End-Member Mixing Analysis Applied
to the Karstic Madison Aquifer Using
Water Chemistry in the Southern
Black Hills, South Dakota
Presented by: Joshua F. Valder
Co Authored by: Andrew J. Long, PhD, Arden D. Davis, PhD,
and Scott J. Kenner, PhD
2010 GSA Rocky Mountain Section Meeting
Western South Dakota Hydrology Conference
Rapid City, SD
April 22, 2009
U.S. Department of the Interior
U.S. Geological Survey
Purpose of the Study
 Develop a multivariate statistical approach
that can characterize potential source areas
and mixing proportions.

Sources are not well understood

Mixing proportions are not well understood

Application to chemical data
Purpose of the Study
Location and Scope of the
Wind Cave Study
 Samples collected

during April, May, and
July of 2007
19 sites




Springs
Sinking Streams
Cave Waters
Wells

Ca, Mg, Na+K, SO4,
HCO3+CO3, Cl, N, Si,
δ2H, δO18
 10 species
Development of the Approach
Three main ideas: (1) Principal Component
Analysis and Cluster Analysis, (2) the Conceptual
Model, and (3) the MIX Model.
Principal Component Analysis
Statistically the PCA:

Is a way of analyzing large and complex
datasets that may otherwise be very
confusing

Reduces multivariate datasets into a lower
dimension (2D or 3D) as to explain the most
relevant information
Principal Component Analysis
Principal components in 3 dimensions – physical representation




Identification of
extreme points
Identify mixing
points that are
proportions of endmembers
Shows the relation
among data points
Location of the
data points directly
relates to the
correlation of the
data
3D Example of Multivariate Space
PC 1
PC 3
PC 2
x2
x1
Cluster Analysis

Applied graphically through the principal component
analysis and statistically using a partition of medoids
cluster analysis.
 Graphically:
The Conceptual Model

The conceptual model
 simple (cartoon
schematics)
 complex (GIS Spatial
Interpretation tools)

The conceptual model can
better define end-members
and interpret mixing
proportions by providing a
hydrological connectivity and
understanding of the data
The MIX Model




Fortran Based Model
Developed by Carrera and others, 2004
The approach uses the MIX model to predict the
likely concentrations and mixing proportions
Input parameters include:
 The extreme values that are identified in the PCA
 The data matrix for all mixing points and
variables
 The mixing proportions for each site (input as
initial guesses)
Wind Cave National Park WaterChemistry Study
South Dakota
Black Hills
Study Area
Clustering the
estimated endmembers (EEMs)
with the data is
done visually
using the PCA
and statistically
using cluster
analysis.
Principal Component 2
Principal Component Analysis
Principal Component 1
Principal Component Analysis
Eastern
Outcrop
Western
Outcrop
Regional
Groundwater
Flow
MIX Model and Verification using
Principal Component Analysis
Eastern
Outcrop
Western
Outcrop
 Calculated mixing
percentages using the 3
end-member waters.
 Example – PW1
Regional
Groundwater
Flow
MIX Model and Verification using
Principal Component Analysis
Eastern
Outcrop
Western
Outcrop
 PW1:
 Eastern Outcrop – 52%
 Western Outcrop – 42%
 Regional GW Flow – 6%
Regional
Groundwater
Flow
Concluding Thoughts
Revisit the purpose of the research:
To develop a multivariate statistical approach that can
characterize potential source areas (end-members)
and mixing proportions.
Has the approach worked?
YES!!
Able to determine end-members
Define quantitatively the proportions of end-members
at a mixing point
Concluding Thoughts

PCA was used to define extreme
points using water chemistry to
define end-member waters.

The MIX model estimated endmember chemistries and mixing
proportions at sample sites.

PCA was used to verify the results of
the MIX model and cluster the
samples to better understand
groundwater flow.
Questions
Supplemental Slide #1
Limitations and Assumptions of the Approach
• Geochemical changes
• Number of variables in the MIX Model
• Conceptual model must be fully understood to put
physical meaning on the mathematics
• Assume complete mixing at a mixing point
• All assumptions of the mathematical models must be
met:
•
•
•
•
Linearity
Mean and variance can be used to characterize the data
Large variance in the data is significant
The PCs are orthogonally oriented
Supplemental Slide #2
South Dakota
Black Hills
Study Area
General Water Quality
Seasonality