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Application of the DEXP Method to the Streaming Potential Data


Abstract and Figures

We interpret the self-potential data related to groundwater flow by the depth from extreme points (DEXP) method; a multiscale method in which the data are upward-continued and scaled by a scaling law depend on the structural index. The depth to the water table is estimated from extreme points of the DEXP image without a priori estimate of the hydraulic coupling coefficient. The method is tested with a synthetic model of the water table and applied to a real self-potential dataset near a pumping well. The obtained results agree well with the known information.
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Near Surface Geoscience
Turin, Italy, 6-10 September 2015
We 21P1 15
Application of the DEXP Method to the Streaming
Potential Data
M.A. Abbas* (Univ. of Naples Federico II & South Valley Univ.) & M. Fedi
(University of Naples Federico II)
We interpret the self-potential data related to groundwater flow by the depth from extreme points (DEXP)
method; a multiscale method in which the data are upward-continued and scaled by a scaling law depend
on the structural index. The depth to the water table is estimated from extreme points of the DEXP image
without a priori estimate of the hydraulic coupling coefficient. The method is tested with a synthetic model
of the water table and applied to a real self-potential dataset near a pumping well. The obtained results
agree well with the known information.
Near Surface Geoscience
Turin, Italy, 6-10 September 2015
The self-potential anomalies have two main mechanisms; the oxidation-reduction associated with ore
deposits and streaming potential (SP) associated with ground water flow through the electrokinetic
coupling (Revil et al., 2004). There are two approaches for interpreting the origin of the SP anomalies
related to ground water flow. The first approach supposes that SP signals could be mainly related to
the thickness of the unsaturated (vadose) zone in which water penetrate vertically until the water table
(e.g., Zablocki, 1978; Jackson &Kauahikaua, 1987). Another approach presumes that the main SP
contribution is located along the water table and that the variations of the hydraulic head are directly
responsible for the electrical potential anomalies measured at the ground surface (Fournier, 1989;
Revil et al., 2004).
In this work, we consider the application of a multiscale method; the Depth from EXtreme Points
(DEXP) method (Fedi, 2007; Fedi and Abbas, 2013), for the interpretation of the streaming potential
data related to the flow of groundwater.
Considering that the SP sources are along the water table (Figure 1), the electric potential U(r) (in
mV) measured at the point P(r) located on the Earth’s surface is related to the hydraulic head h at
point M(rq) at the water table (in m) by a Fredholm equation of the first kind (Fournier, 1989; Revil et
al., 2004),
where n is the unit normal vector over the water table
, dS is a surface element of the water table, c
is an apparent streaming potential coupling coefficient (in mV m-1) given by s
, where C and
Cs are the electric potential coupling coefficients of the vadose and saturated zone respectively.
is the
ratio between the electrical conductivity of the saturated and unsaturated zones (Revil et al., 2004).
For a 2D profile, assuming that the variation in y direction can be neglected, the electric potential is
obtained by (Revil et al., 2004):
r (2)
represents the curvilinear coordinate along the water table line.
Figure 1 Sketch of the geometry of the water table for a pumping well (modified after Rizzo et al.,
If we assume a flat measurement surface, so that n is constant, equation (1) will be equivalent to the
potential field due to a sheet or a simple layer (Baranov, 1975). Hence, as we mentioned in the
previous chapter, the streaming potential has the same mathematical form as the magnetic potential.
In fact, any deviation from the horizontal level of the water table can be interpreted as a one-point
source (point-dipole, line of dipoles, or others). Consequently, we can apply the DEXP method to the
streaming potential anomaly. The DEXP method is a multiscale method in which the potential or its
derivative is continued and scaled with a scaling-law dependent on the geometry of the source. The
Near Surface Geoscience
Turin, Italy, 6-10 September 2015
scaling exponent is not fixed but can be determined directly from the data (Fedi, 2007; Fedi and
Abbas, 2013). An important issue is that we do not need any a priori estimate of the hydraulic
coupling coefficient; one more reason to use the DEXP method is that it is fast and stable.
