Geomagnetic field B = B x e x + B y e y + B z e z (where e_x, e_y, and e_z are unit vectors directed to North, East and downward correspondingly) continuously exists in and around the Earth and is recorded nowadays digitally with a time reading interval Δt. After the conventional processing using Fourier transform a B record of duration t2-t1 is transformed into a superposition of harmonic components with periods T 1 , T 2 , ⋯ , T n .

Response function (RF) is the term widely used in natural sciences and mathematics. In the geoelectromagnetic studies of electrical conductivity σ(x, y, z) of the Earth’s interior [1] [2] [3] the RFs are supposed to be some functions derived from the Earth’s electromagnetic (EM) data that provide us with a possibility to determine the conductivity structure of the Earth. EM RFs are usually frequency/(period T) dependent and then they are complex functions having real (index u) and imaginary (index v) parts. We use two of these functions.

Induction vector C = A e x + B e y (A and B are determined from the linear equation: B z = A B x + B B y ). Real induction vectors C u = A u e x + B u e y possess an important property: in the notation of Wiese used here, they are directed away from a good conductor.

Anomalous horizontal magnetic variation tensor [M] is determined from the linear system of equations B x ( r i ) = M x x B x ( r 0 ) + M x y B y ( r 0 ) , B y ( r i ) = M y x B x ( r 0 ) + M y y B y ( r 0 ) , where r₀ and r_i are coordinates of base (reference) and some other observation place. Tensor [M] reflects the change in geoelectric structures between two places.

The processing of observed geomagnetic field B for the monitoring of geodynamic and other environmental processes implies transformation of 3 component geomagnetic field time-series with time reading interval Δt (1 min or 1 s in our study) into a variety of time-series with temporal resolution Δτ ( Δ τ ≫ Δ t ) of different RF components: Re and Im, x and y at the set of periods T 1 , T 2 , ⋯ , T n of received harmonics ( Δ t ≪ T i ≪ Δ τ ).

The theory of geoelectromagnetic methods is developed for natural source field in the form of vertically incident plane wave (Tikhonov-Cagniard (T-C) model), which usually holds for an external source field of magnetosphere-ionosphere origin (named as magnetotelluric (MT) field) for the periods less than 10⁴ s. Ideally RF depends only on the Earth’s conductivity distribution which is sensitive to the stress variations and therefore to geodynamic processes including the earthquake (EQ) preparation.

Really the observed B(T) is composed of several sources: 1) MT-field, part of which meets the T-C model requirements, the rest does not meet them and makes an error, 2) ever-present noise, both man-made and natural and sometimes B_LE—internal fields of Lithosphere Emission.

RFs and their variations, especially in relation with the occurrence of EQs and EQ preparation, were studied in Japan for many years and were described in many works between which we cite only few [4] [5] [6] [7] . However, in the last two decades this RF approach became less frequently used for EQ studies because of stronger noise at Japanese observatories due to the enhanced human activity.

It is recently agreed that different kinds of phenomena do take place prior to an EQ, including surface deformation, geochemical anomalies, EM radiation, atmospheric and ionospheric perturbations, etc. [8] . Especially, the geomagnetic and ULF (ultra low frequency) magnetic field variations among EM precursors are considered to be a promising candidate for EQ prediction, because significant precursors have been accumulated for several huge EQs [3] . Further some precursors have already been found to be statistically correlated with EQs, such as ionospheric perturbations [3] [8] .

A huge Tohoku EQ (with magnitude of 9) happened offshore of the Tohoku area on 11 March, 2011, and many scientists have tried to find any EQ precursors to this EQ. Different kinds of precursors (middle- and short-term precursors) to this 2011 Tohoku EQ have been reported so far including EM, geochemical anomalies and others, though most were recognized only retrospectively [8] . By paying particular attention again to the RF methods, we have analyzed the available geomagnetic data all over Japan in order to obtain any middle- and short-term EQ precursors in geomagnetic variations for this Tohoku EQ. Some results were already presented in Russian [9] [10] [11] , and the latest results based on the RF methods will be presented in this paper. Finally, our RF results will be compared with the corresponding former results by other researchers in the conclusion part.

