inversions of time-domain spectral induced polarization ...hence the inversion is much more...

Geophys. J. Int. (2019) 219, 1851–1865 doi: 10.1093/gji/ggz401Advance Access publication 2019 September 11GJI Geomagnetism, Rock Magnetism and Palaeomagnetism

Inversions of time-domain spectral induced polarization data usingstretched exponential

Seogi Kang and Douglas W. OldenburgDepartment of Earth, Ocean and Atmospheric Sciences, University of British Columbia, B.C. V6T 1Z4, Canada. E-mail: [email protected]

Accepted 2019 September 4. Received 2019 August 18; in original form 2019 February 4

S U M M A R YWe provide a two-stage approach to extract spectral induced polarization (SIP) informationfrom time-domain IP data. In the first stage we invert dc data to recover the backgroundconductivity. In the second, we solve a linear inverse problem and invert all time channelssimultaneously to recover the IP parameters. The IP decay curves are represented by a stretchedexponential (SE) rather than the traditional Cole–Cole model, and we find that defining theparameters in terms of their logarithmic values is advantageous. To demonstrate the capabilityof our simultaneous SIP inversion we use synthetic data simulating a porphyry mineral deposit.The challenge is to image a mineral body that is hosted within an alteration halo having thesame chargeability but a different time constant. For a 2-D problem, we were able to distinguishthe body using our simultaneous inversion but we were not successful in using a sequential (orconventional) SIP inversion approach. For the 3-D problem we recovered 3-D distributions ofthe SIP parameters and used those to construct a 3-D rock model having four rock units. Threechargeable units were distinguished. The compact mineralization zone, having a large timeconstant, was distinguished from the circular alteration halo that had a small time constant.Finally, to promote the use of the SIP technique, and to have further development of SIPinversion, all examples presented in this paper are available in our open source resources(https://github.com/simpeg-research/kang-2018-spectral-inducedpolarization).

Key words: Electrical properties; Electromagnetic theory; Inverse theory.

1 I N T RO D U C T I O N

Earth materials can accumulate electrical charges when an electricfield is applied and, when this occurs, the material is said to be‘chargeable’. There are different physical models associated withthis effect: electrode polarization, membrane polarization, electricaldouble layer and Maxwell–Wagner polarization (Marshall & Mad-den 1959; Wong 1979; Revil et al. 2015), but they all produce apolarization of electrical charges. Macroscopically, we see this ef-fect as a frequency dependence of the electrical conductivity suchas shown in Fig. 1(a). The shape of the conductivity spectrum, dif-ferences between high and low limits of the real part, the locationof the peak in the imaginary part and width of the transition willdiffer between rocks. Thus, if we are able to evaluate the complexconductivity, we can discriminate between different materials suchas clays, sulfides and graphites. (Pelton et al. 1978).

The potential for achieving this has long been realized and ithas spawned spectral induced polarization (SIP) surveys, whichrecord earth responses over a broad frequency range (e.g. 0.01–1000 Hz). Surveys can also be carried out in the time-domain, andthe response function, obtained by taking the Fourier transform ofthe frequency-domain response, yields time-domain decays like thatshown in Fig. 1(b).

In practice, the challenge is to carry out electromagnetic surveys,and invert the acquired data, to find σ (x, y, z; ω), or its time-domain counterpart, within the earth. These are 4-D functions andhence the inversion is much more challenging than just trying tofind a 3-D frequency-independent conductivity. Usually, the numberof unknowns is reduced by introducing a parametrization such asCole–Cole (Cole & Cole 1941; Tarasov & Titov 2013):

σcc(ω) = σ∞ − ηccσ∞1 + (ıωτcc)ccc

, (1)

where σ∞ is the conductivity at infinite frequency, ηcc = σ∞−σ0σ∞ is

the chargeability, τ cc is the time constant (s), and ccc is the fre-quency exponent; the subscript cc indicates Cole–Cole and σ 0 isconductivity at zero frequency. Depending upon the Cole–Cole pa-rameters, complex conductivity spectra will vary. For instance, inthe frequency-domain, as τ cc decreases, the peak of the imaginarypart moves to higher frequency (Fig. 1a). Similarly, in the time-domain, time conductivity curve decays slowly with increasing τ cc

as shown in Fig. 1(b).With the aid of parametrization, SIP inversion approaches in both

frequency-domain (Kemna et al. 2004) and time-domain (Yuval &Oldenburg 1997; Honig & Tezkan 2007; Fiandaca et al. 2013) havebeen developed. These have ignored EM induction effects but some

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1852 S. Kang & D.W. Oldenburg

Figure 1. Complex conductivity curves: (a) frequency-domain: σ (ω) and (b) time-domain: σ (t). Black and red lines correspondingly indicate real and imaginarypart of the complex conductivity. The Cole–Cole model is used to calculate the complex conductivity. Solid and dashed lines indicate variable time constants:1 and 10 s, respectively.

more recent studies have attempted to deal with the complicationsintroduced by EM coupling. (Commer et al. 2011; Xu & Zhdanov2015; Kang & Oldenburg 2017; Belliveau & Haber 2018).

In this paper, we focus on the time-domain surveys and assumethat there are no EM induction effects, or that they have been re-moved in a pre-processing step (Kang & Oldenburg 2016). Twomain approaches are then commonly used to extract SIP informa-tion. In the first, all four Cole–Cole parameters (Fiandaca et al.2013) are sought at once. In principle, this is appealing but solvingthe large multiparameter system can be challenging because of thenon-uniqueness of the problem and getting trapped in various localminima. A second approach implements a two-stage strategy wherethe dc data are first inverted to provide the conductivity. The sensi-tivities from the final stage of the dc inversion provide a mappingthat linearly relates IP data to a pseudo-chargeability (Oldenburg& Li 1994). This approach is reasonably valid so long as the in-trinsic chargeability is small (say ηcc < 0.2). Pseudo-chargeabilityis often interpreted directly in field applications but, because it isa nonlinear function of the Cole–Cole parameters, it may be fur-ther analysed to reveal the SIP parameters (Yuval & Oldenburg1997; Honig & Tezkan 2007) by inverting time channels individ-ually and then inverting the recovered pseudo-chargeabilities tofind Cole–Cole parameters. That has drawbacks because there isno connection between the individual inversions; each has its owntrade-off parameter and convergence criterion. We anticipate thathigher quality information can be obtained by inverting all IP dataat once.

