Comparison and optimal parameter setting of ...

Comparison and optimal parameter setting of reference-based harmonic noise cancellation in time and frequency domain for surface-NMR Mike M¨ uller-Petke

1

and Stephan Costabel

2

1

2

Leibniz Institute for Applied Geophysics, Stilleweg 2 Hannover, D-30655, Germany. Federal Institute for Geosciences and Natural Resources Stilleweg 2 Hannover, D-30655, Germany.

ABSTRACT The technique of surface nuclear magnetic resonance (surface-NMR) provides information on porosity and hydraulic conductivity that are highly valuable in a hydrogeological context. However, the applicability of surface-NMR is often limited due to bad signal-to-noise ratio. In this paper we provide detailed insight into the technique of harmonic noise cancellation based on remote references to improve the signal-to-noise ratio. We give numerous synthetic examples to study the influence of various parameters such as optimal filterlength for time domain approaches or necessary record-length for frequency domain approaches, all of which evaluated for different types of noise conditions. We show that the frequency domain approach is superior to time domain approaches. We demonstrate that the parameter settings in the frequency domain and the decision whether or not using separated noise measurement depends on the actual noise properties, i.e. frequency content or stability with time. We underline our results using two field examples.

Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

2

INTRODUCTION The technique of surface nuclear magnetic resonance (surface-NMR) as a hydrogeophysical prospection method has shown continuous development towards a frequently used geophysical technique over the last decades. There has been significant improvements in forward modeling (Lehmann-Horn et al., 2011), inversion and interpretation (Hertrich et al., 2009; Mueller-Petke & Yaramanci, 2010a; G¨ unther & M¨ uller-Petke, 2012) during the last years. These improvements extended the application range of the method concerning solutions to hydrogeological tasks. But the applicability of surface-NMR is highly limited due to bad signal-to-noise (S/N) ratio. In many cases, the expected NMR signals are only few tens of nanovolts and therefore often contaminated by significant electromagnetic noise. Consequently, to extend not only the theoretical range of applications, improving S/N ratio is a demanded development. The first approach was based on a single loop antenna layout in a figure-of-eight design (Trushkin et al., 1994). This technique allows to filter the magnetic field produced by a power-line aligned with the axis of the figure-of-eight. However, this approach is limited to a single source, needs double the size of a single loop and flat terrain. Another promising approach is based on the idea of using reference channels to cancel the electromagnetic noise. Such techniques have been used e.g. in magnetotellurics (MT) and transient electromagnetic (TEM) since the 1980s (Gamble et al., 1979a,b; Goubau et al., 1984; Spies, 1988). A similar approach becomes popular with latest development in surface-NMR instrumentation that provide the necessary multi-channel data acquisition systems (Radic, 2006; Walsh, 2008). Since then several methods using a reference loop based system were developed. Radic (2006) presented a remote reference approach operating in frequency domain and using vertical loops as references. Only few information is available about the technique implemented in the Vista-Clara software (Walsh, 2008). Mueller-Petke & Yaramanci (2010b) presented a time domain approach based on a single reference loop and optimal filtering that has been extended to arbitrary number of references by Neyer (2010). An adaptive filter in time domain has been presented by Dalgaard et al. (2012) and compared to the time domain optimal filter. They show that both approaches lead to similar results. All approaches have demonstrated useful capability to improve S/N of surface-NMR data. However, a comparison that allows for determining the performance, i.e., pro and contra of each approach concerning different measurement conditions is missing. Thus, to compare time domain with the frequency domain approach, we developed a frequency domain code beside the existing time domain code and distinguish two strategies within the frequency domain. We present and describe in detail noise cancellation approaches for time and frequency domain. We show how factors, such as noise characteristics, filter length or the use of signal records influence the performance of each approach using both synthetic and field data. There is an increasing interest and discussion on the use of vertical loops or smaller loop sizes for horizontal loops (Costabel & M¨ uller-Petke, 2012). In the presented field case we measured in addition to a horizontal loop, x and y components of the magnetic field using small vertical loops. Unfortunately, there is only few information on the impact of small diameter horizontal loops (Neyer, 2010) and no comparison of vertical and horizontal loops published. According to our experience, measuring all three components can improve the Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

3

performance of noise cancellation. Besides improving data quality using remote references, surface-NMR records may be contaminated by short pulse like event often referred to as spikes. Jiang et al. (2009) presented an approach to identify and remove spikes using a statistical measure. Dalgaard et al. (2012) showed that spike removal before application of noise cancellation is necessary since otherwise the application may fail. Costabel & M¨ ullerPetke (2013) introduced a despiking approach in the wavelet domain. Even though both topics, the use of vertical small loops and spike removal are related to the application of noise cancellation, a detailed discussion of these topics is beyond the scope of this paper. This paper focuses on the impact of time and frequency domain approaches and is structured in the following way. Firstly, we give a brief description of the mathematical realization of both the time and frequency domain approach while an extended formulation is provided within the appendices. Secondly, synthetic data is used to evaluate the approaches and the impact of different parameter settings. Finally, real data sets are used to underline our findings.

PRINCIPLES OF REMOTE REFERENCE TECHNIQUE FOR SURFACE-NMR Basic idea of reference based harmonic noise cancellation The approach of harmonic noise cancellation (HNC) is based on a simple idea to measure the noise contaminated NMR signal at one channel (detection channel - Tx) and to measure the noise at a second or more remote channels (reference channel - Nx). If the noise is spatially homogeneous and the detection and reference loops are of equal size and shape, one could simply subtract the reference from the detection channel. But usually, the noise is not homogeneous and the usage of smaller loops for the references are preferable for logistic reasons. The approach consists of two steps. First, the transfer function (TF) must be established that describes the coherence between signal and reference loops (Chave et al., 1987; Spies, 1988), i.e. incorporates all relevant characteristic properties of the inhomogeneous noise such as amplitude and phase variation and the differences of the loop shapes and sizes. Second, the Nx record is transformed using the TF and subtracted from the Tx recorded signal at the detection loop. The key point is to construct an adequate TF as accurate as possible. In principle, several ways to calculate the TF can be realized.