Considering the time series of geomagnetic field synchronous records ( [11] , Figure 2), we noticed that MT field appears synchronously at all observatories while noise appears locally at each one. The stations most contaminated by noise, are UCU and KNZ which are the nearest to DC railways. But during the after-mid-night time interval from ≈1:30 to ≈4:30 LT (16:30 - 19:30 UT) the strong noise from DC railways almost disappears, although at some other stations (HAR and partly TTK) weaker noise continues all night.

Direct measurements of the strong MT variation amplitudes in each component provide a check (not precise but very reliable) of the results obtained by our processing. So, the enhancement of B_x at KNZ and OTA at approximately 30% - 40% exists and it can be interpreted only by the electrical conductivity anomaly under the observatories, i.e. under the central part of the Boso peninsula.

The relation between M_xx and M_yy anomalies in KNZ defines the WNW-ESE strike of the Boso conductivity anomaly, and geological data [20] [21] presented in Figure 3(b) and Figure 3(c) suggest the existence of an anomalous conductor of WNW-ESE strike in Miura Group sediments of the Kanto plain at depth 0 - 4 km. Relations between M_xx and M_yy in UCU, OTA and KYS are different as seen in Figure 3(a). It means that the direction of anomalous currents is also different under each observatory. Calculations show that at least 50% of anomalous currents should be located near the surface in the sediments of the Kanto plain to match with the received data.

On the other hand, the plate tectonic evidences that the Boso anomaly is located over the Sagami trough, structure at the depth 15 - 20 km in the complex junction of three lithosphere plates (Figure 1(a)). The strike of the trough is the same, WNW-ESE, so some part of the anomalous conductor can be located in the Sagami trough.

The eastern part of both conductors (shallow sediments and deep trough) has a contact with sea water, while the western one suppositionally contacts with a magma reservoir. In such a circuit it can be some unstable area(s) with conductivity strongly dependent on stress and sensitive to stress changes related with EQs.

Figure 3(b) shows the vectors for a period of about an hour (4000 s) at which industrial noise is practically absent and those vectors adequately reflect the heterogeneity of geoelectric structure. In Figure 3(c) vectors at the period 25 s are built with dominated noise field, which is much greater than MT field at four observatories considered. Real and imaginary vectors at periods 16, 25, 50 and partly at 200 s (Figure 3(d)) are directed to the source of noises—the nearest railway. To reduce influence of the noise, nighttime records and remote reference technique were used (Figure 3(f). Received corrected vectors appeared still very scattered as in Figure 3(e) and can be used cautiously for monitoring of geodynamic processes. However, those vectors averaged over a long period of time can be used for clarifying the geoelectric structure. Real vector in KNZ at the shortest periods is found to direct to north. It means that the most conductive part of Boso electrical conductivity anomaly is located to the south of KNZ, apparently near the southern side of the asymmetric “N.8 half graben fills” of the Miura Group sediments [21] .

Results of the processing of single station and nighttime records at KNZ are given in Figure 3(f). We see that full day and nighttime results are significantly different from each other only for B_u and partly B_v components, because a railway is located to the west of KNZ and brings the distortions mainly in the eastern component. The northern component is less affected by noise at periods 100 s and more that opens the possibility to use it for the separation of EQ precursors.

In the time intervals with high MT field the RF corresponds to the conductivity distribution and can be used for the study of precursors arisen from changes of conductivity. As was discussed above the anomalous conductor under KNZ can be a part of the large scale electrical circuit (sea water—magma reservoir) strongly sensitive to stress changes. The shallow geoelectric structure of Kanto plain sediments is the part of this circuit. The changes of electrical currents in the circuit before an EQ can be recorded even with the use of strong dominant DC noise field. This idea is supported by the use of strong noise from DC railways for the study of geoelectrical structure [22] .

The induction vector derived from very noisy records, practically from noise field, has a small stable phase. Therefore, if some other magnetic field which is not so stable (let it be a precursor field), is superimposed on the field of noise, exactly the phase of induction vector will be the most sensitive component for such a precursor separation. The nature of this “other” magnetic field can be B_LE (data of [23] evidence that after February 22, 2011 at an observatory UCU (in the south of the Boso peninsula) B_LE appeared with maximum at periods 30 - 100 s). An alternative explanation of the precursor is the change of anomalous current in the large scale circuit including the Boso anomaly. In any case phase in such a noisy environment can be the most sensitive parameter for precursor separation.