Although the Cole–Cole representation works well for frequency-domain and time-domain data, it is more computationally intensiveto implement for the time-domain because of the need to carry outthe required Fourier transforms. This becomes particularly evidentwhen working in 3-D and using many cells. To reduce the computa-tion cost, we use a stretched exponential (SE) function (Kohlrausch1854; Hilfer 2002) to represent the IP decays. This has an analyticform for the impulse conductivity and hence it is computationally ef-ficient. Recently, Belliveau & Haber (2018) used the SE function tosimulate IP effects in time-domain EM data. They use the SE func-tion to convert Ohm’s law into an ordinary differential equation,whereas we use the SE function to derive an impulse conductivityfunction in time-domain.

The main goal of this paper is to develop a computationallyefficient algorithm that uses a two-stage procedure to invert all ofthe IP data at once to recover the SE parameters that define IPcharacteristics for earth materials. The most interesting structuresare 3-D (e.g. isolated targets and rough terrains) and hence applying

this SIP inversion in 3-D is an important task. For time-domainproblems it is important to incorporate the repetitive input currentwaveform (Fiandaca et al. 2012) and so we address this issue aswell.

Our paper proceeds as follows. First, we introduce the SE conduc-tivity and compare it to the Cole–Cole conductivity (Section 2.1).With the SE conductivity, we then show how time-domain IP dataare simulated (Section 2.2) and develop our SIP inversion method-ology (Section 2.3). Finally the technique is applied to a syntheticexample emulating a porphyry deposit where we attempt to discovera mineralized zone lying in an alteration halo with equal charge-ability. We compare the results obtained from individual inversionswith those from our all-at-once approach. With the latter approachthe mineralized zone can be distinguished from the halo because ofthe different time constant values (Section 3).

2 M E T H O D O L O G Y

2.1 SE and Cole–Cole functions

Using the Cole–Cole parametrization for an IP response in the time-domain is computationally expensive because of the need to carryout all of the required Fourier transforms. It would be advantageousto have a parametrization in the time-domain. In this section we usethe SE and show how its parameters relate to those of the Cole–Colemodel.

The time domain SE function (Kohlrausch 1854) can be definedas

f (t) = exp

(−

( t

τse

)cse)

. (2)

This can be used to define step-off conductivity response:

σ step-offse = σse ⊗ (

1 − ustep-on)

=⎧⎨⎩

t ≤ 0 : σ∞(1 − ηse)

t > 0 : −σ∞ηseexp

(−

(t

τse

)cse)

,(3)

where ustep-on(t) is the Heaviside step function, and subscript sestands for SE. Here ⊗ indicates a convolution operator. For instancea⊗b can be defined as

a ⊗ b =∫ t

−∞a(u)b(t − u)du. (4)

As in the Cole–Cole representation there are three parameters (η, τ

and c). Putting the SE values into the CC model does not yield the

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Inversions of time-domain SIP data 1853

same response as evaluating the Fourier transform of eq. (1) withthese same values. However, the correspondence is close enough sothat intuition previously developed by using CC parametrization isstill useful.

When ccc=1 (often called Debye model), the two response func-tions are equivalent. For different sets of parameters the functionsdiffer and finding explicit a relationship between the SE and theCole–Cole parameters is not possible (Hilfer 2002). However, for agiven time range (e.g. time domain dc-IP: 0.1 ms–4 s) and with anerror threshold (2 per cent standard deviation), there can be goodagreement between the two. To show this, we fix σ∞ = 1 S m–1

and ηcc = 100 mV V– 1, and compute the step-off conductivity forthe Cole–Cole model for variable ccc and τ cc; these are shown inFig. 2 (solid lines). As ccc increases (green to red curves), the timedecay gets broader, and as τ cc decreases, the response decays faster(Figs 2a and b). We carry out a small parametric inversion to fitthese Cole–Cole decays with the SE function to estimate the equiv-alent SE parameters. The results are presented in Fig. 2 and Table 1.Overall, the SE function closely matches the Cole–Cole responsefor a given time range. As shown in Table 1, ηse is coincident withηcc, but τ se overestimates τ cc, and cse underestimates ccc; this dis-crepancy gets smaller as ccc increases. Therefore, when interpretingthe SE parameters, the readers can use their understanding basedupon Cole–Cole parameters qualitatively, but for quantitative inter-pretation extra care might be necessary.

We now want to use this SE function and find an expression forthe SE conductivity, σ se, which can be considered as an impulseresponse. This can be obtained by taking the derivative of σ step-off

with respect to time and multiplying by -1:

σse(t) = σ∞δ(t) − σ∞ηset−1

(t

τse

)cse

exp

(−

(t

τse

)cse)

. (5)

By using 5, in the following section, we derive the linear IP equation,which will allow us to compute time-domain IP responses.

Note that the impulse SE function derived in Belliveau & Haber(2018) is slightly different from eq. (5). This difference arises dueto two reasons. First, we used a Cole–Cole model from Tarasov &Titov (2013), but Belliveau & Haber (2018) used one from Peltonet al. (1978). Secondly, we used a SE function shown in eq. (2),whereas they used:

f (t) = exp

(− tβ

τ

).

Our choice of the SE function results in a closer connection betweenCole–Cole and SE parameters.

2.2 Simulating time-domain IP data with SE function

The linear IP function (Seigel 1959; Oldenburg & Li 1994), whichhas been broadly used to invert IP data, can be written as

d I P (t) = J [σ∞]η(t), (6)

where dIP(t) is an IP datum at time t, J is a sensitivity function relatedto the background conductivity (σ∞(x, y, z)) and η(x, y, z; t) is apseudo-chargeability, which includes information about IP effects.The sensitivity function is:

J = − ∂ Fdc[σ∞]

∂ log(σ∞), (7)

where Fdc[ · ] is the Maxwell dc operator that computes the dcresponse for a given conductivity model. To use the SE function with

this linear IP equation, it is necessary to define pseudo-chargeabilityand derive an explicit form.

