an algorithm for fast 3-d inversion of surface ert...

1

An Algorithm for Fast 3-D Inversion of Surface ERT Data:

Application on Imaging Buried Antiquities

N.G. Papadopoulos(1),(3), P. Tsourlos(1), C. Papazachos (1), G.N. Tsokas(1), A. Sarris(2) and

J. H. Kim (3)

(1) Department of Geophysics, School of Geology, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece.

(2) Laboratory of Geophysical-Satellite Remote Sensing & Archaeo-environment, Institute for Mediterranean Studies. Foundation of Research and Technology-Hellas, P.O. Box 119, Rethymnon, 74100, Crete, Greece

(3) Korea Institute of Geoscience and Mineral Resources (KIGAM), Mineral Resources Research Division, Exploration Geophysics and Mining Engineering Department, 92 Gwahang-no, Yuseong-gu, Daejeon, 305-350, S. Korea

Corresponding author

Nikos Papadopoulos

KIGAM, Exploration Geophysics and Mining Engineering Department

92 Gwahang-no, Yuseong-gu, Daejeon, 305-350, S. Korea

E-mail: [email protected],

Tel: +82-42-868-3418

2

ABSTRACT

In this work a new algorithm for the fast and efficient three-dimensional (3-D) inversion of

conventional two-dimensional (2-D) surface Electrical Resistivity Tomography (ERT) lines,, is

presented. The proposed approach lies on the assumption that for every surface measurement there is a

large number of 3-D parameters with very small absolute Jacobian matrix values, that can be excluded

in advance from the Jacobian matrix calculation, as they do not contribute significant information in

the inversion procedure. A sensitivity analysis for both homogeneous and inhomogeneous earth models

showed that each measurement has a specific region of influence, which can be limited to parameters

in a critical rectangular prism volume. Application of the proposed algorithm accelerated almost three

times the Jacobian (sensitivity) matrix calculation for the data-sets tested in this work. Moreover,

application of the Least SQuares Regression (LSQR) iterative inversion technique, resulted in a new 3-

D resistivity inversion algorithm more than 2.7 times faster and with computer memory requirements

less than half compared to the original algorithm. The efficiency and accuracy of the algorithm was

verified using synthetic models representing typical archaeological structures, as well as field data

collected from two archaeological sites in Greece, employing different electrode configurations. The

applicability of the presented approach is demonstrated for archaeological investigations and the basic

idea of the proposed algorithm can be easily extended for the inversion of other geophysical data.

Key Words: Jacobian matrix, 3-D resistivity inversion, LSQR, Archaeological prospection

3

INTRODUCTION

During the last years the development of new multiplexed and multichannel resistivity instruments

comprised a breakthrough innovation in the geoelectrical exploration (Stummer, et al., 2004). These

automatic devices made possible the collection of full three-dimensional (3-D) surface apparent

resistivity data sets in a relatively reasonable time, by placing a number of electrodes on the nodes of a

rectangular grid and measuring all the possible potentials (Loke and Barker, 1996). Unfortunately due

to the limitations imposed by the logistics, the most common practice to record the 3-D subsurface

apparent resistivity variation is through the application of dense parallel and/or orthogonal surface two-

dimensional (2-D) lines (Chambers, et al., 2002; Gharibi and Bentley, 2005).

The back-projection technique (Petrick, et al., 1981; Shima, 1992) and the probability tomography

method (Mauriello, et al., 1999) consisted the basic options to develop approximate 3-D inversion

algorithms in the past. The development of more powerful computers led to the development and

application of accurate inversion algorithms, based on the Integral Equation Method (Dabas, et al.,

1994; Lesur, et al., 1999), the Finite Difference Method (Ellis and Oldenburg, 1994; Park and Van,

1991), the Finite Volume approach (Pidlisecky, et al. 2007) and the Finite Element Method (Pridmore,

et al., 1981; Sasaki, 1994; LaBrecque, et al., 1996; Tsourlos and Ogilvy, 1999; Pain, et al., 2002; Yi, et

al., 2001; Günther, et al., 2006; Marescot, et al., 2008) in order to solve the 3-D Direct Current (DC)

forward problem.

In archaeological exploration the DC electric method has nowadays become a valuable and efficient

tool in site evaluation and excavation planning (Clark, 1990). Especially 2-D and 3-D DC resistivity

tomography has gained an increased attention for extracting quantitative information concerning the

burial depth, the depth extend and the lateral dimensions of the buried archaeological structures

(Vafidis, et al., 1999; Diamanti, et al., 2005; Osella, et al., 2005; Papadopoulos, et al., 2006; 2007)

The main drawback of standard 3-D resistivity inversion is the time consuming numerical calculations

and it also requires extensive computer memory resources. The calculation of the Jacobian or

4

Sensitivity matrix (J) consumes more than 55% of the computational time of the inversion procedure,

while for very large 3-D resistivity problems the calculation of the generalized inverse (J*) can also

significantly delay the inversion procedure. Günther et al. (2006) proposed a method to further reduce

only the memory requirements of their 3-D resistivity inversion algorithm by storing the sensitivity

values that were over a threshold which is defined after the Jacobian matrix calculations.

This work will present a new inversion algorithm in an attempt to minimize all previously mentioned

drawbacks of the standard 3-D inversion procedure of DC resistivity data. The efforts of optimizing the

inversion algorithm are: a) to find an automatic way of calculating the part of the Jacobian matrix

which contains the maximum amount of information which is useful for the inversion by avoiding the

calculation of very small absolute sensitivity values based on a-priori geometric criteria and b) to take

advantage of the sparseness of the new Jacobian matrix by applying an iterative solver with mush

smaller memory requirements. The Finite Element Method (FEM) is used as a platform for the forward

resistivity calculations and the inversion problem is regularized by imposing smoothness constraints.

The efficiency and stability of the new algorithm was evaluated through the inversion of synthetic and

real data from archaeological structures.

FORWARD AND INVERSE DC RESISTIVITY MODELING

Finite Element Method

The application of the Finite Element Method to 3-D resistivity problem is extensively discussed in

Pridmore et al. (1981) and Tsourlos and Ogilvy (1999), so the method is briefly outlined in this section.

Poisson’s equation describes the partial differential equation that governs the behaviour of the electric

potential

s s s, , , ,V I x x y y z zx y z x y z (1)

5

where σ is the conductivity of the earth, V is the electric potential, I is a point current located at the (xs,

ys, zs) position and δ is a Dirac delta function. Usually the subsurface is subdivided into a large number

of homogeneous and isotropic hexahedral cells. A simple interpolator function approximates the

unknown potential at the nodes of each hexahedral element of the mesh.

The Galerkin weighted residual method is applied in order to minimize the error between the

approximated and real potential, After applying the Galerkin criterion to every element, and since

hexahedral elements will share common nodes, the individual element equations can be assembled into

one global linear system of the form:

KA=F, (2)

where K is the stiffness matrix that includes the nodal coordinates and element conductivities, the

vector A contains the unknown nodal potentials and the vector F contains the current sources and

boundary terms. For equation (2), the boundary conditions (BC) have to be considered: the Neumann

BC (at the air-earth interface there is no current flow perpendicular to the boundary) are included in the

corresponding surface element equations, while the homogeneous Dirichlet BC (the value of the

potential at the model side and bottom boundaries is zero) are included in the linear system of equation

(2) in the form of additional constraint equations.

Solution of equation (2) allows the determination of nodal potentials. As for a given set of current

sources the apparent resistivities along an Electrical Resistivity Tomography (ERT) cross section are

calculated using the potential values at the corresponding electrode nodes and the geometrical factors

determined by the relative positions of the current and the potential electrodes.

Least-Squares Inversion

A fundamental key in solving the 3-D resistivity inversion problem comprises the determination of a

relationship that links the variation of the apparent resistivity data (ραi) in relation to a change in the

6

subsurface resistivity (ρj). The MxN two-dimensional matrix known as Jacobian (Sensitivity) matrix Jij

expresses this link:

1 1

1

1

log log

log log

log log

log log

a a

N

ij

aM aM

N

J =

, (3)

where M, N are the number of the apparent resistivity measurements and the resistivity model

parameters respectively. Notice that since the resistivities and apparent resistivities often span several

orders of magnitude, their logarithms have been used in equation (3). In this work the Jacobian matrix

values were estimated using the adjoint equation approach (McGillivray and Oldenburg, 1990).