Let us assume Cartesian coordinates where x and z are the horizontal and vertical directions,
respectively, and the depth is positive downward. We can discretize equation (2) in terms of the
depths of the dipoles along the water table zq assuming a vertical electric polarization as:
where d is the horizontal step. We can note from equation (3) that the two above-mentioned
approaches for the origin of SP anomalies are in fact equivalent. We have already said that any
deviation from the horizontal level of the water table can be interpreted as a point source (point-
dipole, line of dipoles, or others) so that the DEXP transformation (Fedi, 2007; and Fedi and Abbas,
2013) can be applied. The DEXP transformation p
is given by Fedi (2007) as:
 
zUzz p
where Up is the p-th order derivative of U and z is the scale or altitude. The depth to the water table
can be estimated from the extreme points of the DEXP image p
. In case of non-vertical polarization,
the DEXP transformation can be applied to the analytic signal modulus of the SP anomaly. This is
equivalent to the dipole occurrence probability (DOP) of Revil et al. (2001) but, once more, in an
easier way: we do not need normalization. We also do not use a fixed scaling exponent (Fedi and
Pilkington,. 2012), which makes the DEXP method more suitable to be used in different situations.
The application of the DEXP transformation method requires upward continuation and vertical
differentiation of order p for the computation of Up and p
. Recalling that the upward continuation is
not allowed for SP anomalies, being the air infinitely resistive. However, the upward continuation of
SP data, from a measurement level zm to the level zm+a, is equivalent to the SP data at zm of the same
source located not at the depth z0 but at the deeper depth z0+a. In this way, the concept of vertical
derivative of SP data may be introduced for SP anomalies as well (Fedi and Abbas, 2013).
Synthetic Example
The SP anomaly of the water table model shown in Figure 2b is calculated using equation (3). The SP
anomaly (Figure 2a) was upward continued from 0 to 60 m and its 2nd order vertical derivative is
calculated. Figure 3a shows the continued derivative with the ridges. The scaling function was
calculated along two ridges at horizontal distances 500 m and 100 m (Figure 3b, c). Both scaling
functions yield a scaling exponent equal 0.6. The scaling functions calculated along the two other
ridges in the central part of Figure 3a did not give reasonable values of homogeneity degree due to the
interference effects connected to the complexity of the source in this part. We used the estimated
scaling exponent to calculate the DEXP image shown in Figure 4a. The extreme points of the DEXP
image give a good estimate of the depth of the water table. Also, we used a geometrical method to
estimate the depth to the water table at the intersection of the ridges of the 4th order derivative of the
SP anomaly (Figure 4b), this order warranting a nice degree of resolution to the problem. The results
agree will with the true model of the water table.
Figure 2 The SP anomaly of a water table model. (a) The SP anomaly, and (b) the depth to the water
Near Surface Geoscience
Turin, Italy, 6-10 September 2015
Figure 3 Scaling function analysis. (a) 2nd order derivative of the SP anomaly and its ridges, (b)
scaling function of the ridge at distance 500 m and (c) scaling function of the ridge at distance 100 m.
Figure 4 Application of DEXP method and geometrical method for the estimation of water table
depth. (a) DEXP image; white stars indicate the estimated depths (b) the depth to the water table
estimated by the geometrical method.
Real Example
The DEXP imaging is applied to the streaming potential data collected near a pumping well from
Bogoslovsky and Oglivy (1973). Figure 5 shows the SP anomaly over the pumping well (K-1) and
also the piezometric depths. We note a positive anomaly with a symmetric form over the well location
and other two small negative anomalies related to the infiltration from the surface drainage ditches.
Figure 5 The streaming potential data near a pumping well (a) and the piezometric depths (b) after
Bogoslovsky & Oglivy, 1973.