We rewrite the impulse SE conductivity, σ se as

σse(t) = σ∞δ(t) − σ∞η Ise(t), (8)

where the impulse pseudo-chargeability, η Ise, is

η Ise(t) = ηset−1

(t

τse

)cse

exp

(−

(t

τse

)cse)

. (9)

From Kang & Oldenburg (2016), the pseudo-chargeability, η(t), canbe defined as

η(t) =∫ t

−∞η I (t − b)w(b)db, (10)

where w(t) is the normalized input current waveform. When a step-off current is injected, w(t) is equal to 1 − ustep-on(t) and thereforethe pseudo-chargeability can be written as

η(t) =

⎧⎪⎪⎨⎪⎪⎩

ηse, t ≤ 0

ηseexp

(−

(t

τse

)cse)

, t > 0

.

(11)

With eqs (11) and (6), the observed time-domain dc-IP data canbe written as

d(t) ={

t ≤ 0 : Fdc[σ∞] + J [σ∞]η(t)t > 0 : J [σ∞]η(t)

(12)

Here, t ≤ 0 and t > 0 indicates when the input current is on and off,respectively. Note that when the current is on, there is a dc signal,but it does not exist in the off-time. Hence, any off-time data canbe considered as IP data (assuming of course that there is no EMcoupling). Considering that an IP datum is a small perturbation ofdc datum (Oldenburg & Li 1994), we ignore dIP when t ≤ 0, andhence the time-domain dc-IP data can be written as

d(t) ={

t ≤ 0 : Fdc[σ∞]t > 0 : d I P (t)

(13)

The assumption made here is equivalent to assuming η � 1 resultingin

σ∞ � σ0 = σ∞(1 − η), (14)

where σ 0 is conductivity at zero frequency (or dc conductivity).The explicit form of the pseudo-chargeability (eq. 11) clearly

shows the numerical advantage of the SE conductivity functioncompared to the Cole–Cole function for computing time-domainIP data. If the Cole–Cole conductivity is chosen then to evaluateeq. (10) we first need to compute η I (ω) and w(ω) in the frequencydomain, multiply them, and convert into time-domain for each 3-Dvoxel. Now, for a given distribution of σ∞, ηse, τ se and cse, time-domain IP data during the off-time can be computed directly usingeq. (6).

For general situations where repetitive rectangular pulses areinjected, a linear combination of the step-off response (eq. 11) isrequired to compute IP data. The details for this are described inAppendix A.

2.3 Spectral IP inversion methodology

To obtain a 3-D distribution of the conductivity and SE parametersfrom time-domain dc-IP data, we develop a gradient-based SIP in-version algorithm using the linear IP function. Our dc-SIP inversionis similar to the conventional dc-IP inversion approach (Oldenburg

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Figure 2. The stretched exponential (SE) fit of the Cole–Cole conductivity in the time domain: (a) τ cc = 5 and (b) τ cc = 0.5; here ηcc is fixed to 100 mV V–1.Lines are the step-off response using a Cole–Cole representation and the circles denote the SE responses. Green, blue, and red colors correspondingly indicateswhen ccc are 0.3, 0.5, 0.7. Associated parameters are given in Table 1.

Table 1. Comparison of the Cole–Cole and the resultant SE parameters forvariable ccc and τ cc.

CC CC CC CC CC CC

ηcc 100 100 100 100 100 100τ cc 5 5 5 0.5 0.5 0.5ccc 0.3 0.5 0.7 0.3 0.5 0.7

SE SE SE SE SE SEηse 100 100 100 100 100 100τ se 9 6 5 1 0.7 0.6cse 0.2 0.4 0.7 0.2 0.4 0.5

& Li 1994). The first step is to invert the dc data to recover a con-ductivity model which we assume that it is a good measure of σ∞.The sensitivity function (eq. 7) is evaluated and hence IP signalscan be computed using eq. (6). The second step is to invert multipletime channels of the IP data to recover distributions of the SE pa-rameters: ηse, τ se, cse. Our approach is different from conventionalapproaches (Yuval & Oldenburg 1997; Kemna et al. 2004; Honig& Tezkan 2007; Kang & Oldenburg 2016), which separately invertIP data to obtain pseudo-chargeability at multiple times and thenextract intrinsic IP parameters with a post-processing parametricinversion.

The predicted IP data at ith time channel, dpredi , can be written

as

dpredi [σ∞, ηse, τ se, cse] = J[σ∞]ηi [ηse, τ se, cse]. (15)

where J is the sensitivity matrix and ηi (x, y, z) is the pseudo-chargeability at i-th time channel. For notation, boldface indicatesvectors and matrices which are respectively presented by lower andupper case letters. For multiple time channels of IP data, the linearIP equation shown in eq. (15) becomes a part of a larger system

dpred = Jblk η⎡⎢⎣

dpred1...

dprednt

⎤⎥⎦ =

⎡⎢⎣

J. . .

J

⎤⎥⎦

⎡⎢⎣

η1...

ηnt

⎤⎥⎦, (16)

where Jblk is a sparse block diagonal matrix, and nt is the numberof time channels.

Our main goal is to set up an inverse problem to recover distri-butions of ηse, τ se, cse by inverting observed time domain IP data(dobs). For the inverse problem there is flexibility regarding howbest to define the inversion parameters. An intuitive choice, and one

that arises from the petrophysical work (Pelton et al. 1978) is ηse,log(τ se), and cse, but other combinations of logarithms or scaled rep-resentations might have practical advantages. Rather than makingthose decisions now we generically write our inversion model as mand postpone a final decision until we perform numerical analysesin Section 3.3.

Our inversion model, m, is written as

m =⎡⎣mη

mτ

mc

⎤⎦, (17)

where m with the superscript,: η, τ or c, indicates the model relatedto each of the SE parameters. Note that, if mη is (M × 1) vector,then the inversion model is (m) is (3M × 1).

