pygimli.physics.SIP

Spectral induced polarization (SIP) measurements and fittings.

Overview

Functions

ColeCole(f, R, m, tau, c[, a])

ColeColeEpsilon(f, e0, eInf, tau, alpha)

ColeColeRho(f, rho, m, tau, c[, a])

ColeColeRhoDouble(f, rho, m1, t1, c1, m2, t2, c2)

ColeColeSigma(f, sigma, m, tau, c[, a])

ColeDavidson(f, R, m, tau[, a])

drawAmplitudeSpectrum(ax, freq, amp[, …])

Show amplitude spectrum (resistivity as a function of f).

drawPhaseSpectrum(ax, freq, phi[, ylabel, …])

Show phase spectrum (-phi as a function of f).

load(fileName[, verbose])

Shortcut to load SIP spectral data.

modelColeColeEpsilon(f, e0, eInf, tau, alpha)

Original complex-valued permittivity formulation (Cole&Cole, 1941).

modelColeColeRho(f, rho, m, tau, c[, a])

Frequency-domain Cole-Cole impedance model after Pelton et al.

modelColeColeRhoDouble(f, rho, m1, t1, c1, …)

Frequency-domain Double Cole-Cole impedance model

modelColeColeSigma(f, sigma, m, tau, c[, a])

Complex-valued conductivity Cole-Cole model

modelColeDavidson(f, R, m, tau[, a])

For backward compatibility.

showSpectrum(freq, amp, phi[, nrows, ylog, axs])

Show amplitude and phase spectra in two subplots.

tauRhoToTauSigma(tRho, m, c)

Convert \(\tau_{\rho}\) to \(\tau_{\sigma}\) for Cole-Cole-Model.

Classes

ColeColeAbs(f[, verbose])

Cole-Cole model with EM term after Pelton et al.

ColeColeComplex(f[, verbose])

Cole-Cole model with EM term after Pelton et al.

ColeColeComplexSigma(f[, verbose])

Cole-Cole model with EM term after Pelton et al.

ColeColePhi(f[, verbose])

Cole-Cole model with EM term after Pelton et al.

DebyeComplex(fvec, tvec[, verbose])

Debye decomposition (smooth Debye relaxations) of complex data

DebyePhi(fvec, tvec[, verbose])

Debye decomposition (smooth Debye relaxations) phase only

DoubleColeColePhi(f[, verbose])

Double Cole-Cole model with EM term after Pelton et al.

PeltonPhiEM(f[, verbose])

Cole-Cole model with EM term after Pelton et al.

SIPSpectrum([filename, unify, onlydown, f, …])

SIP spectrum data analysis.

SpectrumManager([fop])

Manager to work with spectra data.

SpectrumModelling([funct])

Modelling framework with an array of freqencies as data space.

Functions

ColeCole

pygimli.physics.SIP.ColeCole(f, R, m, tau, c, a=1)[source]

ColeColeEpsilon

pygimli.physics.SIP.ColeColeEpsilon(f, e0, eInf, tau, alpha)[source]

ColeColeRho

pygimli.physics.SIP.ColeColeRho(f, rho, m, tau, c, a=1)[source]

ColeColeRhoDouble

pygimli.physics.SIP.ColeColeRhoDouble(f, rho, m1, t1, c1, m2, t2, c2)[source]

ColeColeSigma

pygimli.physics.SIP.ColeColeSigma(f, sigma, m, tau, c, a=1)[source]

ColeDavidson

pygimli.physics.SIP.ColeDavidson(f, R, m, tau, a=1)[source]

drawAmplitudeSpectrum

pygimli.physics.SIP.drawAmplitudeSpectrum(ax, freq, amp, ylabel='$\\rho$ ($\\Omega$m)', grid=True, marker='+', ylog=True, **kwargs)[source]

Show amplitude spectrum (resistivity as a function of f).

drawPhaseSpectrum

pygimli.physics.SIP.drawPhaseSpectrum(ax, freq, phi, ylabel='$-\\phi$ (mrad)', grid=True, marker='+', ylog=False, **kwargs)[source]

Show phase spectrum (-phi as a function of f).

load

pygimli.physics.SIP.load(fileName, verbose=False, **kwargs)[source]

Shortcut to load SIP spectral data.

Import Data and try to assume the file format.