The DC linearized inverse problem can be expressed as

dy = Jdx, (4)

where dy = y – F(x) is the vector of differences between observed and calculated data (logarithm of

apparent resistivities), dx is the correction vector to the initial model parameters x0 and J is the

Jacobian matrix for the x parameter (logarithm of resistivity) distribution. A common approach for the

model parameterization is to divide the earth into hexahedral blocks of unknown constant resistivity.

The cell dimensions are related to the minimum electrode spacing (a) and they are set to (X, Y, Z) = (a,

a, a/2).

Because the resistivity problem is ill-conditioned, unstable and non-unique (Tikhonov, et al., 1998), it

is necessary to impose additional constraints on equation (4) in order to obtain reliable results. A

typical approach, also adopted here, is to incorporate additional smoothness constraints on the model in

equation (4):

r = Cdx = 0 (5)

7

where C is a second difference smoothness operator (Sasaki, 1994). Simultaneous solution of equations

(4) and (5) is usually performed by minimization of the objective function:

U = ||dy – Jdx||2 + λ||r||2, (6)

where || . || denotes the Euclidean norm and λ is a Lagrange multiplier. Minimization of equation (6)

leads to the following set of modified normal equations:

(JTJ + λCTC) dx = JTdy. (7)

The solution to the equation (7) is equivalent to the least-squares solution of the joint system of

equations (5) and (6), i.e.:

λ 0

dyJdx

C= . (8)

Solving equation (8) avoids the calculation of the JTJ and CTC matrices and the matrix inversion in

equation (7). Furthermore the use of an appropriate method (e.g. conjugate gradients) can accelerate

the inversion procedure and possibly save computer memory. The vector dx is added to the initial

vector x0 to obtain the updated resistivity parameters. The procedure is repeated until a misfit between

the measured and modelled data is reduced to an acceptable relative Root Mean Square (RMS) level or

a maximum number of the iterations has been reached. The algorithm calculates the relative RMS

misfit between the observed ( obsid ) and the calculated ( cal

id ) apparent resistivity data as

2M

2i 1

1RMS

M.

obs cali i

obsi

d d

d

(9)

The quality of the collected data dictates the choice of the RMS threshold and corresponds to an

estimation (or calculation) of the apparent resistivity data measured error.

8

The Lagrange multiplier weights the model constraints against the data misfit. The strategy of

decreasing λ beginning from a large starting value down to a minimum value was adopted. The

following empirical scheme was employed after several tests with synthetic and real data:

k k

k k

if k 3

if k 3

k 1, 2...,

2

number of iterations

=

= (10)

This scheme was preferred to the one-dimensional (1-D) line search procedure (which tests several λ

values and finds the optimum λ value by interpolation) since the later proved to be quite time-

consuming. A modest line search needs at least three repetitions of the forward modelling and matrix

inversion procedure. Actually Constable et al. (1987) suggested there is no guarantee that the 1-D line

search procedure will produce a model that fits the data better. Thus, there is no reason to believe that

the adopted empirical scheme is inferior to the 1-D line search scheme.

SENSITIVITY ANALYSIS

Since the calculation of the Jacobian matrix consumes more than half of the computational time, the

main effort for optimizing the 3-D inversion algorithm was focused on defining an automated method

for the faster calculation of this matrix. The basic idea of the proposed algorithm relies on the study of

the actual Jacobian matrix values. Considering the specific geometry of collecting any 3-D apparent

resistivity data (e.g. through the application of dense parallel 2-D lines), it is clear that for every surface

measurement there is a number of 3-D resistivity parameters in the model for which the corresponding

absolute Jacobian matrix values are very small. These parameters are typically found in parts of the 3-

D at distant locations from potential and current electrodes.

9

In order to demonstrate the previous statement, the Jacobian matrix for a 10 Ohm-m homogeneous

earth, assuming the pole-pole array, was calculated for a 3-D model grid consisting of 900 parameters

[(X, Y, Z) = (15, 15, 4)]. Figure 1 (A) shows the 3-D sensitivity distribution for a central measurement

with Nsep = 1a (a = unit electrode distance) for different horizontal depth slices. It is clear that the

measurement has a specific region of high sensitivity values (maximum absolute value ~ 0.06),

corresponding to a region of the influence of significant model parameters, which can be approximated

with a rectangular parallelepiped (cuboid) area with specific dimensions. Any large or small variation

in the resistivity of these parameters will significantly affect the behaviour of this specific

measurement. In other words only the Jacobian matrix entries related to the parameters which are

included in this cuboid region can be considered as significant to the measurement. Parameters outside

this region have very small absolute sensitivity values, hence even large changes in these parameters’

resistivity will practically not influence the apparent resistivity value. Practically figure 1 (A) says that

the sensitivity of a geometrically distant parameter with respect to a measurement is very small and

does not influence this measurement.

In order to verify the previous suggestion, the Jacobian matrix was also calculated for an

inhomogeneous earth assuming a resistive (100 Ohm-m) prismatic body embedded in a homogeneous

half-space of 10 Ohm-m (Fig. 1B). The resistive body is outlined at the upper right corner of the model

area and the sensitivity distribution was estimated for the same measurement as in figure 1 (A). The

results show that the sensitivity values and pattern for this central measurement remained practically

unchanged as the mean relative difference of the sensitivity inside the rectangular region for the two

models is ~0.1%, with a maximum absolute value less than 0.3%. Moreover the apparent resistivity

value of this central measurement for the inhomogeneous earth is 10.008 Ohm-m, less than 0.1%

different than the homogeneous earth reference value (10 0hm-m). The previous results illustrate that

the presence of an inhomogeneity outside the sensitivity area of a specific measurement does not

practically affect the value of the corresponding apparent resistivity measurement.

10

ALGORITHM DESCRIPTION

Fast calculation of the Jacobian matrix

In order to incorporate the sensitivity analysis results in the Jacobian matrix calculations an algorithm

was developed on the basis of the relative position and distance of each apparent resistivity

measurement with respect to all model parameters that describe the subsurface resistivity variation.

Assuming that the data used is a set of dense parallel and/or orthogonal 2-D surface ERT lines, each

one measured using a standard array with typical geometry (inter-electrode spacing a and N-separation

= 1a, 2a…,Nmax a), the threshold dimensions X, Y and Z of the critical cuboid area are calculated using

the following procedure:

The complete Jacobian matrix is initially calculated once for the 3-D homogeneous parameter grid.

However, only the rows of the Jacobian matrix corresponding to a set of centrally placed

measurements of different N-separation (Nsep = 1a, 2a,…, Nmaxa) are kept for further processing.

Despite the fact that any measurement would provide the same sensitivity values, since their

estimation is based on a homogeneous earth, the use of centrally placed measurements allows to

study the full 3-D extent of the sensitivity distribution (Fig. 1).

For the first central measurement (Nsep = 1a) the corresponding N absolute values of the Jacobian

matrix row (JR) are sorted in an ascending order. Then the absolute value JR(b) which corresponds

to the bth entry of the sorted row is located and stored as the threshold value for this particular

measurement geometry (Fig. 2). The same procedure is followed for the remaining central

measurements (Nsep = 2a,…, Nmaxa), hence a different JR

Nsep value is defined. The index b

parameter is defined with the following relationship:

b = (integer) (r. N), r = [0, 1]. (11)

11

The meaning of equation (11) is evident: If r = 0, then b = 0 which means that no elements of

the Jacobian matrix are excluded, hence the threshold value J 0R

Nsep for all central

measurements.

Practically in this situation the procedure is identical to the standard inversion algorithm, where

the whole sensitivity matrix is calculated and used. Alternatively, in the case that r = 1 (b = N)

then the maximum Jacobian value of figure 2 would be set as the threshold value which would

reject all the Jacobian values resulting in the non-existence of the inversion procedure. For an

intermediate value e.g. r = 0.5 (b~N/2) a large part (50% in this case) of the Jacobian row

elements would be smaller than the JR(b) cutoff value. In practice values of variable r in the

range of [0.01, 0.99] were tested.

For every different central measurement (Nsep = 1a, 2a,..,Nmaxa) the corresponding absolute

Jacobian matrix rows entries that are over the defined thresholds JRNsep(b) (Nsep = 1a, 2a,…, Nmaxa)

are plotted against the corresponding parameter coordinates. The most distant from the

measurement model parameters that have sensitivity values larger than JR

Nsep define the maximum

dimensions X, Y, Z of the cuboid that is adopted for each Nsep value. Since this estimation is

performed only for the central measurements the computation time for this procedure is

insignificant.