To solve our inverse problem a gradient-based approach is usedto minimize an objective function

φ(m) = φd (m) + βφm(m), (18)

where φd is a measure of data misfit, φm is a model regularization(or model norm) and β is trade-off (or Tikhonov) parameter. We usethe sum of the squares to measure data misfit:

φd = ‖Wd(dpred (m) − d|)‖22 =

N∑j=1

(d pred

j − dobsj

ε j

)2

, (19)

where N is the number of the observed data and Wd is a diagonaldata weighting matrix which contains the reciprocal of the esti-mated uncertainty of each datum (εj) on the main diagonal, d is theobserved data. The uncertainty for the jth datum, εj, is defined as

ε j = 10−2 · percent j |d| + floor j , (20)

where percentj and floorj are percentage error (per cent) and noisefloor, respectively. Percentage error handles error proportional tothe amplitude of the observed data, whereas noise floor capturesconstant noise level (e.g. sensor sensitivity). For instance, in Sec-tion 3, we will use 5 per cent percent error and 1mV V–1 noise floorfor SIP inversions. Because the gradient-based inversion is used,sensitivity of the predicted data with regard to the inversion model,∂dpred

∂m , must be obtained, and its derivation is shown in Appendix B.Note that this sensitivity function is different from J[σ∞], which isthe sensitivity function for the dc inversion.

The model regularization, φm, is a measure of the amount ofstructure in the model and, upon minimization, this will generate a

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smooth model which is close to a reference model, mre f . The inver-sion model includes information about three different SE models in3-D space and φm can be written as

φm =∑

i=η,τ,c

φim, (21)

where the regularization for each SE parameter is

φim = αi

s‖Ws(mi − mire f )‖2

2 +∑

j=x,y,z

αij‖W j (m

i − mire f )‖2

2, (22)

where Ws is a diagonal matrix containing volumetric informationof the cells, and Wx , Wy and Wz are discrete approximations of thefirst derivative operator in x, y and z directions, respectively. Theαis are weighting parameters that balance the relative importanceof producing small or smooth models (Tikhonov & Arsenin 1977;Oldenburg & Li 2005).

Bound constraints, based upon prior knowledge or physical prin-ciples (e.g. 0 ≤ ηse < 1 and 0 ≤ cse ≤ 1) can be included in theoptimization

minimize φd (m) + βφm(m)

subject to ml ≤ m ≤ mu (23)

φd ≤ φ∗d , (24)

where ml and mu are lower and upper bounds. We solve this prob-lem using a projected Gauss–Newton (GN) method (Kelley 1999).The trade-off parameter β, is determined using a cooling techniquewhere β is progressively reduced from some high value (Olden-burg & Li 2005; Kang et al. 2014). The inversion is started withan initial model, m0, and the non-linear iterations are stopped whenthe target misfit, φ∗

d , is reached (Oldenburg & Li 2005). A targetmisfit of φ∗

d = nd , is equivalent to a root-mean-square misfit (rms)of 1. Details of how we are setting up inversion parameters willbe discussed in Section 3.3. Both 2-D and 3-D SIP inversion algo-rithms are developed, although the above description is based upon3-D case. SIMPEG ’s inversion framework (Cockett et al. 2015;Heagy et al. 2017) is used for implementation of the SIP inver-sion algorithms and packaged up as a SimPEG-SIP module. ThisSimPEG-SIP module is available through main SIMPEG package(https://github.com/simpeg/simpeg).

3 S Y N T H E T I C E X A M P L E S

3.1 Setup and physical property

To demonstrate the capability of our dc-SIP inversion procedure, weconsider a 3-D synthetic model of a mineralized porphyry depositshown in Fig. 3. The central stock is surrounded by an alterationunit that forms a halo structure (donut-shape in Fig. 3). The toppart of the porphyry has been eroded and is covered by overburdenin which patches of clay are embedded. A rectangular block-likemineralized zone exists within the halo structure.

We suppose that the halo structure includes fine-grained miner-alization (e.g. disseminated sulphides), whereas the mineralizationzone has a much greater grain size; this results in the two unitshaving distinctive IP characteristics. Table 1 shows the values ofconductivity and IP parameters for each geological unit and Fig. 4shows the vertical section of conductivity and IP parameters at y = 0m. The halo and clay units have lower resistivities (300 and 500 �m,respectively) than the background (∼1000 �m). The porphyry and

the stock have greater resistivity (5000 �m). Hence, the conductivehalo and clay can be distinguished from the resistive porphyry andthe stock based upon their resistivity values. The three chargeableunits: clay, halo and mineralization have the same chargeability of100 mV V–1, and hence the mineralization is not distinguishablefrom the clay and halo units based upon chargeability alone. How-ever, the mineralization has a greater time constant (5s) than theother two chargeable units (≤0.5 s). These results because of itsgreater grain size (Pelton et al. 1978; Revil et al. 2015). The timeconstant in the SIP decays is thus a potentially diagnostic propertythat can be used to distinguish the mineralization from the halostructure. The frequency exponent of the halo and the mineraliza-tion are equal to 0.5, whereas that of the clay is 0.8 and hence itmight also be possible to distinguish between the alteration haloand clay. Thus, if we can recover 3-D distributions of spectral IP in-formation (e.g. τ se, cse), the three chargeable units: mineralization,halo and clay, should be distinguishable.

In the following numerical experiments, we generate synthetic dcand IP data for the porphyry model, and investigate SIP informationin the data by fitting them using the SE function (Section 3.2. Toextract more rigorous information, we first invert the synthetic dcand IP data in 2-D, then in 3-D. The 2-D inversions, in Section 3.3,are used to answer the question about whether we want to invert forthe spectral parameters or their logarithms. It also allows us to com-pare the results from our simultaneous inversion with those froma conventional approach (Yuval & Oldenburg 1997), which sepa-rately inverts multiple time channels of the IP data and then extractsspectral IP information by fitting resultant pseudo-chargeabilities toa Cole–Cole model. Based upon these results we proceed with our3-D dc and SIP inversions and ultimately recover a 3-D rock modelin Section 3.4.

