SEMI-INFINITE-GEOMETRY BOUNDARY-PROBLEM FOR LIGHT MIGRATION IN HIGHLY SCATTERING MEDIA - A FREQUENCY-DOMAIN STUDY IN THE DIFFUSION-APPROXIMATION

We have studied light migration in highly scattering media theoretically and experimentally, using the diffusion approximation in a semi-infinite-geometry boundary condition. Both the light source and the detector were located on the surface of a semi-infinite medium. Working with frequency-domain spectroscopy, we approached the problem in three areas: (1) we derived theoretical expressions for the measured quantities in frequency-domain spectroscopy by applying appropriate boundary conditions to the diffusion equation; (2) we experimentally verified the theoretical expressions by performing measurements on a macroscopically homogeneous medium in quasi-semi-infinite-geometry conditions; (3) we applied Monte Carlo methods to simulate the semi-infinite-geometry boundary problem. The experimental results and the confirming Monte Carlo simulation show that the diffusion approximation, under the appropriate boundary conditions, accurately estimates the optical parameters of the medium.


INTRODUCTION
Light propagation in turbid media is described by transport theory, also called the theory of radiative transfer.", 2 The Boltzmann transport equation, which is a balance relationship, treats light propagation as the transport of photons through a medium containing particles. In most practical cases the equation of transfer cannot be solved exactly. Often it is necessary to consider an approximate approach. One of these simplified approaches is the diffusion approximation, 3 5 which is valid in the strongly scattering regime. 6 The observed optical properties of most biological tissues 7 are typified by a scattering coefficient that far exceeds the absorption coefficient. A number of studies employed the diffusion theory to investigate the optical properties of tissues. These studies used steady-state spectroscopyI' 0 and time-resolved spectroscopy" in both the time domain1 2 '1 4 and the frequency domain.1 5 We present a frequency-domain study of the applicability of the diffusion approximation to the case of a semiinfinite geometry. Both the light source and the detector are placed at the interface between air and a strongly scattering medium; the interface extends indefinitely. The proper solution of this boundary problem has important practical implications because it represents a reasonable model for in vivo, noninvasive applications of light spectroscopy in medicine. When the light source and the detector are placed on a surface separating two media with different optical properties, the diffusion approximation is not rigorously applicable.1 6 Nevertheless, the diffusion approximation has been applied to predict the time-domain and steady-state response in the reflectance geometry from quasi-semi-infinite tissues.10"1 2 We derive the expression for the frequency-domain photon fluence rate and verify its equivalence with the corresponding ex-pression derived in the time domain. 1 2 Experimentally, we test the expression's level of accuracy by performing a systematic study on a macroscopically homogeneous tissuelike phantom. Since the diffusion theory is highly accurate in predicting the results of experiments performed in an infinite geometry, 17 '1 9 we compare our results obtained in the semi-infinite geometry (i.e., at the surface of the medium) with the results of the measurements conducted deep inside the bulk medium (i.e., in the infinite geometry). The comparison of experimental results is carried out for a wide range of values of /ua and ,-l. A Monte Carlo simulation of the boundary problem has been performed.

