN) and natural neighbourhood (NaN) methods (from top to bottom) of the permeability (left) and injected volumes (right) in the “K” reservoir [41].

**Figure 4.**Experimental semivariograms and porosity maps for the “K” reservoir obtained by the ordinary kriging (OK) method: (

**a**) without the “jack-knifed” method and (

**b**) with the “jack-knifed” method [40].

**Figure 5.**The mapping of the Lower Pontian “K” reservoir, the Sava Depression, Northern Croatia. Left—IDW results, right—MSM results. Top—porosity, middle permeability, down—thickness [43].

**Figure 6.**Example of statistical comparison of fauna from different localities using cluster analysis and non-metric multidimensional scaling analyses (after [48]).

**Figure 7.**Example of using growth characteristics as an indicator of the bathymetry [53].

**Figure 8.**Separation of the species and morphospace occupied by genera as shown by principal component analysis (after [54]).

**Figure 9.**Morphometric characteristics and comparison of the planktic gastropods between different localities based on the measured morphometric elements of the shell (after [55]). (

**A**): Measured parameters on the gastropod shell: H (height of the shell), W (width of the shell), α (apical angle), A

_{1}and A

_{2}(aperture diameters). (

**B**) Comparison of planktic gastropod from different areas (black and white triangles) based on the measured values of the shell height and width. (

**C**) Comparison of planktic gastropod species from different areas (dark and light grey columns) based on the measured values of the apical angle of the gastropod shells.

**Figure 10.**Analysis of oyster attachments positions on the ammonite shell [59].

**Figure 11.**Application of geometric morphometrics on the dinosaur footprints analysis [64].

**Table 1.**Summary results of cross-validation (mean square error, MSE) for IDW, NN, NaN and modified Shepard’s method (MSM) methods [41].

Variable | Number of Data | Value of Cross-Validation (MSE) | ||
---|---|---|---|---|

Inverse Distance Weighting | Nearest Neighbourhood | Natural Neighbour | ||

Injected volumes | 3 | 2.86 × 10^{11} | 3.96 × 10^{11} | - |

Permeability | 18 | 480.8 | 1397.4 | 1044.7 |

**Table 2.**Comparison of cross-validation (MSE) values for OK maps based on original and “jack-knifed” semivariograms [40].

Field/Reservoir | OK (Original Semivariogram) | OK (Jack-Knifed Semivariogram) | Recommendation |
---|---|---|---|

“B”/“K” | 0.001320 (linear) | 0.000970 (Gaussian) | OK with jack-knifed semivariogram |

**Table 3.**Cross-validation (MSE) of the IDW and MSM methods applied in reservoir “K” [43].

Description | No Data | Cross-Validation | |
---|---|---|---|

Inverse Distance (IDW) | Modified Shepard’s Method (MSM) | ||

Porosity | 19 | 0.00119 | 0.00345 |

Permeability | 18 | 480.8 | 516.1 |

Thickness | 14 | 40.7 | 60.5 |

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**MDPI and ACS Style**

Malvić, T.; Bošnjak, M.; Velić, J.; Sremac, J.; Ivšinović, J.; Dinis, M.A.P.; Barudžija, U.
Recent Advances in Geomathematics in Croatia: Examples from Subsurface Geological Mapping and Biostatistics. *Geosciences* **2020**, *10*, 188.
https://doi.org/10.3390/geosciences10050188

**AMA Style**

Malvić T, Bošnjak M, Velić J, Sremac J, Ivšinović J, Dinis MAP, Barudžija U.
Recent Advances in Geomathematics in Croatia: Examples from Subsurface Geological Mapping and Biostatistics. *Geosciences*. 2020; 10(5):188.
https://doi.org/10.3390/geosciences10050188

**Chicago/Turabian Style**

Malvić, Tomislav, Marija Bošnjak, Josipa Velić, Jasenka Sremac, Josip Ivšinović, Maria Alzira Pimenta Dinis, and Uroš Barudžija.
2020. "Recent Advances in Geomathematics in Croatia: Examples from Subsurface Geological Mapping and Biostatistics" *Geosciences* 10, no. 5: 188.
https://doi.org/10.3390/geosciences10050188