3.2 DC and IP data

Five lines of dc-IP data are acquired with a dipole–dipole array, with50 m electrode spacing and n-spacing = 10; each sounding has 10stations resulting in maximum spacing of 500 m between sourceand receiver pairs. There are 135 dc data for each line and 675in total. The IP data consist of 21 time channels, logarithmicallysampled from 1 ms to 4 s in the off-time. The input current is ahalf-duty cycle waveform having 4 s on- and off-time (nstack=2; seeAppendix A). There are 2835 IP data for each line and hence 14 175in total. For the discretization of the 3-D tensor mesh, we use 12.5m × 25 m × 10 m cell for the core region; the total number of cellsis 100 × 50 × 25 = 130 000. To generate dc data, the SIMPEG -dc package is used. The IP data are generated using the linear IPfunction (eq. 6). Five percent Gaussian noise is added to both dc andIP data. Fig. 5(a) shows the simulated dc data (apparent resistivity),and Figs 6(b) and (c) show the IP data (apparent chargeability) at 1ms and 4 s, respectively. The IP data show different characteristicsbecause of the spectral polarization effects.

To investigate the SIP information content in the IP data, we fiteach IP decay with the SE parameters. The predicted IP decay withthe step-off input current can be written as

d I P = Vs(t)

V0× 103 = ηseexp

(−

(t

τse

)cse)

× 103, (25)

By fitting all of the IP decays, we obtain three SE parameters: ηse,τ se and cse for each point in the pseudo-section (Fig. 5). Note thateach point in the pseudo-section has an IP decay consisting of 21time channels from 1 ms to 4 s.

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Figure 3. 3-D porphyry model. Left-hand panel: plan map at z = –95 m. Right-hand panel: vertical section at y = 0 m. Seven geological units are marked asdifferent colours. White crosses on the plan map indicates electrode locations used for the dc-IP survey.

Figure 4. Stretched exponential parameters in section view at y = 0 m. (a) conductivity at infinite frequency (σ∞), (b) chargeability (ηse), (c) time constant(τ se) and (d) frequency exponent (cse).

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Figure 5. Pseudo-sections of dc and IP data at the centre line (y = 0 m). (a) apparent resistivity, (b) apparent chargeability at 1 ms and (c) apparent chargeabilityat 4 s.

Fig. 6(a) shows the cross-plot of the resultant τ se and cse from thefitting. Comparison of the observed and predicted data is shown inFig. 6(b). Clustering in Fig. 6(a) is observed with cse (y-axis) beingthe dividing line. We classified the results as two units: (1) small cse

(<0.65) – black and (2) large cse (>0.65) – red. The correspondingIP decays are colored as black and red in Fig. 6(b); they havedifferent character at times t > 200 ms. This analysis shows thatthere is significant SIP information in the IP data and that we needto unravel it with an SIP inversion.

3.3 2-D inversion

Before inverting in 3-D, there are two questions to be answered:(1) How do we define the model parameters m? and (2) How muchresolution can be added by simultaneously inverting time-domainIP data compared to the conventional SIP inversion approach whereeach time channel is inverted separately? These questions will beaddressed by working in 2-D.

For the 2-D model, we take a cross-section of Fig. 4, along thecentreline (y = 0 m) and assume this a 2-D structure. We discretizethis using 25 m × 15 m cells. To answer the questions, we firstinvert dc data to recover a conductivity model. Then we apply theconventional IP inversion approach to recover 2-D distributions ofpseudo-chargeability at individual time channels; estimates of SEparameters are obtained from analysing these. Next, we use ourSIP inversion approach to estimate SE parameters. Finally, the SEparameters from the two approaches are then compared.

We start with the conventional IP inversion approach. dc dataare first inverted to recover the background conductivity. For thedc inversion, a 1000 �m half-space is used for an initial model.Fig. 7(a) shows the recovered conductivity model, σ est. The resistivestock and surrounding conductive halo structures are successfullyimaged. Then by using the recovered conductivity, we compute thesensitivity function (J[σest ]), and proceed with the IP inversion usingthe linear IP function (eq. 6). We invert IP data at 1 ms (Fig. 5b) torecover pseudo-chargeability. Fig. 7(a) shows the recovered pseudo-chargeability, and all chargeable zones corresponding to clay, halo,

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Figure 6. (a) Cross-plot of the estimated time constant (τ se) and frequency component (cse) obtained by fitting IP decays. (b) Comparison of observed andpredicted IP decays as a function of time. Black and red distinguishes two units: cse > 0.65 and cse < 0.65, respectively.

Figure 7. Results of 2-D dc and IP inversions: (a) conductivity, (b) pseudo-chargeability at 1 ms and (c) pseudo-chargeability at 4 s.

mineralization are imaged well. However, distinguishing these threedifferent chargeable units are not possible just with the recoveredpseudo-chargeability model since all of them are imaged with highpseudo-chargeability values.

We would like to extract more polarization information entangledin the multiple time channels of the IP data such as those shownin Fig. 6. Following Yuval & Oldenburg (1997), we successivelyinvert other time channels of the IP data (1 ms–4 s), and obtainpseudo-chargeabilities at multiple time channels. Fig. 7(b) showsthe pseudo-chargeability at the last time channel (4 s). Interestingly,

near surface chargeable materials have disappeared due to theirfast-decaying nature (small τ se and large cse). More quantitativeinformation can be extracted by interpreting the recovered pseudo-chargeability at multiple times.

To extract SE parameters from pseudo-chargeability at each 2-D prism, we set up a small inverse problem that fits the pseudo-chargeability at multiple times to obtain three SE parameters. Ratherthan doing this on a pixel by pixel basis we solve an inverse problemthat fits all prisms of the pseudo-chargeability together and recovera distribution of the SE parameters. Here the observed data are the

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recovered pseudo-chargeability, ηest , and the predicted data can bewritten as

dpred = η(ηse, τse, cse) (26)

This fitting procedure is similar to how we fit the IP decays with theSE function (eq. 25).