Parameters

fileName (str) –

Returns

freqs, amp, phi – Frequencies, amplitudes and phases phi in neg. radiant

Return type

np.array

modelColeColeEpsilon

pygimli.physics.SIP.modelColeColeEpsilon(f, e0, eInf, tau, alpha)[source]

Original complex-valued permittivity formulation (Cole&Cole, 1941).

modelColeColeRho

pygimli.physics.SIP.modelColeColeRho(f, rho, m, tau, c, a=1)[source]

Frequency-domain Cole-Cole impedance model after Pelton et al. (1978)

Frequency-domain Cole-Cole impedance model after Pelton et al. (1978) [PWH+78]

\[\begin{split}Z(\omega) & = \rho_0\left[1 - m \left(1 - \frac{1}{1+(\text{i}\omega\tau)^c}\right)\right] \\ \quad \text{with}\quad m & = \frac{1}{1+\frac{\rho_0}{\rho_1}} \quad \text{and}\quad \omega =2\pi f\end{split}\]
  • \(Z(\omega)\) - Complex impedance per 1A current injection

  • \(f\) - Frequency

  • \(\rho_0\) – Background resistivity states the unblocked pore path

  • \(\rho_1\) – Resistance of the solution in the blocked pore passages

  • \(m\) – Chargeability after Seigel (1959) [Sei59] as being the ratio of voltage immediately after, to the voltage immediately before cessation of an infinitely long charging current.

  • \(\tau\) – ‘Time constant’ relaxation time [s] for 1/e decay

  • \(c\) - Rate of charge accumulation. Cole-Cole exponent typically [0.1 .. 0.6]

>>> import numpy as np
>>> import pygimli as pg
>>> from pygimli.physics.SIP import modelColeColeRho
>>> f = np.logspace(-2, 5, 100)
>>> m = np.linspace(0.1, 0.9, 5)
>>> tau = 0.01
>>> fImMin = 1/(tau*2*np.pi)
>>> fig, axs = pg.plt.subplots(1, 2)
>>> ax1 = axs[0]
>>> ax2 = axs[0].twinx()
>>> ax3 = axs[1]
>>> ax4 = axs[1].twinx()
>>> for i in range(len(m)):
...     Z = ColeColeRho(f, rho=1, m=m[i], tau=tau, c=0.5)
...     _= ax1.loglog(f, np.abs(Z), color='black')
...     _= ax2.loglog(f, -np.angle(Z)*1000, color='b')
...     _= ax3.loglog(f, Z.real, color='g')
...     _= ax4.semilogx(f, Z.imag, color='r')
...     _= ax4.plot([fImMin, fImMin], [-0.2, 0.], color='r')
>>> _= ax4.text(fImMin, -0.1, r"$f($min($Z''$))=$\frac{1}{2*\pi\tau}$", color='r')
>>> _= ax4.text(0.1, -0.17, r"$f($min[$Z''$])=$\frac{1}{2\pi\tau}$", color='r')
>>> _= ax1.set_ylabel('Amplitude $|Z(f)|$', color='black')
>>> _= ax1.set_xlabel('Frequency $f$ [Hz]')
>>> _= ax1.set_ylim(1e-2, 1)
>>> _= ax2.set_ylabel(r'- Phase $\varphi$ [mrad]', color='b')
>>> _= ax2.set_ylim(1, 1e3)
>>> _= ax3.set_ylabel('re $Z(f)$', color='g')
>>> _= ax4.set_ylabel('im $Z(f)$', color='r')
>>> _= ax3.set_xlabel('Frequency $f$ [Hz]')
>>> _= ax3.set_ylim(1e-2, 1)
>>> _= ax4.set_ylim(-0.2, 0)
>>> pg.plt.show()

Examples using pygimli.physics.SIP.modelColeColeRho

modelColeColeRhoDouble

pygimli.physics.SIP.modelColeColeRhoDouble(f, rho, m1, t1, c1, m2, t2, c2, a=1)[source]

Frequency-domain Double Cole-Cole impedance model

Frequency-domain Double Cole-Cole impedance model returns the sum of two Cole-Cole Models with a common amplitude. Z = rho * (Z1(Cole-Cole) + Z2(Cole-Cole))

modelColeColeSigma

pygimli.physics.SIP.modelColeColeSigma(f, sigma, m, tau, c, a=1)[source]