Using the previously described procedure a library of threshold X, Y, Z dimensions for different

electrode arrays and geometries can be created. The calculations of these threshold dimensions are

carried out only once and the inversion program uses these corresponding values for each Nsep

choice as fixed constants. Therefore, depending on the Nsep value of each measurement, the

appropriate threshold dimensions X, Y, Z are used. Using this approach, for every row of the

Jacobian matrix only the sensitivity values for parameters inside the critical cuboid are calculated.

All the other sensitivity values corresponding to model parameters outside the critical cuboid are

12

not estimated and are set to zero. As a result the modified Jacobian matrix becomes quite sparse

depending on the specific choice of r.

A question which arises for the above described procedure is the determination of the percentage of the

Jacobian matrix elements for which the algorithm skips their calculation. Figure 3 shows the variation

of the parameter r value in relation to percentage (%J0) of the Jacobian matrix elements that are not

calculated by the algorithm and are set to zero assuming a given parameterized space (1444

parameters), electrode geometry (Nsep(max) = 4a, 1400 measurements) and configuration (Pole-Pole

array). The percentage of the Jacobian matrix elements (%J0) that is not calculated at all is equal to

40% and remains unchanged if the values of parameter r range from 0.01 to 0.2. On the contrary,

within the space of [0.2, 0.99] the variation of r with respect to %J0 converges to the line %J0 = 100r.

For example when the value of the variable r equals to 0.6 then the new algorithm will avoid the

calculation of almost 64% of the Jacobian matrix elements.

It is clear that the percentage of the Jacobian matrix elements that the algorithm does not calculate is a

function of the electrode geometry (maximum Nsep), the number of the model parameters, the number

of the collected apparent resistivity data that and the value of parameter r. However, this does not

render the algorithm of limited practical interest, as in every case study only a relatively small amount

of computation time will be used in the pre-processing stage for the calculation of the threshold

dimensions.

The speedup of the computation time due to the calculation of the largest magnitude Jacobian values is

an important element of the presented algorithm. Table 1 shows the different data sets used for a

comparison study, regarding the computation time that the original and the new algorithm require for

the Jacobian matrix calculation. In practice only the first iteration was performed by the algorithms as

the purpose was to investigate the performance of the new algorithm in terms of speed and memory

requirements. The pole-pole array, with increasing number of electrodes and horizontal grid

dimensions, was considered. All the possible measurements along parallel and orthogonal 2-D lines for

the data sets (DS) 1 to 8 were created. The basic inter-electrode distance along each line and the inter-

13

line distance was set equal to 1a. For each different data set a parameter mesh consisted of eight layers

of constant thickness (d = a/2) was assumed. The horizontal dimensions of the model parameters along

the X and Y axes were set equal to the basic electrode distance. The value 0.6 was adopted for r and

the last column of Table 1 presents the corresponding percentage of the Jacobian matrix elements that

are not calculated using the aforementioned technique.

Figure 4 (A) shows a significant decrease for the computational time by the new algorithm for the

calculation of the Jacobian matrix, ranging from 64% to 72% depending on to the data set. It is clear

that the described technique can accelerate the Jacobian matrix calculation almost three times for the

specific choice of r, by omitting computations related to the insignificant model parameters for each

measurement. The observed computation reduction is indicative for typical data sets collected in

archaeological prospection and is further improved as the size of the data set increases.

Solving the augmented system

Due to the incorporation of a large number of zero values in the modified Jacobian matrix (sensitivity

values not computed in the previous step), the resulting augmented matrix k

k λ

J

C is sparse since the

smoothness matrix also contains mostly zero elements To further reduce the memory requirements of

the new algorithm, only the non-zero elements of the augmented matrix were efficiently stored in a

column vector with its size equals to the maximum number of the non-zero elements. A second vector

of the same dimension is used to store the column index of each non-zero element for every row of the

augmented matrix. Finally two other vectors with dimensions (M+N) are used to store the maximum

number of the non-zero elements in every row of the augmented matrix and the total sum of the non-

zero elements respectively.

The Least SQuares Regression (LSQR) iterative method (Paige and Saunders, 1982) was incorporated

into the inversion algorithm in order to solve the augmented system of equation (8) since it is

appropriate for the robust and fast calculation of large sparse linear systems. LSQR is a powerful

variant of the CG methods, which originated from a modification of Lanczos method (Lanczos, 1950).

14

For this reason, since more than 20 years it is generally considered as the most efficient method to

solve large and sparse tomographic systems (Nolet, 1983, 1985; Scales, 1987; Van der Sluis and Van

der Vorst, 1990). Moreover, LSQR has intrinsinc damping properties (e.g. Van der Sluis and Van der

Vorst, 1987), hence even if it is applied to a tomographic system without any kind of additional

constrains (e.g. smoothing), it does not lead to unstable solutions of the tomographic system.

Moreover, it introduces no implicit hidden scaling of the data or the model as other methods (e.g. older

fast back-projection methods such as SIRT) and it is easily extended in continuous models with

complex variables (Nolet and Snieder, 1990).

One of the main advantages of LSQR is that although it is a generalized conjugate gradient (CG)

method it operates directly on the augmented matrix of equation (8). It also provides much more stable

solutions, as the singular values of matrix J (equation 8) are much larger than the corresponding

eigenvalues of matrix JTJ (equation 7). Furthermore LSQR involves only a few multiplications of the

augmented matrix of equation (8) with vectors, hence the whole process can be significantly speed up

as by taking advantage of the matrix sparseness in these multiplications. LSQR has routinely been

applied in problems of seismic tomography (Nolet, 1985; Papazachos and Nolet, 1997; Yao, et al.,

1999), but has not yet met a direct application in the inversion of geoelectrical data, as the conventional

Jacobian matrix is not sparse.

Due to the iterative nature of the LSQR technique, suitable termination criteria for the LSQR steps

must be established in advance. Despite the fact that LSQR theoretically converges to the

corresponding least-squares solution (Paige and Saunders, 1982), this often never occurs in practice, as

the algorithm tends to repeat the same conjugate search directions. In general the convergence of the

LSQR depends on the data errors as well as on the singularity level of the Jacobian matrix, which

makes the determination of the maximum LSQR steps (m) in every iteration (k) of the inversion

procedure a cumbersome process (Papazachos and Nolet, 1997). Moreover the unlimited continuation

of the LSQR iterative steps almost always dramatically increases the norm ( kdx ) of solution vector

(Paige and Saunders, 1982), leading to unrealistic geoelectrical models.

15

For a maximum number of 50 steps (m = 1, 2,…, 50) of the LSQR technique, the norm of the

resistivity correction vector ( kmdx ) and the linear misfit k

mr were calculated for a maximum

number of four iterations (k = 1, 2, 3, 4). The linear misfit refers to a quantitative measure of the

quality of the updated solution mdx in every step of the LSQR technique. Furthermore, for every

different resistivity correction vector mdx (m = 1, 2,…, 50 ) an updated resistivity model mx was

estimated and the non-linear misfit ( km

%RMS ) was calculated by using the finite element algorithm to

calculate the synthetic apparent resistivities. Several synthetic and real data sets were used to extract

safe conclusions, which generally are summarized below.

Figure 5 (A) shows that for all the four iterations of the inversion the linear misfit ( k m

r ) decreases

with increasing LSQR steps. In fact the linear misfit remains practically unchanged after 20 and 10

LSQR steps for the first and the second iteration respectively, while for the remaining iterations (3rd

and 4th) it stabilises after the 5th LSQR step. An almost similar situation is presented in the variation of

the non-linear (true) misfit with increasing LSQR steps (Fig. 5B). The fact that the non-linear misfit

appears to have an increasing trend after a specific number of LSQR steps in every iteration (m=20 if

k=1; m=10 if k=2; m=5 if k>2) is very important. In practice, this suggests that the least-squares

solution (where LSQR tries to converge) has a much larger true (non-linear) misfit in comparison with

the true misfit of an intermediate LSQR iteration step. Similar behaviour of increasing non-linear misfit

with LSQR steps has also been reported in problems of seismic tomography (Papazachos and Nolet,

1997).