Basic equations for transfer functions (TF) in time domain (TD) and frequency domain (FD) To obtain a transformation, i.e. a filter h, from a signal r(t) to a signal s(t) in the time domain an optimal filter is an appropriate choice as used in e.g. Dalgaard et al. (2012). The final equation for a single reference loop reads rst =

X

hτ rrt−τ .

(1)

τ

with rr and rs, the auto-correlation of r(t) and the cross-correlation of r(t) with s(t), respectively. Using matrix notation the solution for the filter h reads h = (rT r)−1 · rT s. Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

(2)

M¨ uller-Petke & Costabel

4

Hence, the estimation of h is an inverse problem. Assuming that records contain the correlated harmonic noise but are additionally contaminated by Gaussian noise, the determination of h becomes unstable and includes uncertainties. Therefore, optimal filtering limits the number of coefficients hτ in equation (1). As for any ill-posed inverse problem (e.g Aster et al., 2005) , with decreasing number of unknown in the model domain (coefficients hτ ), but constant number of data (the cross- and auto-correlation rst and rrt−τ ) the estimation in the model domain becomes more precise, i.e, the uncertainty of the coefficients hτ is decreasing. Moreover, using as few coefficients as possible minimizes artifacts at the beginning of the filtered record (Buttkus, 2000). Nevertheless, a certain number of coefficients is needed to sufficiently transform the signal, i.e. more coefficients are necessary with increasing noise complexity. This obtained filter h finally is convolved with the signal recorded at the reference loop and then subtracted from the signal recorded in the detection loop. Equation (1) can be easily extended to an arbitrary number of references, see appendix A. We refer to the filter h that contains the coefficients hτ as transfer function (TF) in the time domain. To obtain a transformation in the frequency domain we rewrite equation (1) for optimal filter using the Fourier transformation rst =

X

hτ rrt−τ

c

τ

s R∗ · S = H · R2 .

(3)

This yields the coefficients of H relating the spectra of R and S for each frequency ωn separately H(ωn ) = S(ωn )/R(ωn ). (4) As from above, records are contaminated by Gaussian noise, thus, the determination of H(ωn ) contains uncertainty due to the variances of R(ω) and S(ω). To decrease uncertainty and to obtain stable filter coefficients, we introduce i repeated measurements R = [R(ωn )1 , · · · , R(ωn )i ]T S = [S(ωn )1 , · · · , S(ωn )i ]T

(5)

and use the least squares solution for H(ωn ) H(ωn ) = (RT R)−1 · RT · S.

(6)

Equation (6) can be easily extended to an arbitrary number of references, see appendix B. We refer to the set of filter coefficients H(ωn ) as TF in the frequency domain.

Local vs. global determination of TF in FD There are two approaches to realize i repeated measurements in the frequency domain. The first approach, which we refer to as local TF approach, subdivides a single (multi)-channel record into several partitions of identical length and calculates a TF based on the set of partitions. Consequently, for each record an independent TF can be derived. As we will describe next, a key property of the local TF approach is a trade-off between frequency resolution and uncertainty of the filter coefficients which is due to this separation of one record into a number of partitions. This trade-off is basically a consequence of the fact that the product of frequency resolution (or bandwidth) times the variance of the estimated Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

5

relative uncertainty of "ilter coef"icients

local TF global TF

4 partitions

4 stacks

0.4 8 partitions 16 partitions 0.3 32 partitions 512 partitions

0.2

16 stacks 256 partitions 0.1

64 partitions 128 partitions

0

−3

10

−2

10

100 stacks −1

10

0

10

frequency resolution, relative to original resolution

Figure 1: Relative uncertainty of filter coefficients and frequency resolution with varying number of partitions derived from one time series (local TF) or varying number of stacks (global TF). For further detail see text.

Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

6

spectrum of a time series is constant (e.g. Buttkus, 2000). It is beyond the scope of this paper to give a detailed derivation, but, it is obvious, that with increasing number of partitions, the length of each single partition becomes shorter. A shorter partition length causes a decreasing frequency resolution, i.e., the number of coefficients H(ωn ) obtained from one partition of the complete time series is decreasing. Whereas the number of coefficients H(ωn ) decreases, the uncertainty of each of the coefficients H(ωn ) is constant. Now, one can calculate i sets of independent coefficients H(ωn ) for i partitions. The uncertainty of H(ωn ) is than decreased by averaging those i sets of coefficients (eqs. 5 and 6). This finally gives the constant relation between the number of coefficients (or frequency resolution) and the uncertainty of these coefficients. Thus, the less coefficients the better they are determined. The second approach, which we refer to as the global TF approach, uses the frequency resolution of the original time series without any decrease in frequency resolution. All (multi-) channel records of the MRS measurement, i.e., all stacks for all pulse-moments are used simultaneously to calculate one (global) TF that is applied to all records. The uncertainty of the resulting TF is according to the amount of stacks used. The above described relations of frequency resolution and uncertainty for local and global TF are depicted schematically in Figure (1). On the x-axis the resolution relative to the original resolution in the frequency domain and on the y-axis the expected uncertainty of the TF [1/sqrt(number of partitions or number of stacks)] is plotted for different numbers of partitions and stacks. We considered 4 to 512 partitions and 4 to 100 stacks. A trade-off between an appropriately high number of partitions and the resulting resolution is shown, which is of importance for the further evaluation. It might be expected that the global TF approach has promising properties in regard to an optimized HNC due to highest possible frequency resolution on the one hand and due to the lowest uncertainty of the TF calculation on the other hand (see also Fig. 1). However, compared to the local TF approach, we anticipate the global TF approach to be less flexible regarding possible time dependent changes of the noise properties. A comparison and assessment of both approaches is done here with synthetic data and field data examples from two test sites with different harmonic noise characteristics. Moreover, we want to investigate if it is advantageous to make separate noise measurements for the TF calculation. In contrast, the use of the signal records together with the simultaneously recorded reference time series could be used without additional measurement effort.

