Integral Parameters¶

Integral parameters extracted from the RTD fall into two categories: chargeability related values and relaxation time related values. The first category extracts information regarding the total or partial polarization strength of the system, while the second extracts information regarding relaxation times, i.e. the time scales on which the polarization processes take place:

Chargeability parameters:

The total chargeability $m_{tot} = \sum_i^N m_i$ is the analogon of the DD to the chargeability as defined by Seigel, 1959: $m_{seigel} = \frac{\epsilon_{\infty} - \epsilon_0}{\epsilon_{\infty}} = \frac{\rho_0 - \rho_{\infty}}{\rho_0}$ (this is also the definition used for $m_{cc}$ ). This is, howoever, only true insofar as the majority of the polarisation response of the system must be located within the measured frequency range for the DD to pick it up, while the original definition of the chargeability extends over the whole frequency domain. Thus, not fully resolved polarization peaks indicate an underestimation of the total polarization of the system by $m_{tot}$ in the DD.
Nordsiek and Weller, 2008 computed chargeability sums for each relaxation time decades, normed by $m_{tot}$ . These so called decade loadings provide frequency (or relaxation time) dependent chargeabilities.
The total, normalized chargeability $m_{tot}^n = \frac{m_{tot}}{\rho_0}$ is obtained by normalizing the total chargeability with the DC resistivity (Scott2003phd, Weller2010g_a). It gives an indication of the total polarization of the measured system without any influence of any occuring resistivity contrasts.

Relaxation time parameters:

Various parameters to determine characteristic relaxation times from the whole RTD were proposed:

Cumulative relaxation times $\tau_x$ denote relaxation times at which a certain percentage $x$ of chargeability is reached (Norsieg and Weller, 2008; Zisser et al. 2010). For example, $\tau_{50}$ is the median relaxation time of a given RTD. (See ref:environ_vars on how to set individual percentages).
Nordsiek and Weller, 2008 introduced the non-uniformity parameter $U_\tau = \frac{\tau_{10}}{\tau_{60}}$ which characterizes the width of the RTD. However, no information regarding the number of siginificant peaks in the RTD can be derived using $U_{\tau}$ .
Tong et al, 2004 use the arithmetic and geometric means of the relaxation times for further analysis:

$\tau_g = \left(\prod_{i=1}^N \tau_i^{m_i} \right)^{\frac{1}{\sum_{i=1}^N m_i}}\\ \tau_a = \frac{\sum_{i=1}^N m_i \cdot \tau_i}{\sum_{i=1}^N m_i}$
Nordsiek et al., 2008 introduced the logarithmic average relaxation time $\tau_{mean}$

$\tau_{mean} = \frac{exp(\sum_i m_i \cdot log(\tau_i))}{\sum m_i}`$

The listed relaxation time parameters do not take into account the specific shape of the RTD, and thus it is also useful to determine local maxima of the distribution, e.g. to extract characteristic relaxation times specific to certain polarisation peaks. This approach has conceptual similarities to the use of (multi-)Cole-Cole models as the produced relaxation times can be directly related to polarization peaks. The relaxation time with the larges corresponding chargeability is called $\tau_{max}$ (Attwa2013hess), and the in the generalized form the relaxation time $\tau_{peak}^i$ , refers to the i-th local maximum of the RTD, starting with the low frequencies (i.e. high $\tau$ values). This approach can recover multiple peaks without any knowlegdge of the exact number of peaks in the data. However, this process can yield multiple small maxima if the smoothing between adjacent chargeabilitiy values is not strong enough. In these cases the corresponding smoothing parameters of the DD should be increased. Integral parameters extracted from the RTD fall into two categories: chargeability related values and relaxation time related values. The first category extracts information regarding the total or partial polarization strength of the system, while the second extracts information regarding relaxation times, i.e. the time scales on which the polarization processes take place:

Chargeability parameters:

The total chargeability $m_{tot} = \sum_i^N m_i$ is the analogon of the DD to the chargeability as defined by Seigel, 1959: $m_{seigel} = \frac{\epsilon_{\infty} - \epsilon_0}{\epsilon_{\infty}} = \frac{\rho_0 - \rho_{\infty}}{\rho_0}$ (this is also the definition used for $m_{cc}$ ). This is, howoever, only true insofar as the majority of the polarisation response of the system must be located within the measured frequency range for the DD to pick it up, while the original definition of the chargeability extends over the whole frequency domain. Thus, not fully resolved polarization peaks indicate an underestimation of the total polarization of the system by $m_{tot}$ in the DD.
Nordsiek and Weller, 2008 computed chargeability sums for each relaxation time decades, normed by $m_{tot}$ . These so called decade loadings provide frequency (or relaxation time) dependent chargeabilities.
The total, normalized chargeability $m_{tot}^n = \frac{m_{tot}}{\rho_0}$ is obtained by normalizing the total chargeability with the DC resistivity (Scott2003phd, Weller2010g_a). It gives an indication of the total polarization of the measured system without any influence of any occuring resistivity contrasts.

Relaxation time parameters:

Various parameters to determine characteristic relaxation times from the whole RTD were proposed:

Cumulative relaxation times $\tau_x$ denote relaxation times at which a certain percentage $x$ of chargeability is reached (Norsieg and Weller, 2008; Zisser et al. 2010). For example, $\tau_{50}$ is the median relaxation time of a given RTD.
Nordsiek and Weller, 2008 introduced the non-uniformity parameter $U_\tau = \frac{\tau_{10}}{\tau_{60}}$ which characterizes the width of the RTD. However, no information regarding the number of siginificant peaks in the RTD can be derived using $U_{\tau}$ .
Tong et al, 2004 use the arithmetic and geometric means of the relaxation times for further analysis:

$\tau_g = \left(\prod_{i=1}^N \tau_i^{m_i} \right)^{\frac{1}{\sum_{i=1}^N m_i}}\\ \tau_a = \frac{\sum_{i=1}^N m_i \cdot \tau_i}{\sum_{i=1}^N m_i}$
Nordsiek et al., 2008 introduced the logarithmic average relaxation time $\tau_{mean}$

$\tau_{mean} = \frac{exp(\sum_i m_i \cdot log(\tau_i))}{\sum m_i}`$

The listed relaxation time parameters do not take into account the specific shape of the RTD, and thus it is also useful to determine local maxima of the distribution, e.g. to extract characteristic relaxation times specific to certain polarisation peaks. This approach has conceptual similarities to the use of (multi-)Cole-Cole models as the produced relaxation times can be directly related to polarization peaks. The relaxation time with the larges corresponding chargeability is called $\tau_{max}$ (Attwa2013hess), and the in the generalized form the relaxation time $\tau_{peak}^i$ , refers to the i-th local maximum of the RTD, starting with the low frequencies (i.e. high $\tau$ values). This approach can recover multiple peaks without any knowlegdge of the exact number of peaks in the data. However, this process can yield multiple small maxima if the smoothing between adjacent chargeabilitiy values is not strong enough. In these cases the corresponding smoothing parameters of the DD should Integral Parameters ===================

Integral parameters extracted from the RTD fall into two categories: chargeability related values and relaxation time related values. The first category extracts information regarding the total or partial polarization strength of the system, while the second extracts information regarding relaxation times, i.e. the time scales on which the polarization processes take place:

Chargeability parameters:

The total chargeability $m_{tot} = \sum_i^N m_i$ is the analogon of the DD to the chargeability as defined by Seigel, 1959: $m_{seigel} = \frac{\epsilon_{\infty} - \epsilon_0}{\epsilon_{\infty}} = \frac{\rho_0 - \rho_{\infty}}{\rho_0}$ (this is also the definition used for $m_{cc}$ ). This is, howoever, only true insofar as the majority of the polarisation response of the system must be located within the measured frequency range for the DD to pick it up, while the original definition of the chargeability extends over the whole frequency domain. Thus, not fully resolved polarization peaks indicate an underestimation of the total polarization of the system by $m_{tot}$ in the DD.
Nordsiek and Weller, 2008 computed chargeability sums for each relaxation time decades, normed by $m_{tot}$ . These so called decade loadings provide frequency (or relaxation time) dependent chargeabilities.
The total, normalized chargeability $m_{tot}^n = \frac{m_{tot}}{\rho_0}$ is obtained by normalizing the total chargeability with the DC resistivity (Scott2003phd, Weller2010g_a). It gives an indication of the total polarization of the measured system without any influence of any occuring resistivity contrasts.