Our inversion methodology is the same as that provided in Sec-tion 2.3 but we have not yet addressed the parametrization for theinversion. As stated in Section 2.3, for each of the SE parame-ters one can choose either a linear or log model. As a first trial, wechoose m = [ηse, log(τ se), cse] and carry out the inversion. The initialmodel, m0 is a homogeneous earth having the uniform SE parame-ters: ηse = 0.01, τ se = 0.1, cse = 1; 5 per cent Gaussian errors areassigned as uncertainties. To ignore any impact from regularization,we set the trade-off parameter β = 0. Unfortunately, this inversiondoes not converge and is trapped in a local minimum. Consideringthat we are using a gradient-based inversion, the reason for this islikely associated with the scale of the gradients. For instance, if thechargeability has much greater sensitivity compared to the others,our gradient-based inversion will try to update the chargeability, butnot the two other parameters. To investigate this, we compute thegradient of the data misfit term, ∇mφd:

∇mφd =(

∂dpred

∂m

)T

r (27)

where the residual, r, is

r = Wd (dpred − dobs) (28)

The gradient term, ∇mφd, can be separated into gradients for eachof three model parameters: mη, mτ , mc. Fig. 8(a) shows the his-togram for log10(|∇mφd|). The gradient of ηse generally dominatesand thus the inversion will preferentially update chargeability, butnot the other two parameters. We believe this is the main reasonwhy the inversion did not converge. This can be rectified by usingthe logarithm of each parameter and hence we try m = [log(ηse),log(τ se), log(cse)]. With this choice, the inversion converges afterfive iterations. Fig. 8(b) shows the gradients for each SE parameter.Character of the gradient distributions (e.g. mean and variance) foreach of three SE parameters shown in Fig. 8(b) are more similarcompared to those shown in Fig. 8(a). This results in more uniformweighting for each SE parameter when computing the gradient ofφd. Based upon this experiment, we chose our inversion model as

m =⎡⎣mη

mτ

mc

⎤⎦ =

⎡⎣log(ηse)

log(τse)log(cse)

⎤⎦ (29)

This choice of the inversion parameter is same as Fiandaca et al.(2012), although the reasons for that choice were not provided intheir paper.

The recovered ηse, τ se, cse are presented in Figs 9(a)–(c), re-spectively. The recovered ηse is similar to the recovered pseudo-chargeability at 1 ms (Fig. 7b). In the chargeable regions, τ se has agreater value than the background, whereas both chargeable anoma-lies at the left- and right-hand sides show similar τ se values. Charge-able regions at the near surface (clay) show large cse values (∼1)compared to the other chargeable regions, and hence the charge-able clay can be distinguished from two other chargeable units. Thecross-plot of the recovered τ se and cse shown in Fig. 11(a) clearlyshows two distinct clusters of points due to different cse values,whereas the recovered τ se shows similar values (∼0.5). Therefore,with this sequential SIP inversion approach we can obtain the loca-tion of chargeable regions, and distinguish the clay from the halo

and mineralization. However, the distinction between the chargeablehalo and the mineralization is not evident.

We now use our approach, which simultaneously inverts 21 timechannels of the IP data, to recover the distribution of ηse, τ se and cse.Although not shown here, a similar analysis for the gradients wasperformed, and based upon that, we chose the same parametrizationof the inversion model by taking the logarithm of all three SEparameters. The same homogeneous halfspace model (ηse = 0.01,τ se = 0.1, cse = 1) was used as the initial and reference model;the other parameters of the inversion are summarized in Table 3 .This SIP inversion converges after nine iterations. Fig. 10 showsthe recovered SE parameters. The recovered ηse is similar to thatfrom the previous inversion (Fig. 9a). However, the recovered τ se

shows significant differences, particularly for the chargeable regionat the right hand side compared to Fig. 9(b); a much greater τ se

value is recovered (∼2 s) as shown in Fig. 10(b). The cross-plot ofthe recovered τ se and cse in Fig. 11(b) differs from the sequentialinversion results (Fig. 11a); another cluster having large τ se andsmall cse is observed. Hence, with the simultaneous SIP inversion,a chargeable region having large τ se can be distinguished from twoother chargeable units. In addition, similarly with the recoveredcse, clay can be distinguished from the halo and mineralized zone(Fig. 11b). In summary, our 2-D analysis has shown that, with aproper setup of the SIP inversion, the simultaneous SIP inversioncan obtain important spectral information about the mineralization(large time constant). We now address the 3-D problem.

3.4 3-D inversion

We now apply our 3-D dc-SIP inversion procedure. It includestwo steps: (a) dc inversion to recover conductivity model and (b)simultaneous SIP inversion to recover SE parameters: ηse, τ se, cse.For both dc and SIP inversions we use the same inversion parametersused in the 2-D inversions; both inversions reached the target misfit.

The 3-D dc inversion recovers a conductivity model, σ est(x, y,z). Fig. 12(a) shows the plan map (z = –95 m) and vertical section(y = 0 m) of the recovered model. In the plan map, the resistivecentre (the stock) and the conductive circular structure (the halo)are successfully imaged. In the vertical section, the conductive nearsurface is also imaged. In the second step we recover three SEparameters by simultaneously inverting 21 time channels of IP data.Fig. 12(b) shows the recovered chargeability model. The chargeablevolumes are colocated with conductive anomalies seen in the dcinversion. The time constants for the halo region are generally highbut the region on the right-hand side, associated with the mineralizedzone, has a significantly higher τ se value. The cross-plot of therecovered τ se and cse values with colored resistivity is shown inFig. 13(a); here ηse values greater than 80 mV V–1 are plotted. Usingthe recovered SE distributions and their cross-plot, we analyse theclustering of SE parameters and distinguish different rock units.

All chargeable volumes shows low resistivity, and spatially thereare two main features: circular features due to halo, and isolatedpatches at near surface. The overall circular volume shows large τ se

and small cse, but the right side shows greater τ se than the left-handside (right-hand panel of Fig. 12c). This distinction is clearly shownon the x-axis of Fig. 13(a). The near surface patches show highercse compared to the rest of the chargeable volumes (right panel ofFig. 12d), and this trend is presented in the y-axis of Fig. 13(a).Therefore, three main rock units: R1–R3 can be distinguished inthe chargeable volumes:

(i) R1: small τ se and small cse

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Figure 8. Histograms of the gradient for the data misfit, ∇mφd. Blue, orange and green colours indicate histogram correspond to each of the inversion models:mη , mτ and mc . Two sets of the inversion models are considered: (a) ηse, log(τ se), cse and (b) (a) log(ηse), log(τ se), log(cse).

Figure 9. Recovered distributions of SE parameters extracted from the pseudo-chargeability at 21 time channels (e.g. Figs 7b and c). (a) Chargeability (ηse),(b) time constant (τ se) and (c) frequency exponent (cse).