The use of noise records vs. signal records Equations (2) to (6) do not specify the signals used. NMR (just as TEM) is an active experiment, i.e. NMR signals are triggered by an excitation pulse and after a short time of a few seconds NMR signals have vanished. Thus, TFs can be calculated based on measurements of pure noise before or after an NMR experiment or based on measurements that contain NMR signal. On the one hand, pure noise records ensure that the filter will not cancel any NMR signal when applying afterwards. However, this filter is less appropriate if noise characteristics change with time. On the other hand, when using records that contain NMR signals the filter is most up-to-date with noise characteristics but may cancel the wanted Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

7

NMR signal itself (Dalgaard et al., 2012). A comparison and assessment is done again with synthetics and field examples.

SYNTHETIC EVALUATION Evaluation scenarios To evaluate the different approaches on the base of synthetic signals we simulate several noise cases and measurement setups. For instance, a power-line as the main source of harmonic noise. We set the higher harmonics to be separated by 100 Hz, each harmonic has the same amplitude and phase relation from Tx to Nx, we assume the frequency content to be constant with time and noise records should be used to calculate TF. To setup a case like this and allow for other cases we distinguish four basic properties (a) The distribution of harmonics within a frequency interval; (b) The relation of amplitudes and phases of each harmonic from Tx to Nx; (c) The signals used for TF calculation; (d) The variability of properties (a–b) for different measurements i; each of these properties (a–d) has two possible settings. For (a) these settings are: 1. Ten frequencies starting at 1500 Hz equally spaced by 100 Hz. 2. Ten frequencies randomly chosen between 1800 Hz and 2200 Hz. The setting for (b) are: 1. The amplitude and phase for each harmonic is the same. In Tx the amplitude of each harmonic equals one while in Nx the amplitude is four. The phase differs by 1 rad. 2. The amplitudes and phase in Tx are unchanged while in Nx the amplitude of each harmonic is set independently between [1, 4] and the phase between [-pi, pi]. The setting for (c) are: 1. Noise records are used to calculate TF. 2. Signal records are used to calculate TF. The setting for (d) are: 1. The settings for the properties (a–b) do not change for all measurements i.

Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

8

1111 1211 1121 1221 2211 2222

TD 0.7 0.8 0.8 1.1 2.0 2.2

global FD 0.7 0.75 0.75 0.75 0.85 5.0

local FD 0.7 0.75 0.75 1.0 0.9 1.2

Table √ 1: NL after HNC approach for different scenarios. A perfect cancellation would lead to 0.52 + 0.52 since both Nx and Tx records have a standard deviation of 0.5 a.u. Gaussian noise.

2. The frequencies of the harmonics and the amplitude and phase relation change for each measurements i. Note, this has no effect if a measurements is split into several partitions, since this property affects a complete record. According to these properties the above scenario of a power-line noise source is classified as (1111). Using this classification of scenarios their impact on the noise cancellation approach can be easily studied. Some further realistic examples are: (1121) would be a power-line source but TF is calculated based on signal records; (2211) indicate a number of different sources; (2212) is the worst case of a number of sources that change their characteristics with time. Note, the given values above a not based on a numerical modeling of the physics. All records have the length of one second. In addition to the noise cases, the records at Tx contain NMR signals. The frequency of the NMR signal is 1905 Hz for the case of equally spaced harmonics. If the harmonics are randomly chosen, the NMR frequency is set to the frequency of the fifth harmonic plus 5 Hz. The offset of 5 Hz between NMR frequency and harmonic is changed later to check whether or not the use of signal records to estimate the TF has an impact on the obtained NMR signal after HNC. The amplitude of the NMR signal is 50% (i.e. 5 arbitrary units - a.u.) of the total amplitude of all harmonics (10 a.u.). Finally, all signals contain additionally Gaussian noise with a standard deviation of 5% (0.5 a.u.) of the total amplitude of the harmonics.

Results of the synthetic evaluation In Figure (2) the results of selected scenarios for TD, global FD and local FD are shown according to the settings explained above. Table (1) summarizes the results in terms of remaining noise level (NL) after HNC. The NL is calculated as standard deviation of the signal that remains after additionally subtraction of the synthetic NMR signal, i.e. it should represent the Gaussian noise part. First, we discuss two important properties of TD and FD transfer functions, the filter-length and number of partitions. Second, the impact of using noise or signal records is shown. Finally the impact of the complexity of the assumed noise is evaluated.

Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

9

(a) !l = 100 samples (0.01 s)

(b) stacks = 32

(c) partitions = 32

(d) !l = 100 samples (0.01 s)

(e) stacks = 32

(f) partitions = 32

(g) !l = 500 samples (0.05 s)

(h) stacks = 32

(i) partitions = 4

(k) stacks = 32

(l) partitions = 4

[a.u.]

10 0 −10

[a.u.]

10 0 −10

[a.u.]

10 0 −10 (j) !l = 500 samples (0.05 s) [a.u.]