First, the 1st derivative of SP anomaly is continued from 0 to 30 m with a 0.2 m vertical step (Figure
6a). Then the scaling exponent is estimated by analyzing the scaling function along vertical ridges at
x={95, 163, 270} m (Figure 6d,e,and f). We found that the central positive anomaly over the pumping
well and the negative anomaly related to the infiltration of the water in the subsurface at x= 270 m
need both a 0.9 scaling exponent, whereas the negative anomaly related to the infiltration of water in
the subsurface at x= 95 m needs a 1.15 scaling exponent. Finally, we obtained two DEXP images with
the two estimated values of scaling exponents (Figure 6b and c). The estimated depths to the water
table, indicated by the white stars, agree well with the measured piezometric depths.
Near Surface Geoscience
Turin, Italy, 6-10 September 2015
Figure 6 Application of the DEXP method to SP data near a pumping well. a) first-order derivative of
SP at different scales and ridges (cyan solid lines); b) DEXP image with 0.9 scaling exponent
overlaid by piezometric depths (white solid line); c) DEXP image with a 1.15 scaling exponent
overlaid by piezometric depths (white solid line); d) scaling function of the rightmost (d) central ridge
(e) and leftmost ridge (f). White stars in b&c indicate the estimated depth to water table.
We applied the DEXP method to the streaming potential data related to groundwater flow. DEXP is a
fast imaging method transforming the field data, or its derivatives, into a quantity proportional to the
source distribution. It depends on the knowledge of the structural index of the source which may be a
priori determined by applying related multiscale methods based on the scaling function. The depth to
water table is estimated by the extreme point of the DEXP image and does not need a priori estimate
of the hydraulic coupling coefficient. We showed the usefulness of DEXP method to SP dataset near a
pumping well. The estimated depths agree well with the known information.
Baranov, V.I. [1975] Potential fields and their transformations in applied geophysics (No. 6).
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fields sources. Geophysics, 72(1), I1-I11.
Fedi, M. and Abbas, M.A. [2013] A fast interpretation of self-potential data using the depth from
extreme points method. Geophysics, 78(2), E107-E116.
Fedi, M. and Pilkington, M. [2012] Understanding imaging methods for potential field data.
Geophysics, 77(1), G13-G24.
Fournier, C. [1989] Spontaneous Potentials and Resistivity Surveys Applied To Hydrogeology In A
Volcanic Area: Case History Of The Chaîne Des Puys (Puy-De-Dôme, France). Geophysical
Prospecting, 37, 647-668.
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Revil, A., Naudet, V. and Meunier, J.D. [2004] The hydroelectric problem of porous rocks: Inversion
of the water table from self-potential data. Geophysical Journal International, 159, 435-444.
Rizzo, E., Suski, B., Revil, A., Straface, S. and Troisi, S. [2004] Self-potential signals associated with
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Summary A new method is described to determine the depth to sources of potential fields. The theory is here developed for sources such as poles or dipoles, but it may be extended to lines of poles, line masses, lines of dipoles, dykes, ribbons and so on. The field is scaled at several altitudes following a specific law analytically determined. The depth to the source is finally obtained by determining the scaled field extreme points. The method is fast and stable. It may be applied to any vertical derivative of a Newtonian potential, by using theoretically determined scaling functions for each order of derivative.
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We used a fast method to interpret self-potential data: the depth from extreme points (DEXP) method. This is an imaging method transforming self-potential data, or their derivatives, into a quantity proportional to the source distribution. It is based on upward continuing of the field to a number of altitudes and then multiplying the continued data with a scaling law of those altitudes. The scaling law is in the form of a power law of the altitudes, with an exponent equal to half of the structural index, a source parameter related to the type of source. The method is autoconsistent because the structural index is basically determined by analyzing the scaling function, which is defined as the derivative of the logarithm of the self-potential (or of its pth derivative) with respect to the logarithm of the altitudes. So, the DEXP method does not need a priori information on the self-potential sources and yields effective information about their depth and shape/typology. Important features of the DEXP method are its high-resolution power and stability, resulting from the combined effect of a stable operator (upward continuation) and a high-order differentiation operator. We tested how to estimate the depth to the source in two ways: (1) at the positions of the extreme points in the DEXP transformed map and (2) at the intersection of the lines of the absolute values of the potential or of its derivative (geometrical method). The method was demonstrated using synthetic data of isolated sources and using a multisource model. The method is particularly suited to handle noisy data, because it is stable even using high-order derivatives of the self-potential. We discussed some real data sets: Malachite Mine, Colorado (USA), the Sariyer area (Turkey), and the Bender area (India). The estimated depths and structural indices agree well with the known information.