This intrinsic limitation of the linear misfit behaviour is well-known and has led to various conclusions,

e.g. Tarantola (1987) suggested that when solving a non-linear problem with a linearized

approximation, all resolution and error estimates should incorporate the final non-linear (and not

linearized) residuals, even if this requires additional work (recomputation of the apparent resistivities

for the final model in our case). The misfit jump observed between non-linear iterations (difference

16

between linear and non-linear misfits), when the new Jacobian and data residuals are re-estimated has

also been observed for different kinds of geophysical data and linear system solution techniques (e.g.

Sambridge, 1990).

The true (non-linear) misfit increase during LSQR iterations has been attributed to the fact that as

LSQR (or any iterative method) iterates, the new model updates are still based on the original Jacobian

estimate. Therefore after a certain number of iterations, when the model is different, the “old” Jacobian

does not contain information that can contribute to the minimization of the non-linear (true) misfit.

Moreover, figure (5C) shows that continuing the LSQR steps also results in significant increase of the

model correction vector, kmdx , hence leading to unusually high solutions, which often correspond to

unrealistic geoelectrical models. Therefore, termination of the LSQR iterations when the linear misfit

shows very small changes is necessary not only to avoid unrealistic (high-amplitude) model corrections

but also since the true (non-linear) data misfit also increases for large LSQR steps.

There is no objective criterion to determine the optimum LSQR premature termination step. In other

words, we do not know when we have the best non-linear misfit (estimates such as those in figure 5 are

not available in routine LSQR application, as they are computationally inefficient), as this will depend

on the non-linearity of the problem: For linear or almost linear problems we can stop LSQR later, for

very non-linear ones earlier. However, the proposed strategy (termination of LSQR when linear misfit

does not show significant changes) is a safe choice. This approach may result in a larger number of

non-linear iterations, but will allow the determination of robust solutions even for very non-linear

problems.

LSQR was initially applied for the solution of equation (7) (original algorithm) which was used to

update the resistivity parameters. Figure 4 (B) shows the relative computation time decrease when the

LSQR is used to solve the augmented system of equation (8) of the new algorithm for r = 0.6. Even for

relatively large data sets, like DS-8, the absolute time LSQR needed to find a solution in both

algorithms, was kept to reasonable time levels. The incorporation of the LSQR for solving the sparse

17

augmented matrix of the new algorithm resulted in a relative decrease in time of 30% in all the data

sets tested in this work, hence the matrix inversion time is faster by a factor of ~1.5.

Figure 6 (A) presents a simple flowchart of the algorithm for calculating the sensitivity cuboid

(threshold dimensions). This pre-processing procedure has to be carried out only once for each data set,

before the inversion routine, which is shown in figure 6 (B).

The overall computation time for the proposed algorithm has been reduced ~3 times in comparison to

the original one, due to the combination of the truncated Jacobian estimation and use of the LSQR

routine (Fig. 4 C). Furthermore the use of the augmented matrix avoids the storage of the JTJ and CTC

matrices. Additional memory saving was achieved by the efficient storage of only the non-zero values

of the augmented matrix, resulting in a relative decrease regarding memory requirements between the

original and the new algorithm, ranging from 26% to 53% (Fig. 4 D). Its has to be pointed out that the

proposed storage of non-zero values in vectors results in a significant memory usage reduction due to

the sparsity of matrix C and the imposed (through the selection of parameter r) sparsity of matrix J. If r

is very small (e.g. 0.1) the proposed matrix storage approach may even result in higher memory

requirements. However, for most practical cases (r=0.6 or larger) the memory save is significant.

EVALUATION OF THE ALGORITHM

Synthetic data

The new algorithm was first evaluated using synthetic data for 3-D geoelectrical models which

simulate buried archaeological structures. Since the geoelectrical method is mainly used to locate

subsurface antiquities, which exhibit higher resistivity than their surrounding environment, similar

features were used for the tested models.

Model 1

A geoelectrical survey using the pole-pole array, consisting of twenty parallel 2-D surface

tomographies along the X-axis (Fig. 7 A) was simulated. Twenty electrodes were placed along each

18

line. The basic inter-electrode spacing and the inter-line distance were equal both to a = 0.5 m, with a

maximum separation Nsep(max) = 4a. Figure 7 (A) shows the survey layout and the model resistivity

distribution at four depths. The model consists of two prismatic bodies with a resistivity of 100 Ohm-

m, buried in a background area with resistivity of 10 Ohm-m. The synthetic apparent resistivity data

were corrupted with 2% random Gaussian noise (Press, et al., 1992), in an effort to simulate

observation errors. Figure (7A) shows in total 20 2-D ERT lines which corresponded to 1440 synthetic

apparent resistivity measurements. Furthermore 1444 hexahedral model parameters (X, Y, Z = 19, 19,

4), modelled he subsurface resistivity hence the Jacobian matrix contained 2021600 (1400x1444)

elements.

Figures 7 (B) and 7 (C) respectively show the inverted 3-D geoelectrical models, as horizontal depth

slices of increasing depth, that resulted using the original and the new algorithm. The initial value for

the Lagragne multiplier was set equal to 0.01 for all the inversions. A maximum of ten iterations or a

convergence rate smaller than 3%, were used as stopping criteria for the inversion procedure.

Furthermore, when the %RMS dropped below the random noise that was originally considered the

inversion was also terminated

The shape and the outline of the prismatic bodies were fully reconstructed by the original algorithm in

the depth of Z = 0.25-0.50 m. Within the depth layer of 0.50-0.75 m the inverted model underestimates

the true resistivity value because of the reduced resolution with depth of the surface pole-pole

measurements. The slight extension of the model in the depth slice Z = 0.75-1.00 m is probably due to

the smoothness constraints imposed in the inversion procedure.

Figure 7 (C1 to C7 presents the final 3-D inverted resistivity models using the new algorithm, for

various values of the variable r. Almost all inverted models are of comparable accuracy with the

inverted model from the original algorithm of Figure 7 (B). Therefore, the new algorithm managed to

successfully reconstruct the original model for all the tested values of the variable r (r = 0.3, 0.4, 0.5,

0.6, 0.7, 0.8, 0.9). Even in the extreme case (r = 0.9), where more than 87% of the Jacobian matrix

elements were not computed (set equal to zero) and only less than 13% of the Jacobian matrix elements

19

were used in the inversion, the new algorithm converged to a comparably acceptable resistivity model

(Fig. 7 C7). Larger values than 0.9 for the variable r were also tested but in these cases the new

algorithm diverged after two iterations.

Model 2

The second model consists of three prismatic bodies (B1, B2, B3) with high resistivity value (500

Ohm-m) which is typical of archaeological structures (e.g. walls). The bodies B2, B3 are located at the

depth of Z = 0.5-1.0 m and they are 0.5 m thick, while the body B1 is located at the depth of Z = 0.0-

1.0 m. A body (B4) with resistivity 10 Ohm-m at the depth of Z = 1.0-2.0 m is used to simulate an area

with increased moisture levels (Fig. 8 A). In order to calculate the synthetic apparent resistivities for

the above model, ten parallel surface 2-D ERT lines along the X-axis were considered. Twenty

electrodes were placed along each line, at one meter interval. The inter-line distance was also set to one

meter. A different than Model 1 array was considered, namely the dipole-dipole array, with a

maximum separation between the current and potential dipoles equals to Nsep(max) = 6m. In total 870

apparent resistivity measurements were considered and a 2% Gaussian noise was also added, while the

subsurface model contains 1026 parameters (19x9x6). As a result the Jacobian matrix contained

892620 elements and 56.3% of them weren’t calculated by the new algorithm when using the value r =

0.6. Figures 8 (B) and 8 (C) show the inverted models using the original and the new algorithm. The

new algorithm reconstructed the locations, the dimensions and the resistivity values of the prisms. In

general the final inversion models show comparable accuracy, although the 3-D image using the new

algorithm has a slightly poorer performance for the deeper body B4.

Real Data

The new 3-D inversion algorithm was also applied to real data collected from archaeological areas with

known buried structures, in order to verify the results obtained from the modelling approach. The real

data-sets were collected from two different archaeological sites in Greece. Sikyon archaeological site is

20

situated in central Greece at 37059’N and 22042’E, while Europos area lies in the northern Greece

(41001’N and 22051’E).

Sikyon archaeological site

Figure 9 (A) shows the 15x10 m2 rectangular grid which was selected from Sikyon archaeological site.

