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Relationship between Cole-Cole modelparameters and spectral decomposition
parameters derived from SIP dataM. Weigand and A. Kemna
Department of Geophysics, Steinmann Institute, University of Bonn
Introduction
We here present a numerical study which compares the resulting parameters ofthe Debye and Warburg decomposition schemes with the Cole-Cole parametersused to generate the SIP signatures. The Cole-Cole (CC) model is commonly usedto analyze the characteristics of SIP data. However, decomposition approachesdescribing a given SIP signature by means of a superposition of a large number of
polarization terms have recently been increasingly adopted. SIP parameters fromdifferent phenomenological models have been jointly used to improve data cov-erage of certain relationships or processes. Slight variations in model parameterscan possibly lead to interpretation errors, which are investigated in this study bymeans of numerical experiments.
Cole-Cole model
frequency [Hz]
σ′ [
S/m
]
σ′′
[µS
/cm
]
The Cole-Cole model describes the complex resistivity (ρ̂) (Pelton et al., 1978):
ρ̂(ω) = ρ0
[1 −m
(1 − 1
1 + (jωτ )c
)],
where ρ0 is the DC resistivity, m the chargeability, τ the characteristic relaxationtime, and c the CC exponent.
Cole-Cole decomposition
The Cole-Cole decomposition (CCD) describes ρ̂ using a superposition of multiplerelaxation terms:
ρ̂(ω) = ρ0
1 −N∑k=1
mk
[1 − 1
1 + (jωτk)c̃
] ,
where mk is the k-th chargeability weight of the k-th relaxation time τk. Theimaginary number is denoted by j and the frequency dispersion of each relax-ation term is controlled by the fixed constant c . The distribution of weights mk
produced by the decomposition is called the relaxation time distribution (RTD).Important settings of the CCD are the number of relaxation times per frequencydecade and the range of relaxation times (Weigand and Kemna, 2016a).
Integral parameters
frequency [Hz]
σ′ [
mS
/m]
frequency [Hz]
σ′′
[S/m
]
τ50ττmean
τ [s]
log 1
0(m
k)[V
/V]
RTD • total chargeability mtot =∑N
k=1 mk
• τ50 is the relaxation time at which 50 %of the total chargeability is reached.
• τmean = exp(∑N
k=1 mk log(τk)∑Nk=1 mk
)frequency [Hz]
σ′ [
mS
/m]
frequency [Hz]
σ′′
[S/m
]
τ50τ
τmean
τ [s]
log 1
0(m
k)[V
/V]
RTD
Reconstruction of mtot
0.1 0.3 0.5 0.7 0.9c [–]
–4–3–2–1
012
log 1
0(τ)[
s])
c̃ = 1.0
0.1 0.3 0.5c [–]
c̃ = 0.5
0.1 0.3c [–]
c̃ = 0.3
–100 –50 0 50 100(m – mtot)/m [%]
The total chargeability mtot increasingly underestimatesthe original CC based m parameter with decreasing cvalue and with τ approaching the relaxation time limits.This behavior is observed for different CC kernels and iscompressed within the range c ≤ c̃ .
Reconstruction of relaxation times
0.1 0.3 0.5 0.7 0.9c [–]
–4–3–2–1
012
log 1
0(τ)[
s])
c̃ = 1.0
0.1 0.3 0.5c [–]
c̃ = 0.5
0.1 0.3c [–]
c̃ = 0.3
–3.0 –1.5 0.0 1.5 3.0log10(τ )[s]) – log10(τmean[s])
0.1 0.5 0.9c [–]
–4–3–2–1
012
log 1
0(τ
[s])
τmean
0.1 0.5 0.9c [–]
τ50
–3.0 –1.5 0.0 1.5 3.0
The deviation between input and recovered relax-ation time shows a similar pattern as observed forthe chargeability. Deviations of up to three orders
of magnitude exist for decreasing c values and τapproaching the relaxation time limits.Reconstruction patterns are similar for τmean andτ50.
Summary
•Cole-Cole chargeability is underestimated by up to 80%, when determined usinga CCD
•Characteristic relaxation times of the CC model and CCD differ by up to threeorders of magnitude
•CC parameters can only be estimated by CCD parameters for mono-modal SIPsignatures when τmean and τ50 are used
These results highlight the importance of a consistent SIP data analysis procedure,especially if results from different studies are to be quantitatively compared.
Software
The CCD software is maintained under:https://github.com/m-weigand/Debye_Decomposition_Tools
•Resistivity and conductivity formulations
•Open-source
•Python 2.7
References
Pelton, W., Ward, S., Hallof, P., Sill, W., and Nelson, P. (1978). Mineral discrimination andremoval of inductive coupling with multifrequency ip. Geophysics, 43(3):588–609.
Weigand, M. and Kemna, A. (2016a). Debye decomposition of time-lapse spectral inducedpolarisation data. Computers and Geosciences, 86:34–45. doi:10.1016/j.cageo.2015.09.021.
Weigand, M. and Kemna, A. (2016b). Relationship between Cole-Cole model parameters andspectral decomposition parameters derived from SIP data. Geophysical Journal International.
Contact: [email protected] 4th International Workshop on Induced Polarization, Aarhus, Denmark, June 6-8, 2016