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Suggested citation:

Torak, L.J., and Painter, J.A., 2019, Geostatistical estimation of the bottom altitude and thickness 

of the Mississippi River Valley alluvial aquifer: U.S. Geological Survey Scientific Investigations 

Map 3426, 2 sheets, https://doi.org/10.3133/sim3426.

ISSN 2329-132X (online)

https://doi.org/10.3133/sim3426

U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Map 3426

Sheet 2 of 2

Figure 4. Isotropic variograms for Mississippi River Valley alluvial 

aquifer data showing semivariance of A, unfiltered bottom altitude and 

B, filtered bottom altitude and log-semivariance of C, thickness.

0

590

1,179

1,769

2,359

0 31,000 62,000 93,000

Semivariance, in feet squared

Separation distance, in meters

Exponential model

n = 9,292

nugget = 631

sill = 2,452

range = 101,100

r

2

 = 0.993

0

395

790

1,184

1,579

Exponential model

n = 6,604

nugget = 114

sill = 1,974

range = 180,000

r2

 = 0.999

0 31,000 62,000 93,000

Semivariance, in feet squared

Separation distance, in meters

0

0.0158

0.0315

0.0473

0.0630

0 46,000 92,000 138,000

Exponential model

n = 6,604

nugget = 0.0112

sill = 0.0601

range = 87,000

r

2

 = 0.995

Separation distance, in meters

ln(Semivariance)

−467.00

−224.68

17.65

259.98

502.30

−467.00 −143.90 179.20 502.30

Actual bottom altitude, in feet

Estimatedbottom altitude, in feet

r

2

 = 0.935

n = 9,292

−427.00

−230.50

−34.00

162.50

359.00

−427.00 −165.00 97.00 359.00

r2 = 0.984

n = 6,604

Actual bottom altitude, in feet

Estimatedbottom altitude, in feet

r2 = 0.926

n = 6,602

Actual thickness, in feet

Estimated thickness, in feet

18.31

122.81

227.30

331.80

436.29

18.31 157.64 296.96 436.29

Figure 5. Cross-validation plots of actual and estimated values of Mississippi River Valley alluvial aquifer 

data for A, unfiltered bottom altitude, B, filtered bottom altitude, and C, thickness.

A A

B B

C C

laf19-0909_fig. 06

0 15 30 45 60 MILES

0 15 30 45 60 KILOMETERS

Base modified from U.S. Geological Survey,

digital data 1:100,000 scale

Albers Equal Area projection;

World Geodetic System of 1984

EXPLANATION

Mississippi Alluvial Plain extent

Estimation variance, in feet squared

130 to 400

410 to 640

650 to 920

930 to 1,200

1,300 to 1,600

91°30' 90°0' 88°30'

37°30'

36°0'

34°30'

33°0'

31°30'

30°0'

laf19-0909_fig. 07

0 15 30 45 60 MILES

0 15 30 45 60 KILOMETERS

Base modified from U.S. Geological Survey,

digital data 1:100,000 scale

Albers Equal Area projection;

World Geodetic System of 1984

91°30' 90°0' 88°30'

37°30'

36°0'

34°30'

33°0'

31°30'

30°0'

EXPLANATION

Mississippi Alluvial Plain extent

Estimation variance, in feet squared

321 to 470

470.01 to 656

656.01 to 792

792.01 to 963

963.01 to 1,190

1,190.01 to 1,574.99

Figure 6. Estimation variance of estimated values 

of bottom altitude of the Mississippi River Valley 

alluvial aquifer.

Figure 7. Estimation variance of estimated 

values of thickness of the Mississippi River Valley 

alluvial aquifer.

Bottom altitude

Thickness

Geostatistical Estimation of the Bottom Altitude and Thickness of the 

Mississippi River Valley Alluvial Aquifer

By

L.J. Torak and J.A. Painter

2019

Clark, B.R., Barlow, P.M., Peterson, S.M., Hughes, J.D., Reeves, H.W., and Viger, 

R.J., 2018, National-scale grid to support regional groundwater availability 

studies and a national hydrogeologic database: U.S. Geological Survey data 

release, accessed July 25, 2018, at https://doi.org/10.5066/F7P84B24.

Clark, B.R., Westerman, D.A., and Fugitt, D.T., 2013, Enhancements to the 

Mississippi Embayment Regional Aquifer Study (MERAS) groundwater‐flow 

model and simulations of sustainable water‐level scenarios: U.S. Geological 

Survey Scientific Investigations Report 2013–5161, accessed August 9, 2018, at 

http://pubs.usgs.gov/sir/2013/5161/pdf/sir2013-5161.pdf.

Csontos, R.M., 2007, Three dimensional modeling of the Reelfoot Rift and New 

Madrid seismic zone [Ph.D. dissertation]: Memphis, Tennessee, University of 

Memphis, 92 p.

Hoffmann, J.H., 2017, Thickness of the Mississippi River Valley alluvium and its 

relationship to the underlying Tertiary formations of northwestern Mississippi: 

Mississippi Department of Environmental Quality, Office of Geology, OpenFile Report 291.

Missouri Department of Natural Resources, 2007, Missouri Environmental 

Geology Atlas (MEGA 2007, Version 2.1): Missouri Department of Natural 

Resources, CD-ROM.

Painter, J.A., and Westerman, D.A., 2018, Mississippi Alluvial Plain extent, 

November 2017: U.S. Geological Survey data release, accessed February 22, 

2018, https://doi.org/10.5066/F70R9NMJ.