10 0 −10 0

0.1 0.2 time [s]

0.3 0

0.1 0.2 time [s]

0.3 0

0.1 0.2 time [s]

0.3

Figure 2: Results of selected scenarios (1211: (a)-(c); 1221 (d)-(f); 2211 (g)-(i); 2222 (j)-(l)) for TD (a,d,g,j), global FD (b,e,h,k) and local FD (c,f,i,l). Gray is the record before and black after HNC; red line is the NMR signal; the title of each sub-figure gives the filterlength (fl) for TD, number of stacks for global FD and number of partitions for local FD to obtain optimal results.

Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

NL after HNC [a.u.]

6

(a)

(b)

5 4 3 2 1 10−4

6 NL after HNC [a.u.]

10

10−3 10−2 "ilter-lenght [s]

10−1

10−4

(c)

10−3 10−2 "ilter-lenght [s]

10−1

(d)

5 4 3 2 1 100

101 number of partitions

102

100

101 number of partitions

102

Figure 3: Noise-level (NL) after HNC for scenario 1211 (a,c) and 2211 (b,d) as a function of filter-length (a,b) for TD and number of partitions (c,d) for local FD.

Influence of filter-length and number of partitions For TD the filter-length is a critical parameter. Figure (2a) displays the result of the simple scenario 1211. Even though the TD filter is only 10 ms long there are artifacts at the beginning of the record after HNC while both global FD and local FD cancel the harmonics very accurately. The filter length should be as short as possible to avoid artifact at the signal beginning but long enough to capture the noise characteristics to perform a noise cancellation (Buttkus, 2000). Figure (3a,b) shows the dependency of the noise level (NL) after applying the filter and subtraction. According to the settings above a perfect harmonic √ cancellation would lead to a NL of 0.52 + 0.52 . As expected there is a minimum length of the filter to transform the noise signal sufficiently. Increasing the filter-length further does not decrease the NL. Thus, the optimal filter-length for the scenario 1211 is about 10 ms while for scenario 2211 it is about 50 ms. The latter is due to the increases complexity of the harmonic noise that needs to be captured by an increased filter length. For FD filtering the number of partitions as indicated in Figure (1) is important. Figure (3c,d) shows the NL as a function of the number of partitions. On the one hand, if using too few partitions (< 4) the NL slightly increases since the filter becomes sensitive to the Gaussian noise. However, the increase is slightly since the Gaussian noise is only 5% of the amplitude of the harmonic noise and therefore the uncertainty of H(ωn ) is small. On the other hand, if using too many partitions, the filter is unable to capture all harmonics appropriately since the frequency resolution is lost. In the case of 2211, the frequency resolution is inadequate as indicated by an increase of NL if more then 6 partitions are used. Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

(a)

rel. NMR canc. [%]

25

∆ f = 0Hz ∆ f = 1Hz ∆ f = 5Hz

20 15 10 5 0 0 10

1

10 number of stacks

10

2

10

2

(b)

25 rel. NMR canc. [%]

11

∆ f = 0Hz ∆ f = 1Hz ∆ f = 5Hz

20 15 10 5 0 0 10

1

10 number of partitions

Figure 4: Cancellation of NMR signal for global FD (a) and local FD (b) as a function of the number of stacks/partitions and ∆f (the frequency difference between NMR signal and harmonic noise) for TF calculations based on signal records.

In this case, amplitudes of the adjacent harmonics cannot be sufficiently distinguished, the calculated TF therefore cannot appropriately transform the Nx record to Tx and harmonic noise remains. In the case of 1211, using more than 50 partition seem to produce an inadequate frequency resolution, even though the increase in NL is less prominent. A maximum number of partition can be roughly estimated since the harmonics are uniformly distributed by a separation of 100 Hz. The complete record is 1 s long, thus 50 partitions lead to a partition length of 20 ms or a frequency resolution of 100 Hz. Hence, using any higher number of partitions leads to a TF that averages the adjacent harmonics. Note that for a case such as 1111, i.e., no change in the amplitude of the harmonics, even inadequate frequency resolution will not cause an increase of NL since the TF is constant. Consequently, the slight increase in Figure (3c) indicates that the amplitude of the harmonics change only slightly. Signal records versus noise records Now we calculate the TF using the signal records, i.e., we add a NMR signal to exactly the same simulated noise and apply the same processing. This is scenario 1221. While the result of the global FD approach remains unchanged , both the TD and local FD cancel the NMR signal to some extend (Fig. 3). Figure (4) provides a more detailed insight at this effect for global and local FD. In principle, using signal records to calculate a TF is possible if the Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