Complex-valued conductivity Cole-Cole model

modelColeDavidson

pygimli.physics.SIP.modelColeDavidson(f, R, m, tau, a=1)[source]

For backward compatibility.

showSpectrum

pygimli.physics.SIP.showSpectrum(freq, amp, phi, nrows=2, ylog=None, axs=None, **kwargs)[source]

Show amplitude and phase spectra in two subplots.

tauRhoToTauSigma

pygimli.physics.SIP.tauRhoToTauSigma(tRho, m, c)[source]

Convert \(\tau_{\rho}\) to \(\tau_{\sigma}\) for Cole-Cole-Model.

\[\tau_{\sigma} = \tau_{\rho}/(1-m)^{\frac{1}{c}}\]

Examples

>>> import numpy as np
>>> import pygimli as pg
>>> from pygimli.physics.SIP import modelColeColeRho, modelColeColeSigma, tauRhoToTauSigma
>>> tr = 1.
>>> Z = modelColeColeRho(1e5, rho=10.0, m=0.5, tau=tr, c=0.5)
>>> ts = tauRhoToTauSigma(tr, m=0.5, c=0.5)
>>> S = modelColeColeSigma(1e5, sigma=0.1, m=0.5, tau=ts, c=0.5)
>>> abs(1.0/S - Z) < 1e-12
True
>>> np.angle(1.0/S / Z) < 1e-12
True

Classes

ColeColeAbs

class pygimli.physics.SIP.ColeColeAbs(f, verbose=False)[source]

Bases: pygimli.core.ModellingBaseMT__

Cole-Cole model with EM term after Pelton et al. (1978)

__init__((object)arg1[, (object)verbose=False]) → object :[source]
C++ signature :

void* __init__(_object* [,bool=False])

__init__( (object)arg1, (object)dataContainer [, (object)verbose=False]) -> object :

C++ signature :

void* __init__(_object*,GIMLI::DataContainer {lvalue} [,bool=False])

__init__( (object)arg1, (object)mesh [, (object)verbose=False]) -> object :

C++ signature :

void* __init__(_object*,GIMLI::Mesh [,bool=False])

__init__( (object)arg1, (object)mesh, (object)dataContainer [, (object)verbose=False]) -> object :

C++ signature :

void* __init__(_object*,GIMLI::Mesh,GIMLI::DataContainer {lvalue} [,bool=False])

response(par)[source]

phase angle of the model

ColeColeComplex

class pygimli.physics.SIP.ColeColeComplex(f, verbose=False)[source]

Bases: pygimli.core.ModellingBaseMT__

Cole-Cole model with EM term after Pelton et al. (1978)

__init__((object)arg1[, (object)verbose=False]) → object :[source]
C++ signature :

void* __init__(_object* [,bool=False])

__init__( (object)arg1, (object)dataContainer [, (object)verbose=False]) -> object :

C++ signature :

void* __init__(_object*,GIMLI::DataContainer {lvalue} [,bool=False])

__init__( (object)arg1, (object)mesh [, (object)verbose=False]) -> object :

C++ signature :

void* __init__(_object*,GIMLI::Mesh [,bool=False])

__init__( (object)arg1, (object)mesh, (object)dataContainer [, (object)verbose=False]) -> object :

C++ signature :

void* __init__(_object*,GIMLI::Mesh,GIMLI::DataContainer {lvalue} [,bool=False])

response(par)[source]

phase angle of the model

ColeColeComplexSigma

class pygimli.physics.SIP.ColeColeComplexSigma(f, verbose=False)[source]

Bases: pygimli.core.ModellingBaseMT__

Cole-Cole model with EM term after Pelton et al. (1978)

__init__((object)arg1[, (object)verbose=False]) → object :[source]
C++ signature :

void* __init__(_object* [,bool=False])

__init__( (object)arg1, (object)dataContainer [, (object)verbose=False]) -> object :

C++ signature :

void* __init__(_object*,GIMLI::DataContainer {lvalue} [,bool=False])

__init__( (object)arg1, (object)mesh [, (object)verbose=False]) -> object :

C++ signature :

void* __init__(_object*,GIMLI::Mesh [,bool=False])