Within this area a known buried feature had previously been identified by a standard Twin Probe

resistance mapping survey (Sarris, et al., 2007). The rectangular grid was surveyed with twenty one

surface 2-D lines parallel to X-axis using the pole-pole array (Fig. 9 B). The inter-line and the basic

inter-electrode distance was a = 0.5 m and the maximum separation between the current and the

potential electrodes was 2 m (Nsep(max) = 4a,). More than 2370 data were collected, while the

subsurface was divided into four layers and 2400 (30x20x4) hexahedral model parameters to describe

the resistivity distribution.

Figure 10 (A) presents the reconstructed resistivity model produced by the original 3-D inversion

algorithm in the form of horizontal slices of increasing depth (λ0 = 0.1, RMS = 2.1%, 8 iterations). The

first depth slice (Z = 0.0-0.25 m) seems to have no indications of architectural remnants. The outline of

an archaeological feature appears in the second and the third depth slices (z = 0.25 - 0.75 m). Two

small walls parallel to Y-axis (Z = 0.25 - 0.50 m, X = 7 m), which divide this building into two main

compartments can be also identified. Furthermore some fade remnants of the structure can be seen in

the depth range of Z = 0.75 - 1.00 m.

Figure 10 (B1 to B7) shows the 3-D geoelectrical inverted models for various values of the variable r

using the same inversion parameters (λ0 = 0.1, Maximum iterations: 8, convergence rate: 3%). For all

the tested values of the variable r, except for the value r=0.9, the new algorithm converged to a final

solution after eight iterations, with practically the same RMS (2.1%). For all the values of r the new

algorithm produced inverted models of comparable accuracy with the original algorithm and the

outline of the buried building as well as its inner details were accurately reconstructed. Even in the

extreme case when only 10.5% (r = 0.9, %J0 = 89.5) of the Jacobian matrix elements were used in the

21

inversion (Fig. 10 B7), the algorithm proved its stability by producing a very reasonable archaeological

structure.

Europos archaeological site

Twenty one 2-D ERT lines along the X-axis, covering a 10x10 m2 square grid at the archaeological site

of Europos, were collected with the dipole-dipole configuration. The inter-line and the basic inter-

probe spacing were a = 0.5 m. The maximum distance between the current and potential dipoles was

3.5 m (Nsep (max) = 7a) (Diamanti et al., 2005). After outlier rejection the dense parallel tomographies

contained more than 2200 measurements, while the subsurface model was divided in 2800 model

parameters.

The original algorithm converged to a 3-D resistivity model after seven iterations with an RMS of

3.8%. Figure 11 (A) shows the horizontal depth slices of increasing depth of the 3-D resistivity

inverted model. A linear anomaly of low resistivity, along the line Y = 7 m is identified in the first

depth slice (Z = 0.00 - 0.25 m). Some features related with archaeological ruins are observed as high

resistivity values in the second depth layer (Z = 0.25-0.50 m). The main resistivity anomaly is

formulated at the depth Z = 0.50-0.75 m, probably corresponding to horizontal and vertical walls of a

buried building. This archaeological structure seems to reach maximum depth of 1.50m below the

surface. A schematic interpretation of the high resistivity anomalies is shown in the 3-D inversion

model of figure 11 (A).

By assigning the value 0.6 to the variable r, only the 42% of the Jacobian matrix elements were used

by the new algorithm for the inversion of the apparent resistivity data. The new algorithm converged to

a final resistivity model after 9 iterations, with an RMS=3.8%. It resulted once more in a 3-D model

that described the resistivity variation with comparable accuracy with respect to the original one, as far

as all the spatial properties (dimensions, burial and depth extend) of the buried archaeological structure

concerned (Fig. 11 B).

22

DISCUSSION AND CONCLUSIONS

In this work a new algorithm for the fast 3-D non-linear inversion of dense parallel 2-D lines was

presented. An empirical procedure for the faster calculation of the Jacobian matrix was described. Only

the larger sensitivity values corresponding to parameters that are included in a critical cuboid were

calculated using geometrical criteria. These criteria are based on the sensitivity of each measurement

with respect to the parameters, as this is determined for a homogeneous starting model. The sensitivity

analysis for both homogeneous and inhomogeneous earth showed the sensitivity cuboids are practically

not affected by distant resistivity changes.

For a specific geometry, variable r (Eq. 10) defines the percentage of the Jacobian matrix elements that

are not calculated by the new algorithm. A range of values for the variable r were tested. Although

even extreme values (e.g. r = 0.9) resulted in reasonable resistivity inversion models, the use of the

value r = 0.6 is finally recommended. This choice seems to be relatively conservative, however this is

derived from the threshold dimensions determined by the range of Jacobian matrix values for

homogeneous earth. It is well known that the Jacobian matrix values depend on the resistivity of the

parameters which change during the iterations. The synthetic and the real examples presented in this

work, as well as a number of other data-sets that have been also tested supported the use of r = 0.6 as a

safe option. This choice minimizes the possibility of removing significant information from the

Jacobian matrix during the later iterations of the inversion procedure.

The incorporation of the new technique for the Jacobian matrix calculation in the new algorithm

significantly decreased the computation time needed to calculate the sensitivity matrix in all the data

sets tested in this work. The Jacobian calculations were accelerated almost three times in relation to the

original algorithm. This observation suggests that the calculation of the relatively small magnitude

Jacobian elements is a very time consuming process for large 3-D resistivity problems, which ca be

safely avoided in order to speed up the inversion procedure.

23

The system of linear equations (Equation 7) was re-arranged to the equivalent augmented rectangular

system (Equation 8), which avoids the calculation and storage of the JTJ and CTC matrices. The use of

the LSQR technique resulted in a 30% decrease in computational time for the inversion of the

augmented matrix. Overall the new algorithm is more than 2.7 times faster than the original one. The

times are indicative for the significant improvement of the new algorithm and can be further improved,

as the size of the 3-D problem increases. Smaller memory requirements were also achieved by the

efficient storage of only the non-zero values of the augmented matrix. These factors resulted in a

relative decrease in memory requirements between the original and the new algorithm ranging from

26% to 53%.

The stability of the new algorithm was evaluated using two synthetic models, assuming two different

electrode configurations. The new algorithm successfully reconstructed the original models and

produced 3-D resistivity inversion images of comparable accuracy with the original algorithm. The

processing of the real apparent resistivity data using the new algorithm, collected from two different

archaeological sites, verified the synthetic modeling results and demonstrated the efficiency of the

proposed 3-D inversion algorithm.

It has to be noted that the presented technique for calculating only the significant part of the Jacobian

matrix, was based on an experimental procedure, as far as the calculation of the optimum threshold

dimensions concerned. This approach concerned the examined models and the real data, which were

used to evaluate the performance of the new algorithm. Hence, typical structures and geometry

configurations, under the framework of a standard procedure of geophysical explorations in

archaeological sites, were considered. Any generalization of the algorithm in order to handle different

kinds of structures and other measurement geometries and dimensions, will require the recalculation of

the optimum threshold values of r according to the current problem parameters, in order to define the

threshold dimensions of the critical sensitivity rectangular. The calculation of the optimum threshold

values of r has to be carried only once at the pre-processing stage of the data consuming small amount

of computational time.

24

The combination of collecting dense parallel 2-D surface ER lines and processing them with the new

and faster 3-D resistivity inversion algorithm comprises a step forward to the routinely application of

the 3-D surface resistivity tomography in the detection of buried archaeological structures. Moreover

the basic idea of the proposed algorithm, for calculating only the significant part of the Jacobian

matrix, can be extended for the faster inversion of other low frequency electrical measurements, like

induced polarization (IP) and spectral induced polarization (SIP). The further generalization of the

proposed technique in other massive geophysical inversion problems (e.g. magnetic, gravity, EM)

comprises a challenging future research topic.

ACKNOWLEDGEMENTS

The first author would like to express his acknowledgments to the Greek State Scholarships Foundation

(GSSF) for their financial support of this work. This research was also supported by the Basic Research

Project of the Korea Institute of Geoscience and Mineral Resources funded by the Ministry of

Knowledge Economy of Korea.

25

APPENDIX

The system of linear equations

T T Tλ d dJ J C C x J y= (A.1)

is equivalent to the least squares solution of the augmented rectangular system

.,0λ

dd

d

=y y F xJ y

xC

= (A.2)

This equivalence can be proven with the following procedure:

If we use the matrices A and b to represent the augmented matrices λ

J

C and

0

d

y respectively,

then the normal equations assuming the matrices A and b will be

TT d A A x A b. (A.3)

The terms TA A και TA b are equal to

T T TT T

JA A J C J J C C

C (A.4)

dd

0T TT T

yA b J C J y. (A.5)

The combination of equations (A.3), (A.4) and (A.5) leads to equation (A.1).