THEORY
The distribution of photons in random media is described by the angular photon density u(r, Qk, t), which is defined so that u(r, f, t)d 3 rdfl is the expected number of photons in d 3 r around r moving in direction fl in solid angle dil at time t. The temporal evolution of the angular photon density in a medium where the processes of absorption and elastic scattering take place is given by the Boltzmann transport equation 4 : u_ v -Vuv(/La + a, 8 )u + j dQ 'v/ 8 p,(Q' fl) at J~l X U(r, 1I', t) + q(r, Q, t), (1) where v is the speed of photons in the medium (and v its modulus), v/a and vlu are the rates of absorption and scattering, respectively, p,(fl I -LI) is the normalized probability for scattering events that carry photons from I' into l, and q is the photon source term. The Boltzmann transport equation is an integrodifferential equation containing both time and spatial derivatives, and 0740-3224/94/102128-11$06.00 ©1994 Optical Society of America its solution requires initial and boundary conditions for u(r, L, t).
In the multiply scattering regime the usual simplification is the diffusion approximation. The approximation assumes that the angular photon flux, defined as ,t(r, LI, t) vu(r, l, t), is quasi-isotropic 3 5 : where T (r, t) f 4 dLQ (r, L, t) is the total photon flux and J(r, t) a f 4 at v dt + VU + ( + As')J = q (4) v at 3 where A,' [defined as (1 -g)ut with g the average cosine of the scattering angle] is the transport scattering coefficient and q 0 and q, are defined by introduction of the following expansion of the angular dependence of the source: If we assume that the photon source is isotropic (q, = 0) and neglect the time derivative of J, which is equivalent to saying that the variations of J occur on a time scale much larger than the time between photon collisions with the scattering particles of the medium, Eq. (4) yields where D = 1(3Aa + 3't) is the diffusion constant. Finally, by using expression (6) for J, we can rewrite Eq. (3) in the form of the photon-diffusion equation: It is important to be clear about the limitations of the diffusion equation. As is discussed, its derivation requires the following approximations: It has been shown that the photon-flux quasi-isotropy condition is well satisfied 6 16 (al) In strongly scattering media (a << L) (a2) Far from boundaries, (a3) Far from sources, where "far" in conditions (a2) and (a3) refers to distances much greater than the photon mean free path. In frequency-domain spectroscopy the intensity of the light source is modulated at a frequency (/21r) typically of tens to hundreds of megahertz, so the photon density is written as where is the phase lag between the source (located at r = 0) and the detector (located at a distance r >> 1/ug' from the source) and x is defined as c/vLa, with v the speed of light in the medium (given by c/n, with n being the index of refraction of the medium). Equations (8)- (10) have been experimentally verified"19-21 and provide a good description of the homogeneous infinite medium problem in the multiply scattering regime.
In most medical applications the method for noninvasive, in vivo spectroscopy measurements is to place both the light source and the detector upon the surface to be examined. It is evident that the infinite-medium scheme is not appropriate for such a geometry. A better approach is to consider a uniform semi-infinite medium and to solve the diffusion equation [Eq. (7)] with the appropriate boundary conditions. Before proceeding, we note that the problem itself is beyond the limits of the diffusion approximation: both source and detector are placed on the boundary, where, as discussed, the diffusion equation does not approximate the transport equation as well as it does deep in the medium. However, it is still a reasonable starting point to treat the problem,1 2 even if the final results must be critically analyzed to verify the extent of acceptability. The validity of the diffusion approximation can be quantified by evaluation of the ratio between the isotropic and the directional photon flux. This ratio should be much greater than 1, as is required by relation (2). In the homogeneous infinite medium, where the diffusion approximation yields accurate results, for typical values of the physical parameters of tissue in the near infrared (a = 0.05 cm-', /utL = 15 cm-', r = 3 cm, v = 2.26 X 1010 cm/s, corresponding to an index of refraction of n = 1.33, and cs = 2ir X 120 MHz) such a ratio is The physical boundary condition required at a vacuum interface is that there be no incoming photons at the boundary. 4 Apparently at the vacuum boundary the diffusion approximation breaks down. The photon flux is nonzero only on half of the range of the solid angle, and the quasi-isotropy condition is not satisfied. On the other hand, a mismatch of the index of refraction at the interface of the strongly scattering medium and the outside nonscattering medium accounts for an inwardly directed component of the photon flux at the boundary. The boundary condition for the mismatch semiinfinite medium can be satisfied when the density of photons U is equal to 0 on an extrapolated boundary at a distance Zb = 2aD, where a is a constant that is related to the relative index of refraction (nrel) of the two media. 