Quantitative Methods Final Project

Laura Burnette

Part I: El Nino and Speleothems

Part II: Directional Data Analysis of Lineaments in Tucker County, West Virginia

Part I: El Nino and Speleothems

Speleothems as paleoclimate indicators

Data

4. Speleothem data - collected by Polyak and Asmerom (2001) - represents annual band data for stalagmite BC2, Bat Cave passage of Carlsbad Cavern, Carlsbad Caverns National Park, New Mexico

1. Carbon isotope data from Actun Tunichil Muknal cave in Belize collected by Frappier, et al. (2002)

2. Calculated monthly Southern Oscillation Index data

3. Reconstructed Annual Southern Oscillation Index data

Date1970 1975 1980 1985 1990 1995 2000 2005

d13C

-14.0-12.0-10.0-8.0-6.0-4.0-2.0

Carbon Isotope data from Actun Tunichil Muknal cave in Belize

Visual Inspection of Correlation

year1970 1975 1980 1985 1990 1995 2000 2005

soi

-4-3-2-10123

Southern Oscillation Index from 1973 to 2001

Frequency (cycles per month)0.0 0.1 0.2 0.3 0.4 0.5

AM

P

0.0

0.10.2

0.30.4

0.5Amplitude Spectrum of the SOI from 1973 to 2001

Frequency (cycles per month)0.000 0.040 0.080 0.120

AM

P

0.0

0.1

0.2

0.3

0.4

0.5Amplitude Spectum of the SOI from 1973 to 2001

f = .0056 cycles/monthP = 14.9 years

f = .018 cycles/monthP = 4.6 years

f = .023 cycles/monthP = 3.6 years

f = .035 cycles/monthP = 2.4 years

f = .041 cycles/monthP = 2.0 years

f = .11 cycles/monthP = 9.1 months

f = .055 cycles/monthP = 18.2 months

Frequency (cycles/month) Period (years)0.0056 14.90.018 4.60.023 3.60.035 2.40.041 20.055 1.50.11 0.76 (9.1 months)

Fourier Analysis of Monthly

Calculated SOI

Year

1700 1750 1800 1850 1900 1950 2000

SOI

-20.0-15.0-10.0

-5.00.05.0

10.0SOI Reconstruction (Stahle, et al.)

Frequency (cycles per year)

0.0 0.1 0.2 0.3 0.4 0.5

AM

P

0.00

0.20

0.40

0.60

0.80

1.00

1.20Amplitude Spectrum of Annual SOI

f = .175 cycles/yearP = 5.7 years f = .24 cycles/year

P = 4.2 years

f = .28 cycles/yearP = 3.6 years

Frequency (cycles/year) Period (years)

0.175 5.7

0.24 4.2

0.28 3.6

Fourier Analysis of

Annual Reconstructed

SOI

Year

1700 1750 1800 1850 1900 1950 2000

SOI

-20.0-15.0-10.0

-5.00.05.0

10.0Band pass filter output of annual SOI (Stahle, et al.) with cutoffs of .15 and .2

Year

1900 1920 1940 1960 1980 2000

SOI

-20.0-15.0-10.0

-5.00.05.0

10.0Band pass filter output of annual SOI (Stahle, et al.) with cutoffs of .15 and .2

Column1

1700 1750 1800 1850 1900 1950 2000

Filte

r9

-20.0

-15.0

-10.0

-5.0

0.0

5.0

10.0Band pass filter output of annual SOI (Stahle, et al.) with cutoffs of .2 and .3

Column1

1950 1960 1970 1980 1990 2000

Filte

r9

-20.0

-15.0

-10.0

-5.0

0.0

5.0

10.0Band pass filter output of annual SOI (Stahle, et al.) with cutoffs of .2 and .3

Bandpass filtering of

annual reconstructed

SOI

date

1970 1975 1980 1985 1990 1995 2000

dC13

-14-12-10

-8-6-4-2

Carbon Isotope Data from Actun Tunichil Muknal cave

Frequency (cycles/year)

0.0 0.1 0.2 0.3 0.4 0.5

AM

P

0.000.200.400.600.801.001.20

Amplitude Spectrum of carbon isotope data from Actun Tunichil Muknal cave

f = .065 cycles/yearP = 15.4 years f = .19 cycles/year

P = 5.3 yearsf = .28 cycles/yearP = 3.6 years f = .46 cycles/year

P = 2.2 years

Frequency (cycles/year) Period (years)0.065 15.40.19 5.30.28 3.60.46 2.2

Fourier Analysis of Carbon

Isotope Data

Years BP150 200 250 300 350 400 450 500Th

ickn

ess

(mm

)

0.0000.0500.1000.1500.2000.2500.300

Speleothem band thickness for Carlsbad Caverns

Frequency (cycles per year)0.0 0.1 0.2 0.3 0.4 0.5A

mpl

itude

0.0000.0050.0100.0150.020

Amplitude spectrum for speleothem band thickness in Carlsbad Caverns over 450 to 161 years BP