Below we apply a new approach developed by V.I.Tregubenko [24] , who used it for processing the data of seismo-prognosis monitoring network in Ukraine. He separated precursors before few strongest (M ≈ 4) Crimean EQs occurred during 15 years, in particular before the Sudak, Crimea EQ M3.9 on 24.01.2005 [24] . We applied this approach to KNZ, KAK and ESA 1-s data, but the precursor was found only in KNZ.

The processing was made with the use of the program developed by Varentsov [12] . Coherences were used as weights estimates for averaging the results. To minimize the effect of noise the estimates with multiple coherence less than 0.7 were ignored that allowed us to obtain minimally shifted estimates of induction vector’s components. The maximum anomalous effect before the Tohoku EQ was observed in the phase of the induction vector northern component—arg (A) for periods of 100 - 200 s. For the longer periods the anomaly gradually decreases, so that we now present the results for T = 100 s. The processing and analysis were performed in two steps:

Step 1. 7 years long (2005-2011) geomagnetic field time series with every 1 s reading was processed for every month as a single unit. arg(A) time series with every month reading were received as in Figure 4(a) and analyzed by a polynomial approximation approach. The most significant first, second and seventh (quasi-annual) order polynomials were extracted from the rough data and we obtained Figure 4(b) which is more perspective for comparison with EQs. But 1 month temporal resolution of RF is not sufficient for such an analysis. As for the Tohoku EQ precursors, we find a 9 month long negative anomaly in 2010 approximately 1 year before the main event.

Step 2. 2 years long (2010-2011) 1 s time series were processed for every day as single unit. A large scatter of every day results was reduced by averaging with moving window of 5 days long with 1 day shift. From these curves, i.e. arg(A) time series with every day reading, we extracted first, second and seventh (quasi-annual) order polynomials determined in Step 1. The result is shown by the grey rough curve in Figure 5(a). The bold solid line is the result of averaging with 29 days window with 1 day shift to suppress monthly variations arising from the Moon rotation around the Earth (gravity effect) [25] and the Sun rotation around its axis (magnetic activity effect). The mean of annual variation during 7 years of 2005-2011 has been subtracted, but we note a residual annual variation in Figure 5(a) (RF annual variation enhancement was recorded at several observatories during 2 - 3 years before Tohoku EQ and can be considered as middle-term precursor). So, such a residual annual variation was determined once more from 2 years data presented in Figure 5(a) and subtracted, the result is presented in Figure 5(b).

The variations of arg(A) given by the bold solid line in Figure 5(b) are cleaned from periodic variations of annual and monthly periods, that make it more convenient to identify EQ precursors. Indeed, we see a strong bay-like variation that begins almost 2 months before the Tohoku EQ and has approximately 2 month duration. This is in good agreement with the finding of Varotsos team [26] [27] . In particular, Varotsos et al. [26] showed that the initiation of an SES activity in Japan appears almost simultaneously with the minimum of the fluctuations of the order parameter of seismicity analyzed in natural time, and such a minimum was clearly observed on January 5, 2011, i.e. almost two months before the Tohoku EQ occurrence [27] . Moreover, the spatio-temporal study of the aforementioned minimum gave a ground for a reliable estimation of the epicentral area of the impending main shock [27] .

The time of beginning of a bay-like variation and its duration depends on the magnitude of an expected EQ. This time is equal to approximately 2 weeks for the processed Crimean EQs with magnitude approximately 4 [24] . But usually the precursors are not so smooth, but they are rather some sequence of separate episodes. And we obtained such precursors a few weeks as well as some days before the Tohoku EQ by the high resolution processing [9] . Really, averaging with 29 days window eliminated not only the monthly variation but also many shorter ones. The rough grey graph (also every day values with only 5 days averaging) in Figure 5(b) can be used for a more detailed separation of precursors, but the impact of superposed background from the noise and magnetic activity should be kept in mind and eliminated as our future work.