Reba, M.L., Massey, J.H., Adviento-Borbe, M.A., Leslie, D., Yaeger, M.A., 

Anders, M., and Farris, J., 2017, Aquifer depletion in the Lower Mississippi 

River Basin: Challenges and solutions: Journal of Contemporary Water 

Research and Education, v. 162, no. 1, p. 128–139, accessed August 9, 2018, at 

https://doi.org/10.1111/j.1936-704X.2017.03264.x.

Robertson, G.P., 2008, GS+: Geostatistics for the environmental sciences: 

Plainwell, Michigan, Gamma Design Software, ISBN 0-9707410-0-6, 176 p.

Saucier, R.T., 1994, Geomorphology and Quaternary geological history of the 

Lower Mississippi Valley, v. I and II: U.S. Army Corps of Engineers Waterways 

Experiment Station, Vicksburg, Miss., 414 p. and 28 pls., accessed April 23, 

2018, at http://lmvmapping.erdc.usace.army.mil/.

Torak, L.J., and Painter, J.A., 2019, Digital surfaces of the bottom altitude and 

thickness of the Mississippi River Valley alluvial aquifer and site data within 

the Mississippi Alluvial Plain project region: U.S. Geological Survey data 

release, https://doi.org/10.5066/P9D9XR5F.

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Coast aquifer systems, south-central United States: U.S. Geological Survey 

Open-File Report 87–677, 213 p.

References

Geostatistical analytics of structural 

analysis, variogram modeling, cross 

validation, kriging, and estimation variance 

(Robertson, 2008) were used to identify 

spatial-autocorrelation structures of the MRVA 

bottom-altitude and thickness data. These 

analytics defined the geospatial framework 

for geostatistical estimation of MRVA bottom 

altitude and thickness as gridded interpolated 

surfaces on the 1-kilometer-resolution USGS 

National Hydrogeologic Grid (Clark and 

others, 2018) (figs. 1 and 3). 

Structural analysis defined spatialautocorrelation properties of the data that 

integrate differences between data values 

at each paired location and their separation 

distances into mathematical expressions of 

semivariance (of bottom altitude, figs. 4A, 

B) and log-semivariance (of thickness, fig. 

4C), which form experimental variograms 

and variogram models (symbols and curves, 

respectively, fig. 4).

Cross validation was used to compare 

estimates of bottom altitude and thickness with 

actual data values (fig. 5). These estimates 

were determined by using averaging (kriging) 

weights developed for, and applied to, the 

data locations based on spatial-autocorrelation 

structures identified by the semivariance and 

variogram models. A result of cross validation 

indicated that filtering to remove 2,619 bottomaltitude values containing estimation errors 

larger than 25 feet would increase the range 

of spatial correlation by 70 percent, compared 

with the range of spatial correlation associated 

with the unfiltered data (compare range values 

on variograms for unfiltered and filtered 

data, fig. 4). Filtering was not applied to data 

from Hoffman (2017), because the spatial 

density of data values provided by this source 

defined local geohydrologic heterogeneity 

in the Mississippi Delta that was preserved 

in the geostatistical estimation process. The 

enhanced accuracy of the variogram model to 

estimate bottom altitude at the data locations 

is manifested in the nearly perfect regression 

(r2 = 0.999) of variogram-model estimates 

with the filtered bottom-altitude data (based on 

6,604 log picks), compared with the regression 

(r2 = 0.993) of the corresponding variogrammodel estimates with the unfiltered data.

Values of MRVA thickness were prepared 

for geostatistical estimation by subtracting 

bottom altitudes of the filtered data from 

DEM values at the data locations. Structural 

analysis, cross validation, and variogram 

modeling of thickness identified two 

outlier data points that were removed from 

the dataset. The resulting 6,602-element 

dataset was used to generate the thickness 

interpolation surface (fig. 3). The complexity 

of the spatial-autocorrelation structure of 

the thickness data, derived from the top 

and bottom surfaces of the MRVA, each 

defined by their own spatial-autocorrelation 

structures, resulted in a range of spatial 

autocorrelation for the thickness variogram 

model of about half of the range associated 

with the filtered bottom-altitude dataset 

(compare range values in figs. 4B and C). 

The spatial variability of the thickness 

data is manifested in the slightly lower 

agreement of the cross-validation regression 

of the variogram model with the thickness 

data (r2 = 0.926, fig. 5C) compared with 

the agreement of the cross-validation 

regression of the corresponding variogram 

model with the filtered bottom-altitude data 

(r2 = 0.984, fig. 5B). 

Kriging based on the autocorrelation of 

data points and their distances established 

from structural analysis and variogram 

modeling was used to generate interpolated 

surfaces for the MRVA bottom altitude 

and thickness (figs. 1 and 3, respectively). 

The averaging weights established during 

variogram modeling form the basis of the 

kriging interpolation process that eliminates 

estimation bias caused by distance of the 

interpolation point from its neighbors, 

clustering, or shadowing of data.

Estimation-variance mapping of the 

interpolated bottom altitude and thickness 

shown in figures 6 and 7, respectively, 

indicates the spatial variation of uncertainty 

in the estimated surfaces. Regions containing 

comparatively large estimation variance 

indicate that additional bottom-altitude or 

thickness data obtained in these areas could 

reduce estimation variance and enhance the 

accuracy of the interpolated surfaces. 

Geostatistical Analytics