Frequency (cycles per year)0.000 0.050 0.100 0.150 0.200

Am

plitu

de

0.0000.0050.0100.0150.020

Amplitude spectrum for speleothem band thickness in Carlsbad Caverns over 450 to 161 years BPf = .0038 cycles/yearP = 263 years

f = .013 cycles/yearP = 77 years

f = .024 cycles/yearP = 42 years f = .038 cycles/year

P = 26 years

f = .065 cycles/yearP = 15 years f = .17 cycles/year

P = 5.9 years f = .18 cycles/yearP = 5.6 years

f = .41 cycles/yearP = 2.4 years

Frequency (cycles/year) Period (years)0.0038 2630.013 770.024 420.038 260.065 150.17 5.90.18 5.60.41 2.4

Fourier Analysis of Speleothem

Band Thickness

Data

Belize carbon isotope data

        15   5.3   3.6 2.2      

Calculated monthly SOI (1973-2001)

        14.9     4.6 3.4 2.4 2 1.5 0.76

Reconstructed annual SOI (1706-1977)

            5.7 4.2 3.6        

Carlsbad speleothem band thickness

263 77 42 46 14 5.9 5.6     2.4      

 

Dataset Periods (in years)

Shared Periodic Components

Part II: Directional Data Analysis of Lineaments in Tucker County, West Virginia

Number of cave entrances within specified distance from the nearest lineament

Distance From Lineament

Number of Cave Entrances

Percentage of Cave Entrances

0 to 50 meters 72 51.1

50 to 100 meters 25 17.7

100 to 150 meters 13 9.2

150 to 200 meters 7 5.0

200 to 250 meters 2 1.4

250 to 300 meters 3 2.1

300 to 350 meters 2 1.4

over 350 meters 1 0.7

  

Statistics for Directional Data

• Rose diagrams may be misleading

• Tests for randomness

• Tests for equality of two samples

• Assumption: Von Mises distributions

• Non-parametric test for equality of two samples: Uniform scores test

Finding the azimuths of the lineaments

x1 y1 x2 y2 length delta x delta y alpha alpha(degrees)90-alpha2 647922.6 4328811.0 646162.7 4320352.0 8640.135 -1759.9 -8459.0 1.4 78.2 11.83 645311.1 4318195.0 642756.3 4312461.0 6277.401 -2554.8 -5734.0 1.2 66.0 24.04 604888.7 4347887.0 602333.9 4341926.0 6485.409 -2554.8 -5961.0 1.2 66.8 23.25 601482.3 4338293.0 599892.6 4330855.0 7605.984 -1589.7 -7438.0 1.4 77.9 12.16 626859.8 4324326.0 629074.0 4322907.0 2629.875 2214.2 -1419.0 -0.6 -32.7 122.77 619592.9 4315129.0 622147.7 4318649.0 4349.414 2554.8 3520.0 0.9 54.0 36.08 614823.9 4321601.0 613120.8 4317059.0 4850.805 -1703.1 -4542.0 1.2 69.4 20.69 612212.4 4314845.0 614142.7 4317457.0 3247.861 1930.3 2612.0 0.9 53.5 36.5

10 615164.6 4323248.0 616697.4 4326370.0 3477.982 1532.8 3122.0 1.1 63.9 26.111 619365.8 4326654.0 625270.2 4328244.0 6114.74 5904.4 1590.0 0.3 15.1 74.912 626292.1 4328641.0 627768.2 4330344.0 2253.681 1476.1 1703.0 0.9 49.1 40.913 639463.4 4322396.0 643324.0 4331196.0 9609.591 3860.6 8800.0 1.2 66.3 23.714 644573.0 4336987.0 643607.9 4331480.0 5590.927 -965.1 -5507.0 1.4 80.1 9.915 636000.3 4314845.0 638838.9 4315016.0 2843.746 2838.6 171.0 0.1 3.4 86.616 607500.3 4326711.0 610736.3 4326541.0 3240.462 3236.0 -170.0 -0.1 -3.0 93.0

Step 1: ArcINFO UNGENERATE command

Step 2: Trigonometry

Landsat lineaments

Figure 12: Rose diagram of the orientations of lineaments digitized from Landsat image of Tucker County.

Figure 13: Rose diagram showing orientation of lineaments digitized from PCA image of Tucker County.

Lineaments digitized from Landsat and PCA

Lineaments digitized from hillshades

Figure 15: Rose diagram showing orientation of lineaments digitized from hillshade of the Mozark Mountain quadrangle with a sun azimuth of 225 degrees.

Figure 16: Rose diagram showing orientations of lineaments digitized from hillshade of the Mozark Mountain quadrangle with a sun azimuth of 315 degrees.