(ii) R2: small τ se and large cse

(iii) R3: large τ se and small cse

This rock classification is also shown in Fig. 13(b) with the cross-plot. The final 3-D rock model is shown in Fig. 14. Consideringthis is a porphyry deposit, we interpret four rock units. The non-chargeable rock unit, R0, can be considered as host rock. R1, whichshows up as a chargeable circular feature in the plan map, is inter-preted as the alteration halo; R3, is interpreted to be a mineralizedzone (the large grain size associated with mineralization should gen-erate a large τ ); R2 having large cse is interpreted as clay. Thresholdvalues of the SE parameters to interpret three rock units are sum-marized in Table 4. This classification of three different rock types:

halo, clay and mineralization would not have been possible if a sin-gle time channel of IP data, or integrated IP data (e.g. Newmontstandard time windows: 0.8 s–1.4 s), was inverted.

3.5 Summary for the synthetic example

By using the synthetic porphyry model as a motivating examplewe have investigated three crucial items. The first concerned thechoice of parameters for the SIP inversion. Choosing logarithmicparameters, m = [log(ηse, τ se, cse)], balanced the gradient (∇mφ)for each SE parameters (Fig. 8). This resulted in a more robustconvergence and the algorithm was not so easily trapped in a local

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Figure 10. Recovered distributions of SE parameters extracted from the SIP inversion, which simultaneously inverts 21 time channels of the IP data. (a)chargeability (ηse), (b) time constant (τ se) and (c) frequency exponent (cse).

Figure 11. Cross-plots of the recovered time constant (τ se) and frequency component (cse) from (a) sequential and (b) simultaneous SIP inversions withcolored chargeability (ηse).

Table 3. Inversion parameters of dc and SIP inversions for syntheticexamples.

dc SIP

Uncertainty (per cent, floor) (5 per cent,10−3.5 V)

(5 per cent, 1 mV V–1)

m0 1000 �m (10 mV V–1, 0.1 s, 1)mre f 1000 �m (10 mV V–1 0.1 s, 1)ml -∞ (0.01 mV V–1, 10−6 s,

0.01)mu +∞ (1000 mV V–1, 10 s, 1)αs 0.1 0.2αx = αz (=αy, when 3-D) 1 1

Table 4. Interpretation of the SE parameters to construct a 3D rock model.

Rock unitσ∞

(S m–1)ηse

(mV V–1) τ se (s) cse Interpretation

R0 Low ηse <80 N/A N/A HostR1 High ηse >80 τ se < 1.1 cse <0.6 HaloR2 High ηse >80 τ se < 1.1 cse >0.6 ClayR3 High ηse >80 τ se > 1.1 cse <0.6 Mineralization

minimum. The second item pertained to the SIP parameters obtainedfrom sequential inversion versus simultaneous inversions. Usingthe 2-D porphyry model, we showed that better resolution can beobtained with the simultaneous inversion. In particular, we were ableto distinguish a chargeable mineralization zone that had large τ se,

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Figure 12. Recovered distributions of SE parameters extracted from the SIP inversion, which simultaneously inverts 21 time channels of the IP data. Plan mapat z = –95 m and vertical sections at y = 0 m are shown: (a) conductivity, (b) chargeability (ηse), (c) time constant (τ se) and (c) frequency exponent (cse).

from a background chargeable halo zone that surrounded the stock.A simultaneous inversion of data at all time channels is also morerobust in the presence of problematic data. For instance low qualitydata at a particular time channel can greatly affect the chargeabilityrecovered from that time channel and therefore cause problemsin a subsequent parametric inversion that recovers IP parameters.However, the effects are ameliorated if those data are incorporatedinto the ensemble of data inverted in a simultaneous approach. Thethird item showed that, by carrying out the SIP inversion in 3-D, wecan identify volumetric regions with different SIP parameters andconstruct a rock model. We applied our simultaneous SIP inversionto the porphyry model and recovered 3-D distributions of ηse, τ se,cse. This allowed us to distinguish three different chargeable volumesidentified as: halo, clay and mineralization in a non-chargeablebackground earth. This information would have not been obtained,

if the usual IP inversion approach, which inverts a single channel ofIP data, or a time-integral of IP data, was used.

4 C O N C LU S I O N S

Earth materials (e.g. clays, minerals and ice) have distinct conduc-tivity spectra and hence their polarization characteristics can bediagnostic in geoscience applications such as mining, groundwater,and environment. In some of these problems it may be sufficientto recognize that there is a difference in conductivity between lowand high frequencies (generally referred to as ‘chargeability’) andall that is necessary is to find regions that are chargeable. For time-domain IP surveys, this could be accomplished by inverting data at asingle time channel, or inverting the integral of the decay curve oversome time gates. Despite the success of that procedure it has been

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Figure 13. Cross-plot of the recovered time constant (τ se) and frequency exponent (cse) from 3-D SIP inversion. (a) Cross-plot of τ se and cse with coloredresistivity. (b) Rock classification result in the cross-plot.

Figure 14. 3-D rock model constructed by using four different physical properties: σ est, ηse, τ se, cse obtained from 3-D dc and SIP inversions. The rock modelincludes four different units, and their features are summarized in Table 3.

a long standing goal to extract more information about the rocksand to discriminate between rocks that have same chargeability. Thefull complex conductivity spectrum is therefore needed and this re-quires high quality data measured over a broad time range; modernsystems now accomplish this. The next stage, and that consideredhere, is to invert these multi-channel data to recover the spectralinformation about the conductivity.

We tackle this problem in the following manner:

(i) We assume that our data are not contaminated with EM effectsand that the Maxwell dc equations are valid. If EM effects areprevalent, then they must be handled in a pre-processing step (e.g.Kang & Oldenburg 2016).

(ii) We parametrize the conductivity response function as a SE.This representation has four parameters (σ∞, η, τ , c). These are thesame parameters as in the frequency Cole–Cole parametrization,but the analytic representation of the SE response function greatlyspeeds up computation.

(iii) Rather than solving for all parameters at once, we adopta two-stage process where we first invert the dc data to estimateσ∞, and use the sensitivity from that solution to generate a linearmapping between IP data and pseudo-chargeability. This requiresthat the chargeability is small (η < ∼0.3).

