In [ ]:
from geoscilabs.dcip.CondUtils import ColeColePelton, vizColeCole
from geoscilabs.dcip.FreqtoTime import transFilt
import matplotlib
from ipywidgets import interact, FloatText, FloatSlider, ToggleButtons
In [ ]:
matplotlib.rcParams['font.size'] = 16

Complex conductivity and resistivity

Purpose

Using a simple Cole-Cole model, we parameterize complex resistivity with four parameters: resistivity at zero frequency ($\rho_0$), chargeability($\eta$), time constant ($\tau$), and frequency dependence ($c$). Based upon those parameters, we understand how resistivity and conductivity changes when medium is chargeable both in frequency domain and time domain.

Set up

Pelton's Cole-Cole model for resistivity and conductivity can be written as

$$ \rho(\omega) = \rho_0 \Big[1 - \eta \Big(1-\frac{1}{1+(\imath\omega\tau)^c}\Big) \Big] $$

and

$$ \sigma(\omega) = \sigma_{\infty}\Big(1-\frac{\eta}{1+(1-\eta)(\imath\omega\tau)^c} \Big) $$

respectively.

Cole-Cole app

Parameters

  • $\sigma_1$: Conductivity of the first layer (S/m)

  • $\sigma_2$: Conductivity of the first layer (S/m)

  • $f$ (Hz): Frequency (Hz)

  • Type:

    • Reflection: Transmission power as a function of incident angle
    • Transmission: Transmission power as a function of incident angle
    • Angle: relationship between $\theta_i$ and $\theta_t$
In [ ]:
interact(vizColeCole, eta=FloatSlider(min=0.1, max=0.5, step=0.05, value=0.4), 
         tau=FloatText(value=0.1), 
         c=FloatSlider(min=0.1, max=1., step=0.1, value=0.5), 
         sigres = ToggleButtons(options=['sigma','resis']), 
         t1=FloatText(value=800), 
         t2=FloatText(value=1400),          
        );
In [ ]: