U.S. Department of the InteriorU.S. Geological Survey

End-Member Mixing Analysis Applied End-Member Mixing Analysis Applied to the Karstic Madison Aquifer Using to the Karstic Madison Aquifer Using Water Chemistry in the Southern Water Chemistry in the Southern Black Hills, South DakotaBlack Hills, South DakotaPresented byPresented by: Joshua F. Valder: Joshua F. Valder

Co Authored byCo Authored by: Andrew J. Long, PhD, Arden D. Davis, PhD, : Andrew J. Long, PhD, Arden D. Davis, PhD,

and Scott J. Kenner, PhDand Scott J. Kenner, PhD

2010 GSA Rocky Mountain Section Meeting2010 GSA Rocky Mountain Section Meeting

Western South Dakota Hydrology ConferenceWestern South Dakota Hydrology Conference

Rapid City, SDRapid City, SD

April 22, 2009April 22, 2009

Purpose of the StudyPurpose of the Study Develop a multivariate statistical approach Develop a multivariate statistical approach

that can characterize potential that can characterize potential sourcesource areasareas and and mixing proportionsmixing proportions..

Sources are not well understoodSources are not well understood

Mixing proportions are not well understoodMixing proportions are not well understood

Application to chemical dataApplication to chemical data

Purpose of the StudyPurpose of the Study

Location and Scope of the Location and Scope of the Wind Cave StudyWind Cave Study

Samples collected Samples collected during April, May, and during April, May, and July of 2007July of 2007

19 sites19 sites SpringsSprings Sinking StreamsSinking Streams Cave WatersCave Waters WellsWells

10 species10 species Ca, Mg, Na+K, SOCa, Mg, Na+K, SO44, ,

HCOHCO33+CO+CO33, Cl, N, Si, , Cl, N, Si, δδ22H, H, δδOO1818

Development of the ApproachDevelopment of the Approach

Three main ideas: (1) Principal Component Analysis Three main ideas: (1) Principal Component Analysis and Cluster Analysis, (2) the Conceptual Model, and and Cluster Analysis, (2) the Conceptual Model, and (3) the MIX Model.(3) the MIX Model.

Principal Component AnalysisPrincipal Component Analysis

Statistically the PCA:Statistically the PCA:

Is a way of analyzing large and complex Is a way of analyzing large and complex datasets that may otherwise be very datasets that may otherwise be very confusingconfusing

Reduces multivariate datasets into a lower Reduces multivariate datasets into a lower dimension (2D or 3D) as to explain the most dimension (2D or 3D) as to explain the most relevant informationrelevant information

Principal Component AnalysisPrincipal Component Analysis

PC 1

PC 2PC 3

Principal components in 3 dimensions – physical representationPrincipal components in 3 dimensions – physical representation

Identification of Identification of extreme pointsextreme points

Identify mixing Identify mixing points that are points that are proportions of end-proportions of end-membersmembers

Shows the relation Shows the relation among data pointsamong data points

Location of the Location of the data points directly data points directly relates to the relates to the correlation of the correlation of the datadata x1

x2

3D Example of Multivariate Space

Cluster AnalysisCluster Analysis Applied graphically through the principal component Applied graphically through the principal component

analysis and statistically using a partition of medoids analysis and statistically using a partition of medoids cluster analysis.cluster analysis. Graphically:Graphically:

The Conceptual ModelThe Conceptual Model The conceptual model The conceptual model

simplesimple (cartoon (cartoon schematics) schematics)

complexcomplex (GIS Spatial (GIS Spatial Interpretation tools)Interpretation tools)

The conceptual model can The conceptual model can better define end-members better define end-members and interpret mixing and interpret mixing proportions by providing a proportions by providing a hydrological connectivityhydrological connectivity and and understandingunderstanding of the data of the data

The MIX ModelThe MIX Model

Fortran Based ModelFortran Based Model Developed by Carrera and others, 2004Developed by Carrera and others, 2004 The approach uses the MIX model to predict the The approach uses the MIX model to predict the

likely concentrations and mixing proportionslikely concentrations and mixing proportions Input parameters include:Input parameters include:

The extreme values that are identified in the PCAThe extreme values that are identified in the PCA The data matrix for all mixing points and The data matrix for all mixing points and

variablesvariables The mixing proportions for each site (input as The mixing proportions for each site (input as

initial guesses)initial guesses)

Wind Cave National Park Water- Wind Cave National Park Water- Chemistry StudyChemistry Study

South Dakota

Black Hills

Study Area

Principal Component AnalysisPrincipal Component Analysis

Principal Component 1

Prin

cip

al C

ompo

nent

2

Clustering the Clustering the estimated end-estimated end-members (EEMs) members (EEMs) with the data is with the data is done visually done visually using the PCA using the PCA and statistically and statistically using cluster using cluster analysis.analysis.

Eastern Outcrop

Western Outcrop

Regional Groundwater

Flow

Principal Component AnalysisPrincipal Component Analysis

Eastern Outcrop

Western Outcrop

Regional Groundwater

Flow

MIX Model and Verification using MIX Model and Verification using Principal Component AnalysisPrincipal Component Analysis

Calculated mixing Calculated mixing percentages using the 3 percentages using the 3 end-member waters.end-member waters. Example – PW1Example – PW1

Eastern Outcrop

Western Outcrop

Regional Groundwater

Flow

MIX Model and Verification using MIX Model and Verification using Principal Component AnalysisPrincipal Component Analysis

PW1: PW1: Eastern Outcrop – 52%Eastern Outcrop – 52% Western Outcrop – 42%Western Outcrop – 42% Regional GW Flow – 6%Regional GW Flow – 6%

Revisit the purpose of the research:Revisit the purpose of the research:

To develop a multivariate statistical approach that can To develop a multivariate statistical approach that can characterize potential source areas (end-members) and characterize potential source areas (end-members) and mixing proportions.mixing proportions.

Has the approach worked?Has the approach worked?

YES!!YES!!

Able to determine end-membersAble to determine end-members

Define quantitatively the proportions of end-members at Define quantitatively the proportions of end-members at a mixing pointa mixing point

Concluding ThoughtsConcluding Thoughts

PCA was used to define extreme PCA was used to define extreme points using water chemistry to points using water chemistry to define end-member waters.define end-member waters.

The MIX model estimated end-The MIX model estimated end-member chemistries and mixing member chemistries and mixing proportions at sample sites.proportions at sample sites.

PCA was used to verify the results of PCA was used to verify the results of the MIX model and cluster the the MIX model and cluster the samples to better understand samples to better understand groundwater flow.groundwater flow.

Concluding ThoughtsConcluding Thoughts

QuestionsQuestions

Supplemental Slide #1Supplemental Slide #1Limitations and Assumptions of the ApproachLimitations and Assumptions of the Approach

•Geochemical changesGeochemical changes

•Number of variables in the MIX ModelNumber of variables in the MIX Model

•Conceptual model must be fully understood to put Conceptual model must be fully understood to put physical meaning on the mathematicsphysical meaning on the mathematics

•Assume complete mixing at a mixing pointAssume complete mixing at a mixing point

•All assumptions of the mathematical models must be All assumptions of the mathematical models must be met:met:

• LinearityLinearity• Mean and variance can be used to characterize the dataMean and variance can be used to characterize the data• Large variance in the data is significantLarge variance in the data is significant• The PCs are orthogonally orientedThe PCs are orthogonally oriented

Supplemental Slide #2Supplemental Slide #2 South Dakota

Black Hills

Study Area

General Water Quality Seasonality