22 ' 23 The distance Zb for nrel = 1.33 (or nrel = 1.4, which is a typical value for a tissue-air interface in the red-near-infrared spectral region 2 4 ) and for typical values of D in tissues is -0.15 cm. Furthermore, it has been shown that a light beam incident upon the surface can be well represented by a single scatter source at a depth zo equal to one effective photon mean free path 0 " 2 [i.e., Zo = 1/( /La + ui4)]. This parameter zo has a value of -0.1 cm in tissues. We observe that this feature accounts for an effective isotropic photon source even if the photons are actually injected in a single direction. Finally, the boundary problem of setting U = 0 on the extrapolated boundary can be treated by introduction of a negative image source of photons above the plane, one that is symmetric with respect to the actual photon source. 2 5 This approach enables one to take advantage of the solution that is valid for the infinite medium. In the semi-infinite-medium model, which is pictorially represented in Fig. 1, the diffusion equation [Eq. (7)] is used with qo = qaq (where a stands for the actual source and i stands for the image) to yield the solution obeying the required boundary conditions in the space z 2 Zb. The solution, by application of the superposition principle, can immediately be written from expressions (8)-(10): where, with the notation introduced in Fig. 1, The new coordinate p is the projection of the sourcedetector distance ra on the interface plane = Zb. The detector coordinate z is at Zb Z Zb + ZO. Assuming that 1 >> (Zb + Zo ± z) 2 /p 2 , in Eq. (12) we carry over expansions to the second order in (Zb + zo ± z)/p. After the necessary calculations, using Eqs. (12), (2), and (6), we find that the dc and ac photon flux along -z (in Fig. 1 the detector fiber receives photons in an inward direction -z) and the phase lag 0s between source and detector are given by the following relationships: z Fig. 1. Semi-infinite-geometry model: Zb is the distance between the extrapolated boundary and the surface of the medium and zo is the depth of the effective single scatter source inside the scattering medium. The strongly scattering medium extends in the space > Zb. The detector optical fiber, which is parallel to the z axis, is immersed in the scattering medium at a depth z ranging from Zb to Zb + zo. The distance between the effective (image) photon source and the tip of the optical fiber is ra (ri).
The projection of the source-detector distance onto the plane z = is p.
In tissues, conditions (16) and (17) are better satisfied than condition (18). For the previously mentioned tissue's optical properties in the near-infrared, the quantities on the left-hand sides of inequalities (16) and (17) are -0.001, and that on the left-hand side of inequality (18) is 0.01.
We have compared our result for the frequency-domain quantities with the expression for the time-domain reflectance in the half-space geometry obtained by Patterson et al.1 2 Since the same boundary conditions have been applied, the two solutions should be related by a Fourier transform with respect to time. We have verified the correspondence of the two results in the limiting case j IJ(p, t)Jexp(iwjt)dt = IJ(p, )l. (19) 1+ ( 2D 1) To verify experimentally the solutions found for the semi-infinite geometry and to use the measurement protocol described in a previous paper, 9 we rewrite Eqs. (13)- (15) to obtain quantities that show a linear dependence on p: where the p-dependent functions FdC, Fac, and F 0 and the p-independent functions Gdc and Gac are defined by Eqs. (13)- (15). The determination of the slopes of the straight lines allows one to recover the values of /La and A/ ofthe medium. That the arguments ofthe logarithms are not dimensionless does not present a problem as far as the slopes of the straight lines are concerned. In fact the particular choice of the units introduces a constant, which does not effect the slopes. We also observe that the particular values of the parameters of the model (namely, Zb, zo, and z) have no influence on the slopes of the lines (z has no effect on their intercept either). This property is important, because the parameters Zb and zo depend on the optical properties of the medium, namely, on /L, ' and n, and the positioning of the tip of the detector optical fiber (which is related to the parameter z) is in practice not exactly reproducible.
We conclude this theoretical section by observing that the isotropy factor defined by Eq. (11), for the same values of the parameters considered in Eq. (11), has a minimum value of -2, which is marginally acceptable compared with the value of 8 in the infinite geometry.
This result indicates that the isotropic term is larger than the directional flux but is not much larger as required by the diffusion approximation. We evaluate the level of accuracy of the semi-infinite-medium expression (13) by performing a series of measurements in a macroscopically homogeneous, strongly scattering medium and by a Monte Carlo simulation.

