Cokriging of the phreatic level using a digital
elevation model, DEM
Mendoza, Edgar1; Herrera, Graciela2 y Díaz, Martín3
UNAM, Av. Cuauhnáhuac, número 8532, Colonia Progreso, Jiutepec, Morelos, México; Tel: (777)
3293600 Ext. 216, Fax: (777) 3293682; edgaryuri@hotmail.com
2Instituto Mexicano de Tecnología del Agua, Av. Cuauhnáhuac, número 8532, Colonia Progreso, Jiutepec,
Morelos, México; Tel: (777) 3293600 Ext. 216, Fax: (777) 3293682; gherrera@tlaloc.imta.mx
3Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas, 152, Col. San Bartolo Atepehuacan, Del.
Gustavo A. Madero, C.P. 07730, México, D.F., Tel: (55) 91-75-8299, Ext. 6993, Fax: (55) 91-75-6993;
mdiazv@imp.mx
1DEPFI,
INTRODUCTION
SEMIVARIOGRAMS
UNIVARIATE
BIVARIATE
•Contour maps of phreatic level in aquifers are used to analyze the
behavior of groundwater levels, and flow directions.
•Interpolation linear methods, like ordinary kriging (KO), have
demonstrated their effectiveness.
•If little information exists, the estimation is smoothed.
•Hoeksema et al. (1989), showed that for geological settings where
the phreatic level is a subdues replica of the ground surface,
cokriging can be used to estimate the phreatic level elevation using
ground surface elevation as a secondary variable.
• This principle is the basis for using secondary information from
digital elevation models (DEM) to supplement sparse water well
observations for mapping phreatic surfaces (Desbarts et al. 2002).
•A spherical model was fitted to the cross semivariogram.
•The residuals semivariogram for the phreatic level and the DEM
data were fitted with a spherical model.
Fig. 4, CrossCross-semivariogram
OBJECTIVE
ESTIMATION
6 7.22
67 .22
95 .51
95.51
44 .22
58 .31
13 .41
3 1. 6 1
03 .31
0 1.91
62.8861.91
80 .41
96 .71
47. 51 87 .31
7 5.61
41.51 11 .21
8 2. 4 1
85 .51
90.41
10.4 1
63 .21
09.31
32 .41
32.41
19 .41
41. 51 11.21
82 .41
•A spherical model was fitted to the cross semivariogram.
08 .21
1 9. 4 1
29 .8
07 .31
07.31
0000853
32. 31
2 0.31
0000 853
32 .31
81. 31
81 .31
5 4.8
54.8
07.01
07. 01
32.41
32. 41
03 .01
03 .01
21 .21
21.21
0000753
0000 753
3359..161
53 .11
15 .21
3359..161
53 .11
15.21
34 .11
94 .11
3 4.11
94.11
0000653
0000 653
32 .21
APPLICATION TO MEXICALI AQUIFER
b
32. 21
a
01 .9
00001 7
•Water table elevation data is available at 90 wells in the valley.
•The aquifer is unconfined in steady-state conditions.
7 5.6 1
28 .31
2 9.8
20.31
0000 953
89.21
84.31
0 8. 2 1
28 .31
•Primary data is water table elevations in meters above sea level in
a set of wells.
•The gridded data for the secondary variable is derived from a
1:50,000 scale, 50 m resolution digital elevation model based on
the NAD27 datum.