Relaxation time parameters:

Various parameters to determine characteristic relaxation times from the whole RTD were proposed:

Cumulative relaxation times $\tau_x$ denote relaxation times at which a certain percentage $x$ of chargeability is reached (Norsieg and Weller, 2008; Zisser et al. 2010). For example, $\tau_{50}$ is the median relaxation time of a given RTD. (See ref:environ_vars on how to set individual percentages).
Nordsiek and Weller, 2008 introduced the non-uniformity parameter $U_\tau = \frac{\tau_{10}}{\tau_{60}}$ which characterizes the width of the RTD. However, no information regarding the number of siginificant peaks in the RTD can be derived using $U_{\tau}$ .
Tong et al, 2004 use the arithmetic and geometric means of the relaxation times for further analysis:

$\tau_g = \left(\prod_{i=1}^N \tau_i^{m_i} \right)^{\frac{1}{\sum_{i=1}^N m_i}}\\ \tau_a = \frac{\sum_{i=1}^N m_i \cdot \tau_i}{\sum_{i=1}^N m_i}$
Nordsiek et al., 2008 introduced the logarithmic average relaxation time $\tau_{mean}$

$\tau_{mean} = \frac{exp(\sum_i m_i \cdot log(\tau_i))}{\sum m_i}`$

The listed relaxation time parameters do not take into account the specific shape of the RTD, and thus it is also useful to determine local maxima of the distribution, e.g. to extract characteristic relaxation times specific to certain polarisation peaks. This approach has conceptual similarities to the use of (multi-)Cole-Cole models as the produced relaxation times can be directly related to polarization peaks. The relaxation time with the larges corresponding chargeability is called $\tau_{max}$ (Attwa2013hess), and the in the generalized form the relaxation time $\tau_{peak}^i$ , refers to the i-th local maximum of the RTD, starting with the low frequencies (i.e. high $\tau$ values). This approach can recover multiple peaks without any knowlegdge of the exact number of peaks in the data. However, this process can yield multiple small maxima if the smoothing between adjacent chargeabilitiy values is not strong enough. In these cases the corresponding smoothing parameters of the DD should be increased. Integral parameters extracted from the RTD fall into two categories: chargeability related values and relaxation time related values. The first category extracts information regarding the total or partial polarization strength of the system, while the second extracts information regarding relaxation times, i.e. the time scales on which the polarization processes take place:

Chargeability parameters:

The total chargeability $m_{tot} = \sum_i^N m_i$ is the analogon of the DD to the chargeability as defined by Seigel, 1959: $m_{seigel} = \frac{\epsilon_{\infty} - \epsilon_0}{\epsilon_{\infty}} = \frac{\rho_0 - \rho_{\infty}}{\rho_0}$ (this is also the definition used for $m_{cc}$ ). This is, howoever, only true insofar as the majority of the polarisation response of the system must be located within the measured frequency range for the DD to pick it up, while the original definition of the chargeability extends over the whole frequency domain. Thus, not fully resolved polarization peaks indicate an underestimation of the total polarization of the system by $m_{tot}$ in the DD.
Nordsiek and Weller, 2008 computed chargeability sums for each relaxation time decades, normed by $m_{tot}$ . These so called decade loadings provide frequency (or relaxation time) dependent chargeabilities.
The total, normalized chargeability $m_{tot}^n = \frac{m_{tot}}{\rho_0}$ is obtained by normalizing the total chargeability with the DC resistivity (Scott2003phd, Weller2010g_a). It gives an indication of the total polarization of the measured system without any influence of any occuring resistivity contrasts.

Relaxation time parameters:

Various parameters to determine characteristic relaxation times from the whole RTD were proposed:

Cumulative relaxation times $\tau_x$ denote relaxation times at which a certain percentage $x$ of chargeability is reached (Norsieg and Weller, 2008; Zisser et al. 2010). For example, $\tau_{50}$ is the median relaxation time of a given RTD.
Nordsiek and Weller, 2008 introduced the non-uniformity parameter $U_\tau = \frac{\tau_{10}}{\tau_{60}}$ which characterizes the width of the RTD. However, no information regarding the number of siginificant peaks in the RTD can be derived using $U_{\tau}$ .
Tong et al, 2004 use the arithmetic and geometric means of the relaxation times for further analysis:

$\tau_g = \left(\prod_{i=1}^N \tau_i^{m_i} \right)^{\frac{1}{\sum_{i=1}^N m_i}}\\ \tau_a = \frac{\sum_{i=1}^N m_i \cdot \tau_i}{\sum_{i=1}^N m_i}$
Nordsiek et al., 2008 introduced the logarithmic average relaxation time $\tau_{mean}$

$\tau_{mean} = \frac{exp(\sum_i m_i \cdot log(\tau_i))}{\sum m_i}`$

The listed relaxation time parameters do not take into account the specific shape of the RTD, and thus it is also useful to determine local maxima of the distribution, e.g. to extract characteristic relaxation times specific to certain polarisation peaks. This approach has conceptual similarities to the use of (multi-)Cole-Cole models as the produced relaxation times can be directly related to polarization peaks. The relaxation time with the larges corresponding chargeability is called $\tau_{max}$ (Attwa2013hess), and the in the generalized form the relaxation time $\tau_{peak}^i$ , refers to the i-th local maximum of the RTD, starting with the low frequencies (i.e. high $\tau$ values). This approach can recover multiple peaks without any knowlegdge of the exact number of peaks in the data. However, this process can yield multiple small maxima if the smoothing between adjacent chargeabilitiy values is not strong enough. In these cases the corresponding smoothing parameters of the DD should Integral Parameters ===================

Integral parameters extracted from the RTD fall into two categories: chargeability related values and relaxation time related values. The first category extracts information regarding the total or partial polarization strength of the system, while the second extracts information regarding relaxation times, i.e. the time scales on which the polarization processes take place:

Chargeability parameters:

The total chargeability $m_{tot} = \sum_i^N m_i$ is the analogon of the DD to the chargeability as defined by Seigel, 1959: $m_{seigel} = \frac{\epsilon_{\infty} - \epsilon_0}{\epsilon_{\infty}} = \frac{\rho_0 - \rho_{\infty}}{\rho_0}$ (this is also the definition used for $m_{cc}$ ). This is, howoever, only true insofar as the majority of the polarisation response of the system must be located within the measured frequency range for the DD to pick it up, while the original definition of the chargeability extends over the whole frequency domain. Thus, not fully resolved polarization peaks indicate an underestimation of the total polarization of the system by $m_{tot}$ in the DD.
Nordsiek and Weller, 2008 computed chargeability sums for each relaxation time decades, normed by $m_{tot}$ . These so called decade loadings provide frequency (or relaxation time) dependent chargeabilities.
The total, normalized chargeability $m_{tot}^n = \frac{m_{tot}}{\rho_0}$ is obtained by normalizing the total chargeability with the DC resistivity (Scott2003phd, Weller2010g_a). It gives an indication of the total polarization of the measured system without any influence of any occuring resistivity contrasts.

Relaxation time parameters:

Various parameters to determine characteristic relaxation times from the whole RTD were proposed:

Cumulative relaxation times $\tau_x$ denote relaxation times at which a certain percentage $x$ of chargeability is reached (Norsieg and Weller, 2008; Zisser et al. 2010). For example, $\tau_{50}$ is the median relaxation time of a given RTD. (See ref:environ_vars on how to set individual percentages).
Nordsiek and Weller, 2008 introduced the non-uniformity parameter $U_\tau = \frac{\tau_{10}}{\tau_{60}}$ which characterizes the width of the RTD. However, no information regarding the number of siginificant peaks in the RTD can be derived using $U_{\tau}$ .
Tong et al, 2004 use the arithmetic and geometric means of the relaxation times for further analysis:

$\tau_g = \left(\prod_{i=1}^N \tau_i^{m_i} \right)^{\frac{1}{\sum_{i=1}^N m_i}}\\ \tau_a = \frac{\sum_{i=1}^N m_i \cdot \tau_i}{\sum_{i=1}^N m_i}$
Nordsiek et al., 2008 introduced the logarithmic average relaxation time $\tau_{mean}$