26

REFERENCES

Clark, A. 1990. Seeing beneath the soil-Prospecting methods in Archaeology, B.T. Batsford Ltd,

London.

Chambers, J.E, Ogilvy, R.D., Kuras, O. and Meldrum, P.I. 2002. 3-D Electrical Imaging of Known

Targets at a Controlled Environmental Test Site. Environmental Geology, 41, 690-704.

Constable, S.C., Parker, R.L. and Constable, C.G. 1987. Occam’s inversion: A practical algorithm for

generating smooth models from electromagnetic sounding data. Geophysics, 52, 3, 289-300.

Dabas, M., Tabbagh, A. and Tabbagh, J. 1994. 3-D Inversion in Subsurface Electrical Surveying-I.

Theory. Geophys. J. Int., 119, 975-990.

Diamanti, N.G, Tsokas, G.N., Tsourlos, P.I. and Vafidis, A. 2005. Integrated Interpretation of

Geophysical Data in the Archaeological Site of Europos (Northern Greece). Archaeological

Prospection, 12, 79-91.

Ellis, R.G. and Oldenburg, D.W. 1994. The Pole-Pole 3-D DC-Resistivity Inverse Problem: A

Conjugate Gradient Approach. Geophys. J. Int. 119, 187-194.

Gharibi, M. and Bentley, L.R. 2005. Resolution of 3-D Electrical Resistivity Images from Inversions of

2-D Orthogonal Lines. JEEG, 10, 4, 339-349.

Günther, T., Rücker, C. and Spitzer, K. 2006. Three-dimensional Modelling and Inversion of DC

Resistivity Data Incorporating Topography-II. Inversion. Geophys. J. Int., 166, 506-517.

Lanczos, C. 1950, An iteration method for the solution of the eigenvalue problem of linear differential

and integral operators, J. Res. Nat. Bur. Standards, 45, 255-282.

LaBrecque, D.J., Miletto, M., Daily, W., Ramirez, A. and Owen, E. 1996. The effects of noise on

Occam's inversion of resistivity tomography data, Geophysics, 61, 538-548.

Lesur, V., Cuer, M. and Straub, A. 1999. 2-D and 3-D Interpretation of Electrical Tomography

Measurements, Part 2: The Inverse Problem. Geophysics, 64, 396-402.

27

Loke, M.H. and Barker, R.D. 1996. Practical techniques for 3-D resistivity surveys and data inversion.

Geophysical Prospecting, 44, 499-524.

Marescot, L., Lopes, S.P., Rigobert, S. and Green A.G. 2008. Nonlinear inversion of geoelectric data

acquired across 3-D objects using a finite-element approach. Geophysics, 73, 3, F121–F133.

Mauriello, P, Monna, D. and Patella, D. 1998. 3-D Geoelectrical Tomography and Archaeological

Applications. Geophysical Prospecting, 46, 543-570.

McGillivray, P., and Oldenburg, D. 1990. Methods for calculating Frechet derivatives and sensitivities

for the non-linear inverse problem: A comparative study: Geophysical Prospecting, 38, 499-524.

Nolet, G. 1983. Inversion and resolution of linear tomographic systems. EOS, Trans., Am. Geoph.

Union, 64, 775-776.

Nolet, G. 1985. Solving or resolving inadequate and noisy tomographic systems. Journal of

Computational Physics, 61, 463-482.

Nolet, G. and Snieder, G. 1990. Solving large linear inverse problems by projection, Geoph. J. Int.,

103, 565-568.

Osella, A., Vega M. and Lascano, E. 2005. 3-D electrical imaging of archaeological sites using

electrical and electromagnetic method. Geophysics, 70, 4, 101-107.

Paige, C. and Saunders, M., 1982. LSQR: An Algorithm for Sparse Linear Equations and Sparse Least

Squares. ACM Transactions on Mathematical Software, 8, 43-71.

Pain, C.C., Herwanger, J.V., Worthington, M.H. and de Oliveira, C.R.E. 2002. Effective

Multidimensional Resistivity Inversion using Finite Element Techniques. Geophys. J. Int., 151, 710-

728.

Papadopoulos, N. G., Tsourlos, P., Tsokas, G. N. and Sarris, A. 2006. Two-dimensional and three-

dimensional resistivity imaging in archaeological site investigation. Archaeological Prospection, 13,

163-181.

28

Papadopoulos, N.G., Tsourlos, P., Tsokas, G.N. and Sarris, A. 2007. Efficient ERT Measuring and

Inversion Strategies for 3-D Imaging of Buried Antiquities. Near Surface Geophysics, 5, 6, 349-362.

Papazachos, C.B. and Nolet, G. 1997. Non-Linear Arrival Time Tomography. Annali di Geophysica,

XL, 1, 85-97.

Park, S.K. and Van, G.P. 1991. Inversion of Pole-Pole Data for 3-D Resistivity Structures Beneath

Arrays of Electrodes. Geophysics, 56, 951-960.

Petrick, W.R. Jr., Sill, W.R. and Ward, S.H. 1981. Three-dimensional Resistivity Inversion using

Alpha Centers. Geophysics, 46, 1148-1162.

Pidlisecky, A., Haber, E. and Knight, R. 2007. RESINVM3-D: A MATLAB 3-D resistivity inversion

package. Geophysics, 72, 2, H1–H10.

Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P. 1992. Numerical Recipes in C: The art

of Scientific Computing. Second Edition, Cambridge University Press.

Pridmore, D.F., Hohmann, G.W., Ward, S.H. and Sill, W.R. 1981. An Investigation of Finite-Element

Modelling for Electrical and Electromagnetic Data in Three Dimensions. Geophysics, 46, 1009-1024.

Sambridge, M.S., 1990. Non-linear arrival time inversion: constraining velocity anomalies by seeking

smooth models in 3-D, Geophys. J. Int., 102, 653-677.

Sarris, A., Papadopoulos, N., Trigkas, V., Lolos, Y., and Kokkinou E. 2007. Recovering the Urban

Network of Ancient Sikyon Through Multi-component Geophysical Approaches. CAA Berlin, April,

2-6, 2007.

Sasaki, Y. 1994. 3-D Inversion using the Finite Element Method. Geophysics, 59, 1839-1848.

Scales, J. 1987, Tomographic inversion via the Conjugate Gradient method, Geophysics, 52, 179-185.

Shima, H. 1992. 2-D and 3-D Resistivity Image Reconstruction using Crosshole Data. Geophysics, 57,

1270-1281.

29

Stummer, P., Maurer, H. & Green, A.G., 2004. Experimental design: Electrical resistivity data sets that

provide optimum subsurface information, Geophysics, 69, 120–139.

Tarantola, A., 1987. Inverse Problem Theory, 613pp., Elsevier, Amsterdam.

Tikhonov, A.N., Leonov, A.S. and Yagola, A.G. 1998. Non-Linear Ill-Posed Problems. Chapman and

Hall, London, UK.

Tsourlos, P. and Ogilvy, R. 1999. An algorithm for the 3-D Inversion of Tomographic Resistivity and

Induced Polarization data: Preliminary Results. Journal of the Balkan Geophysical Society, 2, 2, 30-45.

Vafidis, A., Sarris, A., Sourlas, G., Ganiatsos, Y. 1999. Two and three-dimensional electrical

tomography investigations in the archaeological site of Itanos, Crete, Greece. Second Balkan

Geophysical Congress and Exhibition, Istanbul, Turkey, July, 5-9, 1999.

Van der Sluis, A. and Van der Vorst, H.A. 1987, Numerical solution of large, sparse algebraic systems,

arising from tomographic problems, in Seismic Tomograph (ed. G. Nolet), Reidel, Hingham, MA, pp.

49-84.

Van der Sluis, A. and Van der Vorst, H.A. 1990, SIRT- and CG-type methods for the iterative

solutions of sparse linear least-squares problems, Linear Algebra Appl., 130, 257-302.

Yao, Z.S., Roberts, R.G. and Tryggvason, A. 1999. Calculating Resolution and Covariance Matrices

for Seismic Tomography with the LSQR Method. Geophys. J. Int., 138, 886-894.