12

NMR signal does not show any correlation to the noise signal. Since the global FD approach uses several stacks for calculating TFs the NMR signal always has a random relative phase (∆φ) between NMR signal and a harmonic. Consequently, there is no correlation between harmonics and NMR signal and a TF can be calculated that does not cancel any NMR signal. However, this is true only if the correlations can be calculated sufficiently, i.e., a certain number of stacks are needed. Figure (4a) shows that with increasing number of stacks, i.e. for the global FD, cancellation of NMR signal decreases because the uncertainty of the estimated correlation coefficient decreases. But, in the case of local FD only one record is separated into several partitions. Thus, ∆φ of each partition depends on the frequency relation ∆f between NMR and a harmonic and is not purely randomly distributed as for the global FD. For instance, if ∆f equals zero then ∆φ is constant for all partition and therefore a NMR signal cannot be distinguished from a harmonic signal which leads to cancellation of NMR signals independent of the number of partitions. Figure (4b) shows this for different ∆f ’s. With increasing ∆f the relative phases ∆φ of each partition are no longer constant but change with the partition number. Therefore, NMR signal and harmonic signal can be distinguished. However, this change in ∆φ may still be small for small ∆f . The presented study does not evaluate the impact of changing the ratio(s) of harmonic noise to NMR signals and to the level of Gaussian noise on cancellation of NMR signals. However, the key in using the signal records is that harmonics and NMR can be distinguished by their respective phase relations. If the ratio between NMR amplitude and Gaussian noise is decreasing, i.e., Gaussian part become more prominent, the likelihood to determine this phase relation correctly decreases and therefore an increase in cancellation of NMR can be observed. Nevertheless, the global FD shows to be less affected compared to the local FD due to the randomly distributed phase while for local FD the relative phases depend on ∆f and, thus, may be small. The complex scenario Next we further increase the complexity of the harmonics. As shown in Figure (3b, d) we need to adapt the filter-length to 50 ms and number of partitions to 4. However, the number of stacks for global FD can be still high (32 stacks) since the frequency resolution does not decrease for global FD. For the scenario 2211 (Fig. 2g, h, i) the TD shows very significant artifacts due to the long filter of 500 coefficient while both FD approaches are nearly similar. The slight differences are due to the decreased uncertainty of the TF due to the higher amount of stacks compared to the number of partitions. Furthermore, TF is based on noise records, thus, there is no cancellation of NMR signal. Finally, we assume that noise changes with time and therefore we need to use the signal records. As we observed in scenario 1221 (Fig. 2d, e, f) the results after local FD show slight canceling of NMR signal since ∆f = 5 Hz in this setting. However, the amplitude of canceled NMR signal was low since the number of partitions was high enough. But for scenario 2222 the number of partitions was set to 4 in order to calculate the best TF in terms of canceling the harmonic noise(Fig. 3d). However, one may increase the number of partitions up to 10 to minimize cancellation of NMR signal but by the cost of less effective harmonic noise cancellation. Obviously, we have a trade-off between canceling NMR signal and harmonics for this worst case scenario and the local FD approach. Nevertheless, this Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

13

local FD approach is the only choice since the global FD fails. Note, this approach uses 32 independent stacks each with changed noise properties. Consequently, no global TF exists.

FIELD DATA Noise characteristics We conducted surface NMR measurements at two different test sites in the west of Berlin. First of all, we will discuss qualitatively how the site-specific noise characteristics and dynamics affect the general performance of the HNC in the frequency domain. Later on, we analyze the effect of using whether the noise or the signal records for the TF calculation. We will quantify the difference between the global and the local approach with varying numbers of partitions. The setup for both measurements is the same: A 47-m diameter loop (2 turns) as signal loop (Channel 1) and 3 reference loops 100 m away from signal loop (edge to edge distance). The reference loops are realized as 47-m diameter loop (2 turns) for the zcomponent of the magnetic field and two 1-m square loops (140 turns) for the x- and y-components, respectively. Both soundings were conducted with the NUMIS Poly device (IRIS Instruments) and consist of 8 pulse moments in the range of 0.1 to about 13 As. The stacking rate is 36 and 1 s of signals were recorded. The first site, from now on referred to as test site 1, is located near the village Braedikow. The signal and reference loops were placed in a distance of about 200 m away from the housing estate. In Figure (5a) the amplitude spectra of the measured signal records in the range -/+ 100 Hz around the Larmor frequency are plotted against the recording time to show the time dependence of the interfering harmonics. Obviously, only multiples of the 50-Hz power-line frequency appear at this location due to the vicinity of the housing estate. The amplitude of these harmonics varies between 100 and 1000 nV, while the frequencies at 2050 and 2150 Hz remain constant. The noise situation at the second test site, shown in Figure (5b) is significantly different. This site, referred to as test site 2, is located about 2 km south of the village Barnewitz. It is likely that the noise source at this location is a railroad in about 3 km distance from the position of the signal and reference loops as well as a power-line in about 300 m distance. Figure (5b) shows the interfering harmonics at multiples of 16.7 Hz, which is the mean operating frequency of the railroad, as well as of the 3-phase alternating current in Germany. Test site 2 exhibits a large scattering of +/10 Hz in every frequency band, while the amplitude peaks vary between 1000 and 2000 nV. Regarding the scenarios described for the synthetic evaluation, we characterize the test site 1 as (12x1) and test site 2 as (22x2) while x denote that we are about to demonstrate both strategies, i.e., the use of separated noise measurements (x=1) and the signal records (x=2), respectively.

Local FD compared to global FD In order to compare the performance of HNC, local and global TF’s were calculated for both test sites. For local TF calculation each record was divided into 64 partitions, while in total 288 records were provided for calculating the global TF. Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

14

(a)

t [min]

60

900

40 700

voltage [nV]

20

(b)

t [min]

60 300 40

20 100 2050

2100

2150

frequency [Hz]

Figure 5: Amplitude spectra of signal records as function of recording time for test site 1 (a) and test site 2 (b).

10

3

(a)

(b) V [nV]

3

102

10

102 (c)

10

1

(d)

0

0

Original data Global TF Local TF 20 40 Recording time [min]

60

2

V [nV]

10

V [nV]

V [nV]

2

10

10

0

20 40 Recording time [min]

60

Figure 6: Amplitude of harmonic noise at 2050 (+/- 10) Hz at test site 1 as function of recording time, (a) signal channel before and after noise compensation with global TF and local TF using 64 partitions, (b),(c) and (d) are corresponding reference channels (zcomponent),(x-component),(y-component).

Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

10

4

(a)

(b) Original data Global TF Local TF

V [nV]

10

15

3

10

3

10

2

(c)

V [nV]

V [nV]

102

102

101

101 0

V [nV]

10

20 40 Recording time [min]

60

(d)

3

102 0

20 40 Recording time [min]

60

Figure 7: Amplitude of harmonic noise at 2050 (+/- 10) Hz at test site 2 as function of recording time, (a) signal channel before and after noise compensation with global TF and local TF using 64 partitions, (b),(c) and (d) are corresponding reference channels (zcomponent),(x-component),(y-component).