EXPERIMENTAL METHODS
The experimental arrangement, shown in Fig. 2, is typical for frequency-domain spectroscopy. The light source, a diode laser (Sony SLD104AU) emitting at a wavelength of 780 nm, is intensity modulated at a frequency of 120 MHz by being supplied with the sine-wave output of a frequency synthesizer (Marconi 2022A) by means of an rf amplifier (ENI Model 403 LA). The average emitted light power is -3 mW. In our measurements the laser diode is directly immersed in the medium, and the detected light is collected by a bundle of optical fibers (overall diameter 3 mm) and delivered to a photomultiplier tube (PMT) (Hamamatsu R928). The gain function of the PMT is modulated at a frequency of 120.00008 MHz, which is slightly different from the modulation frequency of the light source. The small frequency difference, which is selectable, produces beating between the detected signal and the gain function of the PMT, giving rise to a signal at the difference frequency (80 Hz in our case), which is sent to a computer card. A digital acquisition technique 2 6 and a fast-Fourier-transform algorithm provide the phase shift relative to a reference signal (), the average intensity (dc), and the amplitude of the intensity oscillations (ac) of the detected light. The signal used as a reference for the phase measurement is a synchronous (with the frequency synthesizer) clock generated by the computer.
The multiply scattering medium used in our measurement is an aqueous solution of Liposyn III 10% [Abbott Laboratories (IL)]. It is an intravenous fat emulsion that was previously used as tissuelike phantom in both steady-state1 0 and time-resolved spectroscopy. 27 We studied four different Liposyn concentrations to test the theoretically derived results in a range of values for A,'. The concentrations employed are 4.5%, 9.0%, 13.5%, and 18% by volume, which correspond to a solids content of 0.45%, 0.90%, 1.35%, and 1.80%, respectively. Consequently the transport scattering coefficient L,' should range from -4 to 16 cm'1, as we verified with measurements in the infinite medium before performing the surface experiment. The aqueous Liposyn solution acts as the host medium, diluting the absorbing substance. For such a substance we chose black India ink, which is soluble in water. We measured its absorption spectrum in a nonscattering regime at the wavelength of the diode laser  sult we decided to increase the concentration of absorber at steps of 0.2 mL/L to increase /La by 0.014 cm'1 per step. After the first 10 steps we increased the amount of absorber added between successive measurements. We performed 24 measurements in 24 different conditions of scatterer and absorber contents. First we increased the transport scattering coefficient (measurements 1-4, ,s' ranging from -4 to 16 cm'l); then we increased the absorption coefficient without changing the scatterer solids content (measurements 5-24, /La ranging from -0.026 to 0.4 cm-'). The solution was held in a cylindrical container (22 cm in diameter by 13 cm in height).
Our measurement protocol consists of two series of measurements for each Liposyn-black-India-ink solution. The first series is conducted in the quasi-infinite geometry (shortened to infinite geometry in what follows), in which both the light source and the detector optical fiber are deeply immersed into the medium (at a depth of -5 cm). The second series is performed in the quasi-semi-infinite geometry (shortened to semi-infinite geometry), in which both the light source and the detector optical fiber are positioned on the surface of the medium. In each one of the two series of measurements we collect data at 5-8 different source-detector separations, ranging from a minimum of 1.6 cm to a maximum of 5.4 cm. The different source-detector separations are accomplished by means of a raster scanning device (Techno XYZ positioning table), which moves the light source with respect to the fixed-detector optical fiber. The uncertainty in the variations of r (or p) is -10 ,um. The experimental configurations in the infinite and semi-infinite geometries are sketched in Fig. 2. We observe that the source-detector separations are measured from the emitting point of the laser diode to the center of the detector fiber bundle. The effect of the finite size of the detector fiber (3 mm in diameter) on the measured values of /La and /t' is negligible when multiple source-detector distances are employed in the data analysis. We have experimentally verified that fibers with different diameters give the same values of /La and /Ls ' The measurement of dc, ac, and phase at several source-detector distances enables us to determine the slopes of the straight lines associated with dc (Sdc) ac (Sac), and phase (So). These straight lines are given by ln(rUdc), ln(rUac), and <) in the infinite geometry 2 0 and by Eqs. (20)- (22) in the semi-infinite geometry. In the infinite geometry the way to recover /Qa and A,' has been described in detail.' 9 In the semi-infinite geometry we treat the problem of recovering La and ,u&, from Eqs. (20)- (22) iteratively: First we neglect the terms containing /ta and /Ls' on the left-hand side of the equations and obtain the slopes S() S), and S(0) from which we determine L(°) and LI(). Then we use these values to obtain SM SM, and S(1) and hence AL) and 41), and we continue applying this procedure recursively until /LCi) and /uL14) reproduce themselves within a given uncertainty of 0.1%. The convergence is reached after few iterations.