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Plots of self-potentials (SP) versus topography and (or) vadose-zone thickness determined from vertical electrical soundings show linear, sometimes segmented, correlations along 4 lengthy SP profiles on Kilauea. The slopes of the correlation lines are used to calculate an apparent water-table elevation along the entire profile. When compared with surface geology and nearby well data, the apparent water table is reasonable. -from Authors
Ore deposits and buried metals like pipelines behave as dipolar electrical geobatteries in which the source is due to (1) variation of the redox potential with depth, (2) oxido-reduction reactions acting at the ore body/groundwater contact, and (3) migration of electrons in the ore body itself between the reducing and oxidizing zones. This polarization mechanism is responsible for an electrical field at the ground surface, the so-called self-potential anomaly. A new quick-look tomographic algorithm is developed to locate electrical dipolar sources in the subsurface of the Earth from the analysis of these self-potential signals. We applied this model to the self-potential anomaly discussed by Stoll et al. [1995] in the vicinity of the KTB-boreholes drilled during the Continental Deep Drilling Project in Germany. The source of this self-potential signal is related to the presence of massive graphite veins associated with steeply inclined fault zones within the gneisses and observed in the KTB-boreholes.
We show that potential fields enjoy valuable properties when they are scaled by specific power laws of the altitude. We describe the theory for the gravity field, the magnetic field, and their derivatives of any order and propose a method, called here Depth from Extreme Points (DEXP), to interpret any potential field. The DEXP method allows estimates of source depths, density, and structural index from the extreme points of a 3D field scaled according to specific power laws of the altitude. Depths to sources are obtained from the position of the extreme points of the scaled field, and the excess mass (or dipole moment) is obtained from the scaled field values. Although the scaling laws are theoretically derived for sources such as poles, dipoles, lines of poles, and lines of dipoles, we give also criteria to estimate the correct scaling law directly from the data. The scaling exponent of such laws is shown to be related to the structural index involved in Euler Deconvolution theory. The method is fast and stable because it takes advantage of the regular behavior of potential field data versus the altitude z. As a result of stability, the DEXP method may be applied to anomalies with rather low SNRs. Also stable are DEXP applications to vertical and horizontal derivatives of a Newtonian potential of various orders in which we use theoretically determined scaling functions for each order of a derivative. This helps to reduce mutual interference effects and to obtain meaningful representations of the distribution of sources versus depth, with no prefiltering. The DEXP method does not require that magnetic anomalies to be reduced to the pole, and meaningful results are obtained by processing its analytical signal. Application to different cases of either synthetic or real data shows its applicability to any type of potential field investigation, including geological, petroleum, mining, archeological, and environmental studies.
The self-potential (SP) method is a fast and cheap reconnaissance tool sensitive to ground water flow in unconfined aquifers. A model based on the use of Green's functions for the coupled hydroelectric problem yields an integral equation relating the SP field to the distribution of the piezometric head describing the phreatic surface and to the electrical resistivity contrast through this phreatic surface. We apply this model to SP data measured on the south flank of the Piton de la Fournaise volcano, a large shield volcano located on Réunion island, Indian ocean. The phreatic surface, inverted with the help of the Simplex algorithm from the SP data, agrees well with the available information in this area [one borehole and electromagnetic (EM) data]. This interpretation scheme, which we call electrography, has many applications to the crucial problem of water supply in volcanic areas where drilling is expensive.