Figures (6a) and (7a), show the amplitude of the harmonic at 2050 Hz at both test sites plotted against the recording time before and after applying the two HNC approaches. Even though we have recorded 3 components of the EM noise, the comparison of the two HNC approaches uses only the z-component as reference. For completeness, Figure (6b) to (6d) for test site 1 (and in Fig. 7b to 7d for test site 2), show the data of all reference channels. In both cases, the two HNC approaches effectively reduce the peak amplitude by more than one decade, while the local TF approach tends to work slightly better. At test site 1 (Fig. 6), the amplitude change with time appears to be less important since the results from both HNC approaches do not differ much. Still the amplitude of the harmonic after HNC at recording times from 30 to 50 min is less for the local FD approach. In agreement to this, Figure 7 shows that the local TF approach yields better results when the frequency of the harmonic exhibits a larger scattering in time. For test site 2, the HNC based on the global TF approach systematically leads to remaining peak amplitudes that are, on the average, two-times higher than the local TF approach.

Dependency of the local FD on number of partitions Regarding a best possible reduction of the noise level when applying HNC, the qualitative analysis above indicates a better performance when using the local TF approach. The next step of our analysis is focused on the optimal number of partitions when using local TFs. For this analysis, both the noise and the signal records to calculate the local TF are considered. In Figure 8 the noise level, i.e., its mean value and standard deviation as errorbars, after HNC is plotted as a function of the used number of partitions. The noise level of the global Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

100

16

(a)

(b)

300

90 250

80 Noise level [nV]

70

200

60 50

150

40 30 20 10 0 0 10

St.dev. (global TF) Global TF Local TF (noise records) Local TF (signal records) 101 102 number of partitions

103

100 50 0 0 10

1

10 10 number of partitions

2

10

3

Figure 8: Noise-level after HNC based on one reference channel (z-component) using the global and local TF approach for test site 1 (a) and test site 2 (b).

TF approach is plotted as well with a horizontal line and with a shaded area representing its standard deviation. When using the noise records for TF calculation, the results show a similar dependency on the number of partitions as expected from the synthetic study (see Fig. 3c and d): Applying the HNC with a very small number of partitions (< 8), the uncertainties of the TFs leads to a higher remaining noise level compared to large numbers of partitions. The lowest noise levels are achieved at partition numbers from 8 to 32 for test site 1 and from 4 to 8 for test site 2. For larger numbers, the remaining noise level increases again due to the loss of resolution in the frequency domain. This effect is much clearer at test site 2 due to the scattering of the frequencies of the interfering harmonics (see Figure 5 b). Test site 2 tends to behave like the synthetic example 2211 (see Figure 3 d). For test site 1 with much less scattering of the harmonic noise frequencies, the increase of the noise level for increasing number of partitions is much smaller. As the synthetic study showed, the noise level would be low and does not change much regardless of the number of partitions only for the case that the harmonic noise frequencies are rather constant and separated by some tens of Hertz. For both test sites, this is not the case. Noise records vs. signal records Note, using noise records for the local TF calculation does not lead to results systematically better than the global TF approach (Fig. 8). For both test sites, the local TF approach leads, on the average, to slightly better results only in the optimal range regarding the number of partitions. Taking the errorbars into account, we note that the apparent benefit of using the local TF approach together with the noise traces is insignificant. Actually, we conclude that there is no systematic benefit when using the local TF approach at the basis of the additional noise measurements compared to the global TF calculation. In contrast, the resulting noise level after the HNC, when using the signal records for TF Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

(a)

100 80 70

180 160 140

60

120

50

100

40

80

30

60

20

40

10

20

0 0 10

1

10 10 number of partitions

2

10

(b)

200

St.dev. (Global TF) Global TF Local TF (noise records) Local TF (signal records)

90

Noise level [nV]

17

3

0 0 10

1

10 10 number of partitions

2

10

3

Figure 9: Noise-level after HNC based on all three reference channel using the global and local TF approach for test site 1 (a) and test site 2 (b).

calculation, is correlated with the number of partitions over the whole range: The smaller the number, the lower the noise level. At both test sites, for partition numbers smaller than 32 the noise level falls below the value achieved with the global TF approach. The direct comparison with the results from calculating the TF from the noise records shows that the results at test site 2 are always superior when using the signal records. This is because of the unstable nature of the noise at this test site (see Fig. 5). As we already learned from the synthetic study above, the more unstable the noise conditions are, the more inaccurate are the results when using separate noise measurements for TF calculation. Of course, when the number of partitions increases, also for the approach using the signal records, the decreasing resolution of the TF in the frequency domain leads again to an imperfect HNC and consequently to higher noise levels. The use of three component references We demonstrate the HNC results for the usage of all the three measured reference channels, i.e., we take now the x and y-components of the EM noise into account. These results are shown in Figure (9). In particular for test site 2, the effect of using x, y, and z instead of just the z-component of the EM noise is clearly visible. We found that the noise level is reduced to almost one half compared incorporating x and y components (Fig. 9b and 8b). Apart from this general reduction, the noise level as function of the partition number for using all EM components shows similar properties like the curve for the z component only (see Fig. 8). This is true for both test sites. For the approach using separated noise records, the noise level increases for small and higher partition numbers, while for using the signal records, the noise level continuously increases. As for Figure (8), the approach based on separated noise measurements does not show any benefit compared to the global TF calculation. In contrast, the approach based on the signal records leads again to the best results. Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

(a)

400 Signal voltage [nV]

18

(b)

200 0 −200 −400 0

0.2

0.4 0.6 time [s]

0.8

orig. data, SNR:0.1, noise level:248 nV comp. data, SNR:2.8, noise level:19 nV "it:E 0 = 47 ± 8 nV, T2* = 123 ± 29 ms

1

0

0.2

0.4 0.6 time [s]

0.8

1

orig. data, SNR:0.5, noise level:245 nV comp. data, SNR:2.2, noise level:40 nV "it: E 0 = 148 ± 15 nV, T2* = 153 ± 21 ms

Figure 10: Stacked NMR signal before (gray) and after HNC (black) application of local TF approach using 4 (a) and 64 partitions (b).