EXPERIMENTAL RESULTS
On the basis of the discussion conducted in an earlier paper' 9 we have recovered /La and /ju' from the data pairs of dc and phase and ac and phase. In what follows we present only the results obtained from dc and phase data, but we note that similar results are obtained from ac and phase data.

Infinite Geometry
The values of Ls' and /La measured in the infinite geometry are plotted in Fig. 3 as a function of Liposyn and blackink concentrations. In Fig. 3(al), j' shows a linear dependence on the scatterer-solids content, in agreement with linear transport theory. 3 By contrast, L is essentially insensitive to the increase in the black-ink concentration [ Fig. 3(a2)]. In the absence of black ink, the measured value of /La for diluted Liposyn (0.026 0.001 cm-') is essentially due to water. In fact, the reported value 2 8 of ua for water at 780 nm, which is 0.023 cm-', is in good agreement with our measurement. The linear dependence of /La on black-ink concentration [ Fig. 3(b2)] is also in agreement both with the theory (/a = e[c], where e is the extinction coefficient and [c] is the chromophore concentration) and with other experiments.' 7 "1 9 The slope of the straight line calculated with the points relative to ink concentrations smaller than 2 mL/L [(64.5 ± 0.4) X 10-3 cm'1 mL' L], for which the diffusion theory provides an excellent approximation to the transport theory, is very close to the slope obtained spectrophotometrically in a nonscattering regime [(66.1 ± 0.3) X 10-3 cmnl mL'1 L]. The measured values of u,, relative to ink concentrations greater than 2 mL/L deviate from the values measured on the spectrophotometer by less than 6%. On the basis of these observations we assume that the infinite-geometry measurements provide accurate results for the optical parameters of the medium. We therefore use these results as reference values for the semi-infinite-geometry measurements

Semi-infinite Geometry
We have analyzed the surface data in two ways: (i) considering Eqs. (20)- (22), thereby taking into account the appropriate boundary conditions, and (ii) using the infinite-geometry equations (8)- (10). In these ways we quantify the corrections yielded by the application of the proper boundary conditions with respect to the infinitegeometry results. The results for I-tu' and j.t in the 24 media variations examined are shown in Fig. 4, where they may be compared with the results of the infinitegeometry measurements. We have also compared the values of the slopes related to dc (Sde), ac (Sac), and phase (S<) in the three cases considered (referred to as the infinite geometry, the semi-infinite geometry with boundary conditions, and the semi-infinite geometry without boundary conditions). This comparison, plotted in Fig. 5, provides information on the behavior of the frequencydomain parameters, namely, on their deviation from the accurate infinite-model predictions.
The sensitivity of the semi-infinite-geometry results to the positioning of the source and the detector relative to the surface plane can be evaluated by comparison of the data presented in Table 1. We measured the values of /j, and Au' for slightly different positions of the laser diode and the tip of the detector optical fiber. That is, assigning to the medium surface a coordinate = 0, we have examined two positions relative to the boundary plane, i.e., a surface position (-0) and 1 mm into the medium ( = 1). We then obtained four possible configurations for the source-detector system, that is, (0, 0), (1, 0), (0, 1), and (1, 1), where the first coordinate is relative to the source and the second is relative to the detector. Table 1 shows the results obtained for alla and juL' in the solution with 1.  The results of the Monte Carlo simulation for a modulation frequency of 120 MHz (to match the experimental modulation-frequency condition) are shown in Fig. 6 and Table 2. In Fig. 6 we show a comparison of the straight lines associated with dc, ac, and phase in the case of the infinite geometry, the semi-infinite geometry with boundary conditions, and the semi-infinite geometry without boundary conditions. In Table 2 we list the values obtained for Iua and A,' in the three cases.

Infinite-Geometry Results
The infinite-geometry results shown in Fig. 3 have been used as a framework to provide the correct values of the optical parameters in the multiply scattering medium. Several arguments have been presented above justify this designation: (i) The linear dependence of ,ut' on Lyposyn solids content; (ii) The independence of p.L,' from black-India-ink concentration; (iii) The independence of Qia from Liposyn solids content; (iv) The linear dependence, quantitatively similar to the one obtained spectrophotometrically, of ILua on black-Indiaink concentration.
Whereas conditions (i) and (iii) are certainly well satisfied, conditions (ii) and (iv) hold rigorously only for black-Indiaink concentrations smaller than -3 mL/L. However, the deviations of the measured values of /ice' and /l-a at the maximum ink concentration examined (6.1 mL/L) from the values that would satisfy conditions (ii) and (iv) are small (-6%). We neglected these deviations in comparing the semi-infinite-geometry results. From a general standpoint these deviations are a sign of the shortcomings of the diffusion approximation for higher absorption coefficients. As discussed in Section 2, the diffusion approximation requires tLs'/Itta to be much greater than 1.
The results of our measurements permit us to quantify this requirement: the values of the optical parameters of our medium are consistent with Mie theory and with spectrophotometric measurements when /i-ta > 80, and they deviate by -6% for Us'/Ma 40.