__init__( (object)arg1, (object)mesh, (object)dataContainer [, (object)verbose=False]) -> object :

C++ signature :

void* __init__(_object*,GIMLI::Mesh,GIMLI::DataContainer {lvalue} [,bool=False])

response(par)[source]

phase angle of the model

ColeColePhi

class pygimli.physics.SIP.ColeColePhi(f, verbose=False)[source]

Bases: pygimli.core.ModellingBaseMT__

Cole-Cole model with EM term after Pelton et al. (1978)

Modelling operator for the Frequency Domain Cole-Cole impedance model using pygimli.physics.SIP.ColeColeRho after Pelton et al. (1978) [PWH+78]

  • \(\textbf{m} =\{ m, \tau, c\}\)

    Modelling parameter for the Cole-Cole model with \(\rho_0 = 1\)

  • \(\textbf{d} =\{\varphi_i(f_i)\}\)

    Modelling eesponse for all given frequencies as negative phase angles \(\varphi(f) = -tan^{-1}\frac{\text{Im}\,Z(f)}{\text{Re}\,Z(f)}\) and \(Z(f, \rho_0=1, m, \tau, c) =\) Cole-Cole impedance.

__init__(f, verbose=False)[source]

Setup class by specifying the frequency.

response(par)[source]

Phase angle of the model.

DebyeComplex

class pygimli.physics.SIP.DebyeComplex(fvec, tvec, verbose=False)[source]

Bases: pygimli.core.ModellingBaseMT__

Debye decomposition (smooth Debye relaxations) of complex data

__init__(fvec, tvec, verbose=False)[source]

constructor with frequecy and tau vector

createJacobian(par)[source]

linear jacobian after Nordsiek&Weller (2008)

response(par)[source]

amplitude/phase spectra as function of spectral chargeabilities

DebyePhi

class pygimli.physics.SIP.DebyePhi(fvec, tvec, verbose=False)[source]

Bases: pygimli.core.ModellingBaseMT__

Debye decomposition (smooth Debye relaxations) phase only

__init__(fvec, tvec, verbose=False)[source]

constructor with frequecy and tau vector

response(par)[source]

amplitude/phase spectra as function of spectral chargeabilities

DoubleColeColePhi

class pygimli.physics.SIP.DoubleColeColePhi(f, verbose=False)[source]

Bases: pygimli.core.ModellingBaseMT__

Double Cole-Cole model with EM term after Pelton et al. (1978)

Modelling operator for the Frequency Domain Cole-Cole impedance model using pygimli.physics.SIP.ColeColeRho after Pelton et al. (1978) [PWH+78]

  • \(\textbf{m} =\{ m_1, \tau_1, c_1, m_2, \tau_2, c_2\}\)

    Modelling parameter for the Cole-Cole model with \(\rho_0 = 1\)

  • \(\textbf{d} =\{\varphi_i(f_i)\}\)

    Modelling Response for all given frequencies as negative phase angles \(\varphi(f) = \varphi_1(Z_1(f))+\varphi_2(Z_2(f)) = -tan^{-1}\frac{\text{Im}\,(Z_1Z_2)}{\text{Re}\,(Z_1Z_2)}\) and \(Z_1(f, \rho_0=1, m_1, \tau_1, c_1)\) and \(Z_2(f, \rho_0=1, m_2, \tau_2, c_2)\) ColeCole impedances.

__init__(f, verbose=False)[source]

Setup class by specifying the frequency.

response(par)[source]

phase angle of the model

PeltonPhiEM

class pygimli.physics.SIP.PeltonPhiEM(f, verbose=False)[source]

Bases: pygimli.core.ModellingBaseMT__

Cole-Cole model with EM term after Pelton et al. (1978)

__init__((object)arg1[, (object)verbose=False]) → object :[source]
C++ signature :

void* __init__(_object* [,bool=False])

__init__( (object)arg1, (object)dataContainer [, (object)verbose=False]) -> object :

C++ signature :

void* __init__(_object*,GIMLI::DataContainer {lvalue} [,bool=False])

__init__( (object)arg1, (object)mesh [, (object)verbose=False]) -> object :

C++ signature :

void* __init__(_object*,GIMLI::Mesh [,bool=False])

__init__( (object)arg1, (object)mesh, (object)dataContainer [, (object)verbose=False]) -> object :

C++ signature :

void* __init__(_object*,GIMLI::Mesh,GIMLI::DataContainer {lvalue} [,bool=False])

response(par)[source]

phase angle of the model

SIPSpectrum

class pygimli.physics.SIP.SIPSpectrum(filename=None, unify=False, onlydown=True, f=None, amp=None, phi=None, k=1, sort=True, basename='new')[source]

Bases: object

SIP spectrum data analysis.