(iv) The IP data at multiple time channels are inverted simulta-neously to recover the parameters (η, τ , c). We use a gradient-basedGauss–Newton algorithm to solve the inverse problem and define

the logarithms of η, τ , c as parameters in the inversion. This equal-izes the distribution of the gradients and results in better conver-gence.

To test our algorithm we used simulated data from a 3-D porphyrymodel. Application of the developed SIP inversion showed that wecan distinguish a chargeable mineralized zone that has a large timeconstant, from other chargeable volumes (clay and an alterationhalo). We would have not been able to accomplish this by invertinga single time channel of IP data.

In conclusion, the inversion procedure we proposed in this pa-per provides a methodology for extracting not only chargeability,η, from the IP decays but also other spectral parameters: τ and c.As shown by the test example, knowing these parameters may playan essential role in answering geologic questions. In addition to theimportant next step of applying this to a field data set, there are stillfurther items to be worked on at the algorithmic level. The first itemis non-uniqueness issue due to increased number of parameters (τand c), and how it can be addressed. The second item is what to dowhen η is large (say η >0.5) and there is potential breakdown of ourassumptions. Careful analysis of these items, and applying the 3-DSIP inversion to field data sets, are required to demonstrate the prac-ticality of our SIP technique. In spite of these missing ingredients,our work in this paper constructs a cornerstone to make the time-domain SIP inversion technique useful in practical applications. Topromote useability of the methodology, and to invite others to con-tribute to this goal, the codes used for the examples presented in this

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paper are open-sourced, and available through: https://github.com/simpeg-research/kang-2018-spectral-inducedpolarization (Kang etal., 2019

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A P P E N D I X A : H A N D L I N G R E P E T I T I V EWAV E F O R M S

In Section 2.2, we have derived an explicit form of pseudo-chargeability assuming the input current is a step-off (1 − ustep-on)as shown in eq. (10). However, in practice we use periodic half-duty cycle waveforms having on- and off-time, then stack themto obtain both dc (on-time) and IP data (off-time). Fiandaca et al.(2012) showed the importance of considering this repetitive wave-form when simulating and inverting time-domain dc-IP data. Thisis important when τ se is large and IP data do not reach steady-statein the prescribed on-time. This consideration of waveform is impor-tant when quantitative analyses are required (e.g. actual values of IPparameters), whereas if one’s goal is obtaining anomalous charge-able volumes and their distinction with other SE parameter (τ se andcse), this consideration of actual waveform is less important. OurSIP inversion code is designed so that the user can input the numberof stacks (nstack). In this section, we illustrate how this functionalityis implemented.

With a normalized current waveform, w(t) = I(t)/max(|I|),pseudo-chargeability, η(t), can be written as

η(t) = η I (t) ⊗ w(t). (A1)

Here I(t) is the input current. Then IP data is

d I P (t) = Jη(t). (A2)

So, if we can obtain stacked pseudo-chargeability, we can computestacked IP data. Now, consider half-duty cycle waveform havingon- and off-time with period of T as shown in Fig. A1. In a simplemanner, by convolving this with η I and stacking off-times T/4–T/2 and 3T/4–T, we can obtain stacked pseudo-chargeability, ηstack.Here the number of stacks (nstack) is 2, but if we have more cyclesthen nstack will be increased. Pseudo-chargeability for the first pulse,η1(t), can be written as

η1(t) = ηoff(t − T/4) − ηoff(t), (A3)

Figure A1. Half-duty cycle waveform having a period of T.

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where ηoff is a step-off pseudo-chargeability (eq. 11). Similarly, forthe second pulse, η2(t),

η2(t) = −(ηoff(t − T/4) − ηoff(t)

)(A4)

+(ηoff(t − T/2) − ηoff(t − T/4)

). (A5)

Hence, ηstackt) can be defined as

ηstack(t) = 1

2(η1(t) − η2(t)). (A6)

This can be extended to nstack case with a series form:

ηstack(t) = 1

nstack

(nstack−1∑

i=0

ηoff(t − T/2i) · (−1)i · (nstack − i)

). (A7)

Similarly, for on-time, stacked pseudo-chargeability, ηstack, can bewritten as

ηstack(t)

= 1

nstack

(nstack−1∑

i=1

ηoff(t + T/4 + T/2(i − 1) · (−1)i · (nstack − i)

).

(A8)

Therefore, to obtain stacked pseudo-chargeability we just need acapability to evaluate ηoff. With the stacked pseudo-chargeability,IP data can be written as

d I P (t) = Jηstack(t). (A9)

Suppose t0 is the moment current is turned off. With the assumption(eq. 14), IP data for the on-time is ignored resulting in observeddc-IP data:

d(t) ={

Fdc[σ∞] , t ≤ t0

d I P (t) = Jηstack(t) , t > t0.(A10)

Although not shown here, we have found that assuming nstack = 2 isoften enough, even when the actual number of stack is much greaterthan 2 (e.g. 16). Hence we have used nstack = 2 for both syntheticand field examples in Section 3.

A P P E N D I X B : S E N S I T I V I T Y

Because we are using gradient-based inversion, obtaining a sensi-tivity function for each model parameter is necessary. The predictedIP data can be written as

d I P (t) = J η(t). (B1)

Required sensitivity functions are the derivative of dIP with respectto each SE parameter: ηse, τ se, cse. Here, we show the derivation forthese sensitivity functions when input current is a step-off.

∂d I P (t)

∂ηse= J

∂η(t)

∂ηse= exp

(−

(t

τse

)cse)

(B2)

∂d I P (t)

∂τse= J

∂η(t)

∂τse= cseηse

τse

(t

τse

)cse

exp

(−

(t

τse

)cse)

(B3)

∂d I P

∂cse= J

∂η

∂cse= ηse

(t

τse

)cse

exp

(−

(t

τse

)cse)

log

(t

τse

).(B4)

By using these sensitivity functions for the step-off case with thestacked functions shown in eq. (A7), we can obtain the needed sensi-tivity functions. If logarithmic variables are used the correspondingchain rule is applied (e.g. ∂log(ηse )

∂ηse= 1

ηse).

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inversions of time-domain spectral induced polarization ...hence the inversion is much more...

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