 

Test for Randomness

90-alpha angle*2 cos sin11.8 23.5 0.917022 0.398837 Xr sum cos 16.8167524.0 48.0 0.668728 0.743507 Yr sum sin 22.654123.2 46.4 0.689638 0.724155 theta mean 53.4125 true mean 26.7062512.1 24.1 0.912633 0.408781 R 28.21368

122.7 245.3 -0.41773 -0.90857 standardized R 0.290863 mean resultant length36.0 71.9 0.309947 0.950754 values near zero indicate dispersed vectors20.6 41.1 0.753463 0.657491 circular variance 0.70913736.5 72.9 0.293545 0.95594526.1 52.3 0.61154 0.791213 With an alpha level of .05, this distribution is statistically 74.9 149.9 -0.86477 0.502166 different from random.40.9 81.8 0.142021 0.98986423.7 47.4 0.677203 0.735796

9.9 19.9 0.940405 0.34005586.6 173.1 -0.99277 0.12004693.0 186.0 -0.9945 -0.1047841.1 82.1 0.137344 0.990523

141.5 283.0 0.224951 -0.97437

Landsat lineaments

Results of the test for Randomness

At an alpha level of .05,

• Landsat lineaments are not random

• PCA lineaments are random

• Both sets of hillshade lineaments are not random

Figure 13: Rose diagram showing orientation of lineaments digitized from PCA image of Tucker County.

Uniform Scores Test

Spreadsheet used to calculate test for equality of samples

lins225 lins315 225rank 315rank 225scores cosscores sinscores100.9917 1.96725 99 211 4.507503 -0.20346 -0.97908263.1426 6.479598 43 208 1.957804 -0.37742 0.926043 sum cos -3.65792

69.0461 81.57158 138 117 6.283185 1 -2.5E-16 sum sin 36.98944229.1251 61.13629 50 149 2.276516 -0.64858 0.761145 R1^2 1381.599293.4964 323.144 23 9 1.047198 0.5 0.866025 n1 138264.9601 117.271 40 85 1.821213 -0.24781 0.968809 n2 76

327.251 280.1307 7 28 0.318712 0.94964 0.313344 n 21440.75865 79.30992 169 120 7.694625 0.158683 0.98733 Rstar 56.11759

0 89.94435 212 109 9.65243 -0.9742 -0.22569272.7372 19.50828 33 189 1.502501 0.068242 0.997669177.5845 90.69536 64 108 2.913941 -0.9742 0.22569 According to appendix 2.13 from Mardia and Jupp177.7267 321.2069 63 11 2.868411 -0.96292 0.269797 these two datasets are statistically different353.8982 291.0488 1 24 0.04553 0.998964 0.045515296.5343 204.446 22 53 1.001667 0.538899 0.8423717.241016 69.6231 206 137 9.379248 -0.99896 0.045515

Results of Uniform Scores Test

• The Landsat lineaments and PCA lineaments are statistically different.

• The two sets of hillshade lineaments are statistically different.

Landsat PCA

Sun azimuth 225 Sun azimuth 315

References

Allan, R.J., J. Lindesay and D. Parker. 1996. El Nino/Southern Oscillation & Climatic Variability. CSIRO Publishing. 408 pp.

  Allan, R.J., Nicholls, N., Jones, P.D. and Butterworth, I.J., 1991: A further extension of the Tahiti-Darwin SOI, early SOI results and Darwin pressure. J. Climate 4, 743-749.

  Cook, E.R. 1985. A Time Series Approach to Tree Ring Standardization. PhD dissertation. Tucson, AZ: U. of Arizona. 171 p. University Microfilms International, 300 N. Zeeb Rd, Ann Arbor, MI 48106

 Davis, J.C., 2002. Statistics and Data Analysis in Geology, John Wiley & Sons, New York, New York.  Frappier, A., D. Sahagian, L.A. Gonzalez, and S.J. Carpenter, 2002, El Nino Events Recorded by Stalagmite Carbon Isotopes, Science 298 (5593), pp.565, 18 October 2002.

  Können, G.P., Jones, P.D., Kaltofen, M.H. and Allan, R.J., 1998. Pre-1866 extensions of the Southern Oscillation Index using early Indonesian and Tahitian meteorological readings. J. Climate 11, 2325-2339.

  Lattman, L., 1958. Technique of mapping geologic fracture traces and lineaments on aerial photographs, Photogrammetric Engineering, 24(4): 569-576. 

Mardia, K.V., 2000. Directional Statistics. John Wiley & Sons, Chichester, West Sussex, England.

NOAA/National Weather Service, The El Nino/La Nina Cycle (Tutorial), National Weather Service Climate Prediction Center, www.nws.noaa.gov.

 Polyak, V.J. and Asmerom, Y. 2001 Late Holocene Climate and Cultural Changes in the Southwestern United States, Science 294: 148-151.  

Ropelewski, C.F. and Jones, P.D., 1987: An extension of the Tahiti-Darwin Southern Oscillation Index. Monthly Weather Review 115, 2161-2165.

  Stahle, D.W., R.D. D'Arrigo, P.J. Krusic, M.K. Cleaveland, E.R. Cook, R.J. Allan, J.E. Cole, R.B. Dunbar, M.D. Therrell, D.A. Gay, M.D. Moore, M.A. Stokes, B.T. Burns, J. Villanueva-Diaz and L.G. Thompson. 1998. Experimental dendroclimatic reconstruction of the Southern Oscillation. Bull. American Meteorological Society 79: 2137-2152.

White, W., 1988. Geomorphology and Hydrology of Karst Terrains. Oxford University Press, New York, NY.

WVASS, unpublished. Caves and Karst of Tucker County, West Virginia.