__init__(filename=None, unify=False, onlydown=True, f=None, amp=None, phi=None, k=1, sort=True, basename='new')[source]

Init SIP class with either filename to read or data vectors.

Examples

>>> #sip = SIPSpectrum('sipexample.txt', unify=True) # unique f values
>>> #sip = SIPSpectrum(f=f, amp=R, phi=phase, basename='new')
checkCRKK(useEps=False, use0=False, ax=None)[source]

Check coupling removal (CR) by Kramers-Kronig (KK) relation

cutF(fcut=1e+99, down=False)[source]

Cut (delete) frequencies above a certain value fcut.

determineEpsilon(mode=0, sigmaR=None, sigmaI=None)[source]

Retrieve frequency-independent epsilon for f->Inf.

Parameters
  • mode (int) –

    Operation mode:

    =0 - extrapolate using two highest frequencies (default) <0 - take last -n frequencies >0 - take n-th frequency

  • sigmaR/sigmaI (float) – real and imaginary conductivity (if not given take data)

Returns

er – relative permittivity (epsilon) value (dimensionless)

Return type

float

epsilonR()[source]

Calculate relative permittivity from imaginary conductivity

fit2CCPhi(ePhi=0.001, lam=1000.0, mpar=0, 0, 1, taupar1=0, 1e-05, 1, taupar2=0, 0.1, 1000, cpar=0.5, 0, 1, verbose=False)[source]

fit a Cole-Cole term to phase only

Parameters
  • ePhi (float) – absolute error of phase angle

  • lam (float) – regularization parameter

  • taupar, cpar (mpar,) – inversion parameters (starting value, lower bound, upper bound) for Cole-Cole parameters (m, tau, c) and EM relaxation time (em)

fitCCEM(ePhi=0.001, lam=1000.0, remove=True, mpar=0.2, 0, 1, taupar=0.01, 1e-05, 100, cpar=0.25, 0, 1, empar=1e-07, 1e-09, 1e-05, verbose=False)[source]

Fit a Cole-Cole term with additional EM term to phase

Parameters
  • ePhi (float) – absolute error of phase angle

  • lam (float) – regularization parameter

  • remove (bool) – remove EM term from data

  • taupar, cpar, empar (mpar,) – inversion parameters (starting value, lower bound, upper bound) for Cole-Cole parameters (m, tau, c) and EM relaxation time (em)

fitCCPhi(ePhi=0.001, lam=1000.0, mpar=0, 0, 1, taupar=0, 1e-05, 100, cpar=0.3, 0, 1, verbose=False)[source]

fit a Cole-Cole term to phase only

Parameters
  • ePhi (float) – absolute error of phase angle

  • lam (float) – regularization parameter

  • taupar, cpar (mpar,) – inversion parameters (starting value, lower bound, upper bound) for Cole-Cole parameters (m, tau, c) and EM relaxation time (em)

fitColeCole(useCond=False, **kwargs)[source]

Fit a Cole-Cole model to the phase data

Parameters
  • useCond (bool) – use conductivity form of Cole-Cole model instead of resistivity

  • error (float [0.01]) – absolute phase error

  • lam (float [1000]) – initial regularization parameter

  • mpar (tuple/list (0, 0, 1)) – inversion parameters for chargeability: start, lower, upper bound

  • taupar (tuple/list (1e-2, 1e-5, 100)) – inversion parameters for time constant: start, lower, upper bound

  • cpar (tuple/list (0.25, 0, 1)) – inversion parameters for Cole exponent: start, lower, upper bound

fitDebyeModel(ePhi=0.001, lam=1000.0, lamFactor=0.8, mint=None, maxt=None, nt=None, new=True, showFit=False, cType=1, verbose=False)[source]

Fit a (smooth) continuous Debye model (Debye decomposition).