It seems that the lowest noise level can be achieved with the local TF approach using the signal records and using a small number of partitions. However, the synthetic study shows that this conclusion cannot be generalized, because the HNC at the base of the signal records using a small number of partitions can also cause the cancellation of NMR signal (see Fig. 4). Thus, we will have a detailed look at the resulting NMR signal to analyze the probability of its cancellation. Cancellation of NMR signal As expected beforehand, the NMR signal can be canceled to some extent when using the signal records for TF calculation, if an interfering harmonic occur in the vicinity of the Larmor frequency. The Larmor frequencies at our test sites are approximately 2099 Hz at test site 1 and 2100 Hz at test site 2. We expect a significant NMR signal cancellation due to the presence of a harmonic of the 50-Hz power frequency at 2100 Hz. Obvously, also even multiples can appear as interfering harmonic noise, even though they have much smaller amplitudes than the odd multiples. Figure (10) shows the stacked NMR signal (stacking rate: 36) at test site 1 at a pulse moment of 12.6 As before and after HNC and the corresponding mono-exponential fit. The TF calculation based on the signal records was realized using 4 (Fig. 10a) and 64 partitions (Fig. 10b). As shown above, the resulting noise level is clearly smaller when using 4 partitions, but the NMR signal in this case appears with significantly reduced energy. It was canceled by almost one third compared to the HNC with 64 partitions. To calculate a relative NMR signal cancellation, the amplitude of a mono-exponential fit (after stacking 36 HNC-processed single records) was related to the fit amplitude for the global TF approach using the noise records that is not affected by any NMR cancellation. We further calculate the mean value (and standard deviation with the errorbars) from the Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

19

(b)

(a) 60

60 TF with noise records TF with signal records

rel. NMR canc. [%]

50

50

40

40

30

30

20

20

10

10

0

0

−10

−10

−20

−20

−30 0 10

1

2

10 10 number of partitions

10

3 −30 0

10

101 102 number of partitions

103

Figure 11: Cancellation of NMR signal as a function of number of partitions. Relative cancellation is calculated in relation to the amplitude of a mono-exponential fit of the global TF approach using noise record.

relative NMR signal cancellation of all pulse moments of the soundings. Fig. 11 shows that the reduction of signal amplitude due to the NMR signal cancellation is a function of partition number using signal records. As we expected from the synthetic analysis, cancellation of NMR signal must be expected for the small number of partitions. The signal cancellation for using only two partitions is about 40 to 50%. For increasing partition numbers, the NMR signal cancellation decreases asymptotically down to zero. On the average, zero signal cancellation is achieved for partition numbers higher than 32, whereas the errorbars indicate a scattering of fit amplitudes depending on the actual remaining white, i.e., non-harmonic, noise level.

SUMMARY AND CONCLUSION We have presented the basic equations and concepts of applying remote reference based noise cancellation for surface-NMR. Using synthetic examples we demonstrated that the frequency domain approach (FD) is superior to the time domain approach since artifacts due to the convolution of large filter are inevitable. We introduced a local and a global FD approach using the individual record or all records, respectively. We found that the performance of a frequency domain approach depends on the frequency resolution archived from the record length versus the complexity of the harmonic noise, i.e. the frequency content. On the one hand, the performance of the local FD approach is limited since a single record is separated into several shorter partitions. On the other hand, the performance of the global FD approach is limited if noise rapidly changes with time. A series drawback, however, of the local FD when using signal records is a cancellation of NMR signals if the NMR frequency is close to a harmonic and complexity of the noise conditions allow only for few partitions. Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

20

We have validated our finding of the synthetic study by field examples of different complexity. The second test site showed that changing noise properties can occur. This case underlines the need of the local FD due to it’s flexibility and shows that the global approach is only superior if noise properties do not change with time. To our experience and field case this is rarely the case. We conclude that there is no perfect strategy that can be applied as a black box. There is a trade-off between noise complexity and stability which then determines using the global or local approach with optimal parameter settings. Nevertheless, we recommend the global FD as the default processing approach due to its robustness and especially to avoid signal cancellation and the local FD for the more experienced user.

REFERENCES Aster, R., Borchers, B. & Thurber, C., 2005. Parameter Estimation and Inverse Problems, Elsevier Academic Press. Buttkus, B., 2000. Spectral Analysis and Filter Theory in Applied Geophysics, SpringerVerlag Berlin. Chave, A. D., Thomson, D. J. & Ander, M. E., 1987. On the Robust Estimation of Power Spectra, Coherences, and Transfer Functions, Journal of Geophysical Research, 92(B1), 633–648. Costabel, S. & Mueller-Petke, M., 2012. MRS noise investigations with focus on optimizing the measurement setup in the field, in Proceedings of Magnetic Resonance in the Subsurface 2012 - 5th International Workshop on Magnetic Resonance, Hannover, Germany. Costabel, S. & Mueller-Petke, M., 2013. Despiking of magnetic resonance signals in time and wavelet domain,Near Surface Geophysics, this issue , submitted. Dalgaard, E., Auken, E., & Larsen, J., 2012. Adaptive noise cancelling of multichannel magnetic resonance sounding signals, Geophysical Journal International, 191(1), 88– 100. Gamble, T. D., Goubau, W. M., & Clarke, J., 1979. Magnetotellurics with a remote magnetic reference, Geophysics, 44(1), P53–P68. Gamble, T. D., Goubau, W. M., & Clarke, J., 1979. Error analysis for remote reference magnetotellurics, Geophysics, 44(5), P959–P968. Goubau, W. M., Maxton, P. M., Koch, R. H., & Clarke, J., 1984. Noise correlation lengths in remote reference magnetotellurics, Geophysics, 49(4), P433–P438. G¨ unther, T. & M¨ uller-Petke, M., 2012. Hydraulic properties at the North Sea island of Borkum derived from joint inversion of magnetic resonance and electrical resistivity soundings, Hydrology and Earth System Sciences, 16, 3279-3291. Hertrich, M., Green, A. G., Braun, M., & Yaramanci, U., 2009. High-resolution surfaceNMR tomography of shallow aquifers based on multi-offset measurements, Geophysics, 74(6), G47–G59. Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