12.01
30. 11
0 0.31
68.3017. 11
05 .41
0000953
89 .21
22 .21
0 9.31
30.1 176.41
1 4.51
35 .41
62 .41
00 .31
85 .51
84 .31
6 3.21
58 .31
45 .31
8 2.41
01. 91
30.11
62.41
68. 3017.11
90 .41
10 .41
0000 063
04.41
42 .51
12.01
30.1176 .41
14 .51
35 .41
0 5.41
22.21
000007
BIVARIATE
04 .61
48 .11
4 4.41
63. 21
94 .51
2 0.21
08. 71
0000063
04 .41
03.31
4 5. 3 1
82 .41
8 0. 4 1
96.71
47 .51 87.31
1 8. 3 1
9 3.61
13. 41
3 1.61
4 2. 5 1
0 1.91
528.17.171
4 3. 6 1
35.61
20 .51
92. 61
04. 61
48. 11
44 .41
63.21
01. 91
6 2.8 1
6.91
Fig. 8, a) Phreatic level semivariogram;
semivariogram; b) DEM semivariogram
0000 163
76.51
1 0. 5 1
80.51
67.81
1 8.31
20.21
0 8.71
5 4.81
528.17.171
43.61
3 5.61
20.51
93 .61
8 8. 4 1
51. 61
0000163
76 .51
1 0.51
9 2.61
9 4. 5 1
22 .91
54.02
95.02
7 4.61
54.81
THE PHREATIC LEVEL AND DEM DATA
27 .61
88 .41
51.61
80 .51
9 4.91
32 .41
22.91
54. 02
95 .02
74.61
67.81
44.22
94 .91
32.41
2 7. 6 1
•To apply cokriging using topographic information of DEMs, to
estimate phreatic level elevations in two different aquifers.
000096
000086
0000553
0000 553
01.9
000076
0000 17
3610000
3610000
3600000
3600000
3590000
3590000
000007
00009 6
000086
0000 76
Fig. 9, CrossCross-semivariogram
2295000
3580000
ESTIMATION
2295000
1890.41
3580000
1890.41
1894.43
1893.61
1894.43
1893.61
1892.80
1892.80
2290000
2290000
1832.29
3570000
1832.29
San Vicente
de Ferrer
1873.76
1857.15
1855.48
C . El Na bo
1821.96
3550000
1724.02
1703.57
1705.53
1675.69
1716.74
1722.52
1715.91 1707.06
1718.28
1717.65
2280000
1728.29
1717.85
1711.76
680000
690000
700000
710000
670000
680000
690000
700000
2275000
1713.26
1713.24
1709.91
1711.53
1714.33
1730.10
2275000
Vi lla del
M arqués
1713.26
LAGUNA EL SALITRE
1705.701712.75
1712.44
1704.60
C. Gordo
1711.58
1702.16
1713.24
1709.91
1711.53
1714.33
1711.73
1702.16
1711.73
1720.24
V. Corregidora
C. El Cim atario
2270000
1744.35
1744.35
1760.01
1760.01
1781.96
1781.96
2265000
2265000
1890.94
•Unconfined aquifer with local confinements.
•The effects of withdrawal by pumping wells produced a cone of
depression centered upon the City of Queretaro.
•The hypothesis that groundwater flow follows the topography is
only true in the areas of Obrajuelos and Salitre.
•Cokriging was used only on those zones.
SCATTERGRAM AND STATISTICAL DATA
Villa
C ayetano R.
C . La Cruz
1702.17
1711.80
C. El Cim atario
2270000
1798.99
1721.94
QUERETARO
1701.69
C. Las Cam panas
1713.16
1720.24
V. Corregidora
APPLICATION TO QUERETARO AQUIFER
a
2260000
335000
338000
341000
b
1890.94
344000
347000
350000
353000
356000
2260000
359000
335000
2295000
338000
341000
344000
347000
350000
353000
356000
359000
2295000
2290000
2290000
San Vicente
de Ferrer
San Vicente
de Ferrer
C . El Na bo
2285000
C. El Nabo
2285000
Jurica
Jurica
C. Arbor Acres
2280000
C . Arbor Acres
2280000
Ae ropuerto
S.P.Martir
Aero pue rto
S.P.Martir
Villa del
M arqués
QUERETARO
Vi lla del
M arqués
Villa
Ca yetano R.
QUERETARO
C . Las Campanas
C . La Cruz
•The residuals semivariogram of the phreatic level and of the DEM
data were fitted with a spherical model.
C. Gordo
V. Corregidora
V. Corregidora
C. El Cim atario
CITY OF
QUERÉTARO
C. El Cim atario
2270000
2270000
2265000
2265000
c
2260000
AQUIFER
d
2260000
335000
Obrajuelo zone
338000
341000
344000
347000
350000
353000
356000
359000
335000
338000
341000
344000
347000
350000
353000
356000
359000
Fig. 10, a) OK; b) COK; c) COK variance;
variance;
d) OK variance/
variance/ COK variance.
variance.