Parameters
  • ePhi (float) – absolute error of phase angle

  • lam (float) – regularization parameter

  • lamFactor (float) – regularization factor for subsequent iterations

  • mint/maxt (float) – minimum/maximum tau values to use (else automatically from f)

  • nt (int) – number of tau values (default number of frequencies * 2)

  • new (bool) – new implementation (experimental)

  • showFit (bool) – show fit

  • cType (int) – constraint type (1/2=smoothness 1st/2nd order, 0=minimum norm)

getKK(use0=False)[source]

Compute Kramers-Kronig impedance values (re->im and im->re).

getPhiKK(use0=False)[source]

Compute phase from Kramers-Kronig quantities.

loadData(filename, **kwargs)[source]

Import spectral data.

Import Data and try to assume the file format.

logMeanTau()[source]

Mean logarithmic relaxation time (50% cumulative log curve).

omega()[source]

Angular frequency.

realimag(cond=False)[source]

Real and imaginary part of resistivity/conductivity (cond=True).

removeEpsilonEffect(er=None, mode=0)[source]

remove effect of (constant high-frequency) epsilon from sigma

Parameters
  • er (float) – relative epsilon to correct for (else automatically determined)

  • mode (int) – automatic epsilon determination mode (see determineEpsilon)

Returns

er – determined permittivity (see determineEpsilon)

Return type

float

saveFigures(name=None, ext='pdf')[source]

Save all existing figures to files using file basename.

showAll(save=False)[source]

Plot spectrum, Cole-Cole fit and Debye distribution

showData(reim=False, znorm=False, cond=False, nrows=2, ax=None, **kwargs)[source]

Show amplitude and phase spectrum in two subplots

Parameters
  • reim (bool) – show real/imaginary part instead of amplitude/phase

  • znorm (bool (true forces reim)) – use normalized real/imag parts after Nordsiek&Weller (2008)

  • - use nrows subplots (default=2) (nrows) –

Returns

fig, ax

Return type

mpl.figure, mpl.axes array

showDataKK(use0=False)[source]

Show data as real/imag subplots along with Kramers-Kronig curves

showPhase(ax=None, **kwargs)[source]

Plot phase spectrum (-phi over frequency).

showPolarPlot(cond=False)[source]

Show data in a polar plot (imaginary vs. real parts).

sortData()[source]

Sort data along increasing frequency (e.g. useful for KK).

totalChargeability()[source]

Total chargeability (sum) from Debye curve.

unifyData(onlydown=False)[source]

Unify data (only one value per frequency) by mean or selection.

zNorm()[source]

Normalized real (difference) and imag. z [NW08]

SpectrumManager

class pygimli.physics.SIP.SpectrumManager(fop=None, **kwargs)[source]

Bases: pygimli.frameworks.methodManager.MethodManager

Manager to work with spectra data.

__init__(fop=None, **kwargs)[source]

Constructor.

createForwardOperator(**kwargs)[source]
createInversionFramework(**kwargs)[source]
invert(data=None, f=None, **kwargs)[source]
setData(freqs=None, amp=None, phi=None, eAmp=0.03, ePhi=0.001)[source]

Set data for chosen sip model.

Parameters
  • freqs (iterable) – Array-like frequencies.

  • amp (iterable) – Array-like amplitudes to work with.

  • phi (iterable) – Array-like phase angles to work with.

  • eAmp (float|iterable) – Relative error for amplitudes.

  • ePhi (float|iterable) – Absolute error for phase angles.

setFunct(fop, **kwargs)[source]
showResult()[source]
simulate()[source]

SpectrumModelling

class pygimli.physics.SIP.SpectrumModelling(funct=None, **kwargs)[source]

Bases: pygimli.frameworks.modelling.ParameterModelling

Modelling framework with an array of freqencies as data space.

Variables
__init__(funct=None, **kwargs)[source]
Variables
  • fop (pg.frameworks.Modelling) –

  • data (pg.DataContainer) –

  • modelTrans ([pg.trans.TransLog()]) –

Parameters

**kwargs – fop : Modelling

property complex
drawData(ax, data, err=None, **kwargs)[source]
property freqs
response((object)arg1, (object)model) → object :[source]
C++ signature :

GIMLI::Vector<double> response(GIMLI::ModellingBase {lvalue},GIMLI::Vector<double>)

response( (object)arg1, (object)model) -> object :

C++ signature :

GIMLI::Vector<double> response(ModellingBase_wrapper {lvalue},GIMLI::Vector<double>)