M¨ uller-Petke & Costabel

21

Jiang, C., Lin, J., Duan, Q., Sun, S. & Tian, B., 2011. Statistical stacking and adaptive notch filter to remove high-level electromagnetic noise from MRS measurements, Near Surface Geophysics, 9 , 459–468. Lehmann-Horn, J. A., Hertrich, M., Greenhalgh, S. A., & Green, A. G., 2011. ThreeDimensional Magnetic Field and NMR Sensitivity Computations Incorporating Conductivity Anomalies and Variable-Surface Topography, IEEE Transactions on Geoscience and Remote Sensing, 49, 3878 – 3891. Mueller-Petke, M. & Yaramanci, U., 2010a. QT inversion — Comprehensive use of the complete surface NMR data set, Geophysics, 75(4), WA199–WA209. Mueller-Petke, M. & Yaramanci, U., 2010b. Improving the Signal-to-noise Ratio of SurfaceNMR Measurements by Reference Channel Based Noise Cancellation, in Proceedings of Near Surface 2010 - 16th European Meeting of Environmental and Engineering Geophysics. Neyer, F., 2010. Surface Nuclear Magnetic Resonance: Processing of Full Time Series, Multichannel Surface NMR Signals, Master’s thesis, ETH Zurich, Department of Earth Sciences. Radic, T., 2006. Improving the Signal-to-Noise Ratio of Surface NMR Data Due to the Remote Reference Technique, in 12th European Meeting of Environmental and Engineering Geophysics. Spies, B. R., 1988. Local noise prediction filtering for central induction transient electromagnetic sounding, Geophysics, 53(8), 1068–1079. Trushkin, D., Shushakov, O., & Legchenko, A., 1994. The potential of a noise-reducing antenna for surface NMR groundwater surveys in the earth’s magnetic field, Geophysical Prospecting, 42(8), 855–862. Walsh, D. O., 2008. Multi-channel surface NMR instrumentation and software for 1D/2D groundwater investigations, Journal of Applied Geophysics, 66(3-4), 140–150, Resonance Sounding – a Reality in Applied Hydrogeophysics.

Appendix A — TF in TD for arbitrary number of references To obtain a transformation from a signal r(t) to a signal s(t) in the time domain and for a single reference, a filter h is defined as st =

n X

hτ rt−τ .

(A-1)

τ =0

with n the number of filter coefficients. We extend this to arbitrary number c of reference channels r(t)c by simply adding all channels together each with its own filter, i.e., we add the results after individual convolution for each reference channel st =

n X

τ =0

1 h1τ rt−τ +

n X

τ =0

2 h2τ rt−τ + ··· +

n X

c hcτ rt−τ .

τ =0

Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.

(A-2)

M¨ uller-Petke & Costabel

22

Again we can write this using the matrix notation for the convolution s = r1 h 1 + r2 h 2 + · · · + rc h c .

(A-3)

This way we have extended the number of unknown filter coefficients largely but still have only one equation. The solution of this equation would fail. However, we can get use of a possible correlation between the channels. Therefore we multiply equation A-3 with each reference channel and introduce the correlation (r1 )T s = (r1 )T r1 h1 + (r1 )T r2 h2 + · · · + (r1 )T rc hc (r2 )T s = (r2 )T r1 h1 + (r2 )T r2 h2 + · · · + (r2 )T rc hc .. . (rc )T s = (rc )T r1 h1 + (rc )T r2 h2 + · · · + (rc )T rc hc

(A-4)

and rewrite into matrix notation   

(r1 )T s .. . (rc )T s





   =  

(r1 )T r1 · · · (r1 )T rc (r2 )T r1 · · · (r2 )T rc .. .. . . ... (rc )T r1 · · · (rc )T rc



    ·  

h1 ..  . .  hc 

(A-5)

This has now to be solved for the individual filter h which are then used for the convolution, i.e., transformation of the recorded noise records.

Appendix B — TF in FD using multiple loops The derivation in frequency domain is similar to the time domain derivation. Here we extend Equation 3 to an arbitrary number c of reference channels S = H 1 R1 + H 2 R2 + · · · + H c Rc

(A-6)

and yield separate equations for each frequency ωn . Introducing repeated measurements and correlation between all channels (R1 )T S = (R1 )T R1 H 1 + · · · + (R1 )T Rc H c

(R2 )T S = (R2 )T R1 H 1 + · · · + (R2 )T Rc H c .. . (Rc )T S = (Rc )T R1 H 1 + · · · + (Rc )T Rc H c and than write   

(R1 )T S .. . (Rc )T S





   =  

(R1 )T R1 · · · (R1 )T Rc (R2 )T R1 · · · (R2 )T Rc .. .. . ... . c T 1 c (R ) R · · · (R )T Rc



 1  H    .  · .  .   Hc

Near Surface Geophysics, 4/2014, 10.3997/1873-0604.2013041.