Semi-infinite-Geometry Results
The method used to recover the values of Ada and ' from the measured data is based on the determination of the slopes of the straight lines associated with dc, ac, and phase. In the semi-infinite geometry this method presents the advantage of being insensitive to the values of the distance parameters of the model, Zb, Zo, and z. This topic was discussed in Section 2 on the basis of the derived expressions for the dc, the ac, and the phase slopes. The results presented in Table 1 experimentally confirm the theoretical predictions relative to the parameter z. Therefore the combined theoretical and experimental results show that the relative index of refraction (influencing Zb) the value of the photon mean free path in the multiply scattering medium (related to z), and  line parameters considered in this paper are obtained by least-squares fits. In all cases considered the linear fits are very good; the correlation coefficients typically exceed 0.999.
The comparison of the measured values of Mua and Ms' in the three cases considered (infinite geometry, semi-infinite geometry with boundary conditions, and semi-infinite geometry without boundary conditions) is presented in Fig. 4. A more quantitative comparison is made by analysis of the deviations of the semi-infinite geometry results from the infinite-geometry results. These deviations are shown in Fig. 7. With proper boundary conditions the semi-infinite measurements yield values of Ma that differ by less than 4% from the values determined with the infinite geometry. The deviations relative to A' are larger, ranging from -5% to 15%, but r, p (cm) Fig. 6. Straight lines associated with (a) dc, (b) ac, and (c) phase as a function of r (infinite geometry) or p (semi-infinite geometry), obtained from the Monte Carlo simulation. The different symbols refer to the three conditions examined (dc* and ac* refer to values relative to the maximum source-detector distance and V)* refers to a value relative to the minimum source-detector distance). Circles, infinite-medium simulation, infinite-geometry equations: dc* = ln(rUdc), ac* = ln(rUac), D* = . Squares, semi-infinitemedium simulation, semi-infinite-geometry equations: dc*, ac*, and * given by the left-hand sides of Eqs. (20)- (22). Triangles, semi-infinite-medium simulation, infinite-geometry equations: dc* = ln(pq('dc 8

T . i -A-~~~~~F
antini et al. 1 the required independence of Mus' from absorber concentration is retained. On the other hand, the analysis of the semi-infinite-measurement data with the infinitegeometry equations yields poor results for both Ma and As'.
Ma typically deviates by 15% from the accurate values, whereas Mas' shows a dependence on the absorber concentration. Obviously the use of the infinite-geometry model for analyzing the semi-infinite-geometry data is not expected to yield good results. Nevertheless the comparison presented in Figs. 4 and 5 allows us quantitatively to evaluate the correction that is due to the semiinfinite geometry model. From Fig. 5 one can see that for absorber concentrations smaller than 3 mL/L, which correspond to MA'/Ma > 80, the use of the semi-infinitegeometry boundary conditions gives rise to corrections in the right direction: the dc, the ac, and the phase slopes are systematically closer to the correct value. For ink concentrations higher than 3 mL/L (A'/Ma < 80) the corrections are less precise, especially in the case of the ac slopes.
The Monte Carlo simulation provides an independent test of the semi-infinite-geometry boundary problem. The results presented in Fig. 6 and Table 2 are similar to the ones obtained experimentally. Use of the semi-infinite-geometry boundary conditions yields better accuracy for Ma than for Ms'. The corrections provided by the boundary conditions are particularly evident and effective in the evaluation of Ma. The slopes of the straight lines associated with dc, ac, and phase are closer to the accurate ones when the boundary conditions are applied.

CONCLUSIONS
In this paper a systematic study of the applicability of the diffusion approximation to the semi-infinite-geometry boundary problem has been presented. The principal result is that in a macroscopically homogeneous, multiply scattering medium reasonably good estimates of the optical parameters are obtained from the diffusion theory, provided that the appropriate boundary conditions are applied. The fact, also shown in this paper, that the measurements are quite insensitive to the precise geometrical configuration at the surface, namely, the positions of the source and the detector relative to the surface plane of the medium, suggests that slightly different boundary geometries could be equally well represented.