Yi. M.J., Kim, J.H., Song, Y., Cho, S.J., Chung, S.H. and Suh, J.H. 2001. Three-Dimensional Imaging

of Subsurface Structures using Resistivity Data. Geophysical Prospecting, 49, 483-497.

30

HOMOGENEOUS EARTH INHOMOGENEOUS EARTH

0 3 6 9 12 15X (a)

0

3

6

9

12

15

Y (

a)

0 3 6 9 12 15X (a)

0

3

6

9

12

15

Y (

a)

Z = (0.0 - 0.5)a Z = (0.5 - 1.0)a

0 3 6 9 12 15X (a)

0

3

6

9

12

15

Y (

a)

0 3 6 9 12 15X (a)

0

3

6

9

12

15

Y (

a)

Z = (1.0 - 1.5)a Z = (1.5 - 2.0)a

A.

0 3 6 9 12 15X (a)

0

3

6

9

12

15

Y (

a)

0 3 6 9 12 15X (a)

0

3

6

9

12

15

Y (

a)

Z = (0.0 - 0.5)a Z = (0.5 - 1.0)a

0 3 6 9 12 15X (a)

0

3

6

9

12

15

Y (

a)

0 3 6 9 12 15X (a)

0

3

6

9

12

15

Y (

a)

Z = (1.0 - 1.5)a Z = (1.5 - 2.0)a

B.

0.00

0.02

0.04

0.06

Sensitivity

Figure 1: Three-dimensional (3-D) sensitivity distribution (absolute values) for a pole-pole array over

A) a 10 Ohm-m homogeneous half-space B) an inhomogeneous model consisted of a 100 Ohm-m

prismatic body (hatched polygon) embedded in a 10 Ohm-m half-space. The white dots indicate the

surface position of the current (left) and potential (right) electrodes.

31

0 200 400 600 800 1000 1200 1400 1600Parameters (N)

10-6

10-5

10-4

10-3

10-2

10-1

Sens

itivi

ty (

JR)

Nsep = 1a

0 200 400 600 800 1000 1200 1400 1600

b elementThreshold Value JR(b)

10-6

10-5

10-4

10-3

10-2

10-1

Figure 2: Schematic presentation of the bth index number for the N parameters of the ascending sorted

absolute sensitivity values of a Jacobian matrix row, which corresponds to a central measurement with

Nsep = 1a. The threshold value JR(b) which corresponds to the bth entry of the sorted row is also shown.

The b parameter is given by the equation b = (integer) (r. N). The variable r controls which bth element

of the sorted row will be chosen.

32

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1r

0

10

20

30

40

50

60

70

80

90

100

% J

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

10

20

30

40

50

60

70

80

90

100

Figure 3: Relation of the variable r and the percentage of the Jacobian matrix elements (%J0) that the

algorithm assumes zero for a specific electrode geometry (Pole-Pole array, Nsep(max) = 4a, 20x20 =

400 electrodes, 1400 measurements, 1444 parameters). The variation of %J0 with respect to r gradually

converges to the line %J0 = 100r for r larger than 0.2-0.3. It can be seen that 64% of the Jacobian

matrix elements will not be calculated for a value of the variable r = 0.6.

33

Jacobian Time Calulation

5.7 11.8

55.9

199.3

311.8

1.6 3.4 7.0

41.8

109.5

206.7

22.9

123.3

578.7

68.9

18.1

0

50

100

150

200

250

300

350

400

450

500

550

600

DS-1 DS-2 DS-3 DS-4 DS-5 DS-6 DS-7 DS-8

Data Sets

Tim

e (m

in)

Original Algorithm

New Algorithm

-70% -68%

-66%

-71%-72%

-65%

-65%

-64%

A

Matrix Inversion Time

0.04 0.09

0.54

2.17

3.49

0.03 0.06 0.14

0.90

2.43

4.65

6.66

1.29

0.200.37

1.51

0

1

2

3

4

5

6

7

DS-1 DS-2 DS-3 DS-4 DS-5 DS-6 DS-7 DS-8Data Sets

Tim

e (m

in)

Original Algorithm

New Algorithm

-31%

-31%

-31%

-31%-29%

-31%

-30%

-30%

B

Toral Inversion Time per Iteration

5.9 12.2

57.2

204.0

318.9

1.7 3.8 7.6

44.1

115.5

216.3

23.6

126.0

590.3

73.0

19.3

0

100

200

300

400

500

600


Tim

e (m

in)

Original Algorithm

New Algorithm

-68% -66%

-65%

-69%-70%

-64%

-64%

-63%

C

Memory

103162

436

1013

1373

76 110 156

387

659

954

2099

734

245 250

508

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2200


Byt

es (

x106 )

Original Algorithm

New Algorithm

-37%

-43%

-47%

-32%-26%

-50%

-52%

-55%

D

Figure 4: Relative percent decrease of the computational time needed by the original and the new algorithms respectively: A) to calculate the Jacobian matrix, B) to invert the matrix for the updated resistivity model, C) Representation of the time needed by the original and the new algorithm to complete one iteration. D) Memory that the two algorithms occupy. The value r = 0.6 was considered for the new algorithm. Table 1 holds the details for the data sets (DS) used in this comparative study.

34

0

1

2

3

4

0 10 20 30 40 50

LSQR steps (m)

Lin

ear

mis

fit

||r m||

Iteration 1Iteration 2Iteration 3Iteration 4

A

0

4

8

12

16

20

24

0 10 20 30 40 50

LSQR steps (m)

|| dx

m ||


C

0

4

8

12

16

20

24

0 10 20 30 40 50

LSQR steps (m)

Non

line

ar m

isfi

t


B

Figure 5: Variation of the linear misfit (A), the non-linear misfit (B) and the norm of the resistivity

correction vector (C) for m=50 LSQR steps, corresponding to four non-linear iterations of the inversion

procedure

35

Calculate the %RMS

YESENDStore Data

1. Read apparent resistivity data2. Number of electrodes and max Nsep 3. Generate mesh of 3D parameters4. Generate Smoothness matrix Store non-zero values5. Set initial resistivity model6. Set initial value for Lagrangian multiplier

INITIAL STEPS

From k = 1 to kmax iterationsSTART ITERATIONS

Is one of the stoppingcriteria satisfied?

Calculate model data3D Finite Element

Jacobian Matrixcalculation

Store non-zero values

NOThreshold Dimensions Χ, Υ, Ζ of

Cuboid Sensitivity Rectangular

3-D INVERSION ALGORITHM

LSQR

If k = 1 => LSQR steps m = 20If k = 2 => LSQR steps m = 10if k > 3 => LSQR steps m = 5

Find ResistivityCorrection vector

Update Resistivity Modeldxkxk+1 xk= +

Adjust Lagrangian multiplier

B)

dxk

If k < 3 => λ = λ /2If k > 3 => λ = λ

k k-1k k-1

ALGORITHM OF CALCULATING THETHRESHOLD DIMENSIONS Χ, Υ, Ζ

Calculate Threshold DimensionsΧ, Υ, Ζ of the Critical Sensitivity

Cuboid for every Central Measurement with Nsep=1a, 2a,...,Nmax a

INITIAL STEPS

1. Read apparent resistivity data2. Number of electrodes and max Nsep 3. Generate 3D parameter mesh

Calculate the entireJacobian MatrixAdjoint equation techniqueHomogeneous earth

1. Choose central measurements with Nsep = 1a, 2a,..., Nmax a

2. Determine the variable value b = (integer) ( r N ), r = 0.6

3. Calculate the threshold values J(b) for every central measurement

A)

Create Augmented System

Jk

Cλk 0

y-F(x )kdxk=

Figure 6: A) Flowchart of the algorithm for the calculation of the threshold dimensions X, Y, Z of the

critical sensitivity cuboid. B) Flowchart of the proposed 3-D resistivity inversion algorithm.

