Randomly distributed tensor resistivity measurement method and system

Abstract

Disclosed is a randomly distributed tensor resistivity measurement method and system. In the method, electrode deployment can be randomly arranged based on site-specific grounding conditions. The current supply station sequentially energizes two orthogonal current injection channels, while all potential measurement stations simultaneously and in parallel acquire potential differences across two measurement channels. The current and potential difference data are recorded with GPS timestamps, enabling synchronization of current supply and potential measurement station data based on corresponding time. Using the recorded data, current density vectors and electric field intensity vectors are calculated for each supply-measurement station combination, from which the corresponding apparent resistivity tensor is derived.

Claims

1. A randomly distributed tensor resistivity measurement method, comprising the following steps: S1: randomly arranging a plurality of potential measurement stations P across a survey area according to surface conditions to cover the entire survey area; S2: arranging four potential electrodes M1, N1, M2, and N2 around each potential measurement station P, wherein each potential measurement station P is equipped with a potential measurement unit, and a line connecting M1 and N1 intersects with a line connecting M2 and N2; a distance between M1-N1 and a distance between M2-N2 are defined as electrode spacing a, where a=( 1/10- 1/20) H, and H represents the exploration depth; S3: deploying a plurality of current injection points within the survey area; selecting any four of the plurality of current injection points as A1, B1, A2, and B2, and ensuring that the line between A1-B1 intersects with the line between A2-B2; S4: supplying power to electrodes at A1 and B1 using a supply station, while all electrodes at potential measurement stations P simultaneously performing potential measurement; the supply station recording a supply current I.sub.1 of electrodes at A1 and B1, and each potential station measuring the potential differences $U_{M1N1}^{\left(1\right)}\mathrm{and}{U}_{M2N2}^{\left(1\right)}$ across electrode pairs M1-N1 and M2-N2, respectively, corresponding to current injection at A1 and B1; subsequently, supplying power to electrodes at A2 and B2, while all electrodes at potential measurement stations P simultaneously perform potential measurement; the supply station recording a supply current I.