CONCLUSIONS
Figure 6, Queretaro Valley and region in which cokriging was
Applied.
SEMIVARIOGRAMS
UNIVARIATE
LAGUNA EL SALITRE
C . Gordo
State limit
Querétaro
b
2275000
LAGUNA EL SALITRE
Salitre zone
Fig. 2, a) Scattergram;
Scattergram; b) Histogram of phreatic level residuals..
Villa
C ayetano R.
C. Las Cam panas
C. La Cruz
2275000
•The linear relation between the primary and secondary variables
was checked and a good correlation was obtained.
•The minimum and maximum water level elevation are 6.93 m and
22.76 m, respectively.
•In the variables analyzed a drift effect was observed and it was
necessary to adjust a polynomial of second order to estimate it. The
polynomial was substracted from the original data and the
geoestatistical analysis was done with the residuals.
a
1814.68
1813.95
1763.63
1712.46
1706.87
1721.02
1717.85
1711.76
1710.66
C . Gordo
Fig. 5, a) OK; b) COK; c) COK variance;
variance;
d) OK variance/
variance/ COK variance.
variance.
Aero pue rto
1666.36
1640.42
1658.08
1706.55
1716.83
1728.29
1717.70
1712.44
1704.60
1711.58
1722.66
1714.83
1708.84
1708.28
1702.74
1704.62
1711.80
1705.701712.75
1713.48
Villa
Ca yetano R.
C. La Cruz
1702.17
1713.16
LAGUNA EL SALITRE
1716.74
1728.55
1704.69
Villa del
M arqués
1721.94
QUERETARO
1701.69
C . Las Campanas
1710.66
1730.10
710000
1707.251699.86
S.P.Martir
1712.45
1700.54
1705.96
1798.99
1706.871721.02
1717.70
1714.09
1705.22
1814.68
1813.95
1763.63
1712.46
1708.84
1708.28
1702.74
2280000
Ae ropuerto
1666.36
1640.42
1658.08
1706.55
1716.83
1728.55
1713.48
1675.69
1722.52
1715.91 1707.06
1718.28
1717.65
1714.83
S.P.Martir
1712.45
1700.54
1705.96
1704.69
C . Arbor Acres
1713.24
1722.66
1707.25
1699.86
1714.09
1705.22
1703.57
1705.53
1714.50
C. Arbor Acres
1713.24
d
1704.62
670000
1709.23
Jurica
1724.02
1714.50
c
3550000
1821.96
1722.18
2285000
1709.23
Jurica
3560000
1873.76
1857.15
1855.48
C. El Nabo
1722.18
2285000
3560000
Figure 1. Mexicali valley and location of 90 wells
with preatic level data.
San Vicente
de Ferrer
3570000
STATISTICAL DATA
•Phreatic levels available at 85 wells.
•Minimum and maximum water level elevations: 1640.4 m and
1894.4 m.
•The correlation coefficient is 0.85.
•The drift effect was estimated using the same procedure as for the
Mexicali aquifer.
For both applications, the improvement in the contour map for the
phreatic level obtained by cokriging indicated by figure 5d and
figure 10d is of 14%, also the values of the error variances is
reduced.
For the Mexicali case the COK estimation and the OK estimation
are very similar. This could be due to the low variability of both
variables.
For the Queretaro case, in contrast, the COK estimation has greater
changes that define in a beter way the directions of flow in the
Obrajuelo and Salitre zones.
REFERENCES
Desbarats, A.J., Logan, C.E., Hinton, M.J., Sharpe, D.R., 2002. On
the kriging of water table elevations using collateral information
from a digital elevation model. Journal of Hydrology. 255, 25-38.
Hoeksema, R.J., Clapp, R.B., Thomas, A.L., Hunley, A.E., Farrow,
N.D., Dearstone, K.C., 1989. Cokriging model for estimation of
water table elevation, Water Resources Research. Vol. 25, No. 3,
429-438.
ACKNOWLEDGMENTS
To CNA, CONACYT and UNAM that supported this project by means
of scholarships.
a
b
Figure 3, a) Phreatic level semivariogram;
semivariogram; b) DEM semivariogram
a
b
Figure 7, a) Scattergram;
Scattergram; b) Histogram of phreatic level residuals.
residuals.