36

Z=0.00-0.25 m

Z=0.25-0.50 m

Z=0.50-0.75 m

Z=0.75-1.00 m

Z=0.00-0.25 m

Z=0.25-0.50 m

Z=0.50-0.75 m

Z=0.75-1.00 m

Z=0.00-0.25 m

Z=0.25-0.50 m

Z=0.50-0.75 m

Z=0.75-1.00 m

Z=0.00-0.25 m

Z=0.25-0.50 m

Z=0.50-0.75 m

Z=0.75-1.00 m

Z=0.00-0.25 m

Z=0.25-0.50 m

Z=0.50-0.75 m

Z=0.75-1.00 m

Z=0.00-0.25 m

Z=0.25-0.50 m

Z=0.50-0.75 m

Z=0.75-1.00 m

Z=0.00-0.25 m

Z=0.25-0.50 m

Z=0.50-0.75 m

Z=0.75-1.00 m

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

10 Ohm-m

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9

X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9

X (m)

0123456789

Y (

m)

B) ORIGINAL ALGORITHM

100 Ohm-m

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

Ohm-m

Resistivity

r = 0.9%J = 87.7A) MODEL

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)0 1 2 3 4 5 6 7 8 9

X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

Z=0.00-0.25 m

Z=0.25-0.50 m

Z=0.50-0.75 m

Z=0.75-1.00 m

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0r = 0.8

%J = 79.20r = 0.7

%J = 69.60r = 0.6

%J = 63.60r = 0.5

%J = 58.50r = 0.4

%J = 52.50

Z=0.00-0.25 m

Z=0.25-0.50 m

Z=0.50-0.75 m

Z=0.75-1.00 m

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

0 1 2 3 4 5 6 7 8 9X (m)

0123456789

Y (

m)

r = 0.3%J = 46.30

C) NEW ALGORITHM

C1) C2) C3) C4) C5) C6) C7)0 1 2 3 4 5 6 7 8 9

X (m)

0123456789

Y (

m)

X-Survey

10

20

30

40

50

60

70

Figure 7: A) Model 1 used in the synthetic tests. B) 3-D resistivity inversion model from the original algorithm. C) 3-D resistivity inversion models from

the new algorithm for various values of r. The percentage of the Jacobian matrix elements (%J0) that are not calculated is also shown for each values of r.

The initial value for the Lagragne multiplier was set to 0.01 for all inversions.

37

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

Z = 0.0-0.5 m Z = 0.5-1.0 m Z = 1.0-1.5 m

Z = 1.5-2.0 m Z = 2.0-2.5 m Z = 2.5-3.0 m

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8Y

(m

)

Z = 0.0-0.5 m Z = 0.5-1.0 m Z = 1.0-1.5 m

Z = 1.5-2.0 m Z = 2.0-2.5 m Z = 2.5-3.0 m

500 Ohm-m

100 Ohm-m

A. 3-D MODEL

B. 3-D INVERSION (Original Algorithm)

Ohm-m

10 Ohm-m

B1

B3

B1

B2 B4

B4

C. 3-D INVERSION (New Algorithm, = 0.6)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

0 2 4 6 8 10 12 14 16 18X (m)

0

2

4

6

8

Y (

m)

Z = 0.0-0.5 m Z = 0.5-1.0 m Z = 1.0-1.5 m

Z = 1.5-2.0 m Z = 2.0-2.5 m Z = 2.5-3.0 m

r 10

70

130

190

250

310

370

430

490Resistivity

Figure 8: A) Model 2 used in the synthetic tests.. B) Inverted 3-D electrical model using the original

algorithm (Dipole-Dipole array, 10 iterations, %RMS=2.0, λ0 = 0.05). C) 3-D inverted model using the

new inversion algorithm (Dipole-Dipole array, 10 iterations, %RMS=2.5, λ0 = 0.05, r = 0.6, %J0=56.3).

38

Figure 9: A) Twin Probe array apparent resistivity map from the archaeological site of Sikyon. B)

Connection settings of the RM15 resistance meter, the MPX15 multiplexer and the PA5 multi probe

frame. The inter-electrode distance A-M1, A-M2, A-M3 and A-M4 is 0.5, 1, 1.5 and 2 meters,

respectively. A and M stands for the current and the potential electrode respectively for the pole-pole

array. The instrument is moved along parallel 0.5 m spaced profiles in the survey area. The sampling

interval along each line is also 0.5 m. In every station the instrument simultaneously records four

measurements, which correspond to the increasing A-M distance,. By this field strategy, an area can be

covered by dense parallel 2-D pole-pole lines with maximum electrode separation Nmax = 4a (a = 0.5m

is the unit electrode spacing).

39

Z=

0.00

-0.2

5 m

Z=

0.25

-0.5

0 m

Z=

0.50

-0.7

5 m

Z=

0.75

-1.0

0 m

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

B) NEW ALGORITHM

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10Y

(m

)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

8 iterations%RMS=2.1

A) ORIGINALALGORITHM


r = 0.3, %J = 51.20

B1) 8 iterations%RMS=2.1

r = 0.4, %J = 54.20


r = 0.5, %J = 58.30


r = 0.6, %J = 60.50


r = 0.7, %J = 70.40

B5)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)


r = 0.9, %J = 89.50

B7)8 iterations%RMS=2.1

r = 0.8, %J = 79.60

B6)

Log

10 R

esis

tivi

ty

Ohm-m

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0 2 4 6 8 10 12 14X (m)

02468

10

Y (

m)

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

Figure 10: Sikyon Archaeological Site. A) Inverted 3-D resistivity model from the study area using the original algorithm. B) 3-D resistivity models form

the application of the new algorithm for various values of r. The percentage %J0 of the Jacobian matrix that is not calculated is also shown.

Papadopoulos, et al.

40

A) ORIGINAL ALGORITHM

Log10 Res.

0 2 4 6 8 10X (m)

0

2

4

6

8

10

Y (

m)

0 2 4 6 8 10X (m)

0

2

4

6

8

10

Y (

m)

0 2 4 6 8 10X (m)

0

2

4

6

8

10

Y (

m)

0 2 4 6 8 10X (m)

0

2

4

6

8

10

Y (

m)

0 2 4 6 8 10X (m)

0

2

4

6

8

10

Y (

m)

0 2 4 6 8 10X (m)

0

2

4

6

8

10

Y (

m)

0 2 4 6 8 10X (m)

0

2

4

6

8

10

Y (

m) 9 iterations

%RMS=3.8

Z = 0.00-0.25 m Z = 0.25-0.50 m Z = 0.50-0.75 m Z = 0.75-1.00 m

Z = 1.00-1.25 m Z = 1.25-1.50 m Z = 1.50-1.75 m

B) NEW ALGORITHM r = 0.6, %J = 58.10

0 2 4 6 8 10X (m)

0

2

4

6

8

10

Y (

m)

0 2 4 6 8 10X (m)

0

2

4

6

8

10

Y (

m)

0 2 4 6 8 10X (m)

0

2

4

6

8

10

Y (

m)

0 2 4 6 8 10X (m)

0

2

4

6

8

10

Y (

m)

0 2 4 6 8 10X (m)

0

2

4

6

8

10

Y (

m)

0 2 4 6 8 10X (m)

0

2

4

6

8

10

Y (

m)

0 2 4 6 8 10X (m)

0

2

4

6

8

10

Y (

m) 7 iterations

%RMS=3.8

Z = 0.00-0.25 m Z = 0.25-0.50 m Z = 0.50-0.75 m Z = 0.75-1.00 m

Z = 1.00-1.25 m Z = 1.25-1.50 m Z = 1.50-1.75 m

1

1.2

1.4

1.6

1.8

2

2.2

Ohm-m

Figure 11: Europos Archaeological site: A) 3-D resistivity inversion model from the original

algorithm. The schematic interpretation of the high resistivity anomalies is overlaid on the

inversion model. B) Final 3-D resistivity model from the new algorithm for r = 0.6 (The

initial value for the Lagragne multiplier was λ0 = 0.1 for both inversions).

3-D Resistivity Inversion

41

Electrodes Measurements Parameters %J0

DS-1 169 (13x13) 2028 1152 67DS-2 225 (15x15) 3150 1568 72DS-3 289 (17x17) 4624 2048 68DS-4 400 (20x20) 7600 2888 71DS-5 529 (23x23) 11638 3872 68DS-6 625 (25x25) 15000 4608 69DS-7 729 (27x27) 18954 5408 67DS-8 900 (30x30) 26100 6728 68

Table 1: Data sets (DS-1 to DS-8) used for the comparative study between the original and the

new algorithms. All the possible electrode combinations along 2D orthogonal lines were

created using the pole-pole array. The third and fourth columns show the number of

measurements and parameters respectively for the corresponding electrode geometry of the

second column. The grid dimensions are also shown in the parenthesis. The percentage of the

Jacobian matrix elements (%J0) that are not calculated by the new algorithm assuming r =

0.6is shown in the fifth column.

an algorithm for fast 3-d inversion of surface ert...

Documents