$\tau_{mean} = \frac{exp(\sum_i m_i \cdot log(\tau_i))}{\sum m_i}`$

The listed relaxation time parameters do not take into account the specific shape of the RTD, and thus it is also useful to determine local maxima of the distribution, e.g. to extract characteristic relaxation times specific to certain polarisation peaks. This approach has conceptual similarities to the use of (multi-)Cole-Cole models as the produced relaxation times can be directly related to polarization peaks. The relaxation time with the larges corresponding chargeability is called $\tau_{max}$ (Attwa2013hess), and the in the generalized form the relaxation time $\tau_{peak}^i$ , refers to the i-th local maximum of the RTD, starting with the low frequencies (i.e. high $\tau$ values). This approach can recover multiple peaks without any knowlegdge of the exact number of peaks in the data. However, this process can yield multiple small maxima if the smoothing between adjacent chargeabilitiy values is not strong enough. In these cases the corresponding smoothing parameters of the DD should be increased. Integral parameters extracted from the RTD fall into two categories: chargeability related values and relaxation time related values. The first category extracts information regarding the total or partial polarization strength of the system, while the second extracts information regarding relaxation times, i.e. the time scales on which the polarization processes take place:

Chargeability parameters:

The total chargeability $m_{tot} = \sum_i^N m_i$ is the analogon of the DD to the chargeability as defined by Seigel, 1959: $m_{seigel} = \frac{\epsilon_{\infty} - \epsilon_0}{\epsilon_{\infty}} = \frac{\rho_0 - \rho_{\infty}}{\rho_0}$ (this is also the definition used for $m_{cc}$ ). This is, howoever, only true insofar as the majority of the polarisation response of the system must be located within the measured frequency range for the DD to pick it up, while the original definition of the chargeability extends over the whole frequency domain. Thus, not fully resolved polarization peaks indicate an underestimation of the total polarization of the system by $m_{tot}$ in the DD.
Nordsiek and Weller, 2008 computed chargeability sums for each relaxation time decades, normed by $m_{tot}$ . These so called decade loadings provide frequency (or relaxation time) dependent chargeabilities.
The total, normalized chargeability $m_{tot}^n = \frac{m_{tot}}{\rho_0}$ is obtained by normalizing the total chargeability with the DC resistivity (Scott2003phd, Weller2010g_a). It gives an indication of the total polarization of the measured system without any influence of any occuring resistivity contrasts.

Relaxation time parameters:

Various parameters to determine characteristic relaxation times from the whole RTD were proposed:

Cumulative relaxation times $\tau_x$ denote relaxation times at which a certain percentage $x$ of chargeability is reached (Norsieg and Weller, 2008; Zisser et al. 2010). For example, $\tau_{50}$ is the median relaxation time of a given RTD.
Nordsiek and Weller, 2008 introduced the non-uniformity parameter $U_\tau = \frac{\tau_{10}}{\tau_{60}}$ which characterizes the width of the RTD. However, no information regarding the number of siginificant peaks in the RTD can be derived using $U_{\tau}$ .
Tong et al, 2004 use the arithmetic and geometric means of the relaxation times for further analysis:

$\tau_g = \left(\prod_{i=1}^N \tau_i^{m_i} \right)^{\frac{1}{\sum_{i=1}^N m_i}}\\ \tau_a = \frac{\sum_{i=1}^N m_i \cdot \tau_i}{\sum_{i=1}^N m_i}$
Nordsiek et al., 2008 introduced the logarithmic average relaxation time $\tau_{mean}$

$\tau_{mean} = \frac{exp(\sum_i m_i \cdot log(\tau_i))}{\sum m_i}`$

The listed relaxation time parameters do not take into account the specific shape of the RTD, and thus it is also useful to determine local maxima of the distribution, e.g. to extract characteristic relaxation times specific to certain polarisation peaks. This approach has conceptual similarities to the use of (multi-)Cole-Cole models as the produced relaxation times can be directly related to polarization peaks. The relaxation time with the larges corresponding chargeability is called $\tau_{max}$ (Attwa2013hess), and the in the generalized form the relaxation time $\tau_{peak}^i$ , refers to the i-th local maximum of the RTD, starting with the low frequencies (i.e. high $\tau$ values). This approach can recover multiple peaks without any knowlegdge of the exact number of peaks in the data. However, this process can yield multiple small maxima if the smoothing between adjacent chargeabilitiy values is not strong enough. In these cases the corresponding smoothing parameters of the DD should be increased.

Integral Parameters¶

Time-lapse Cole-Cole decomposition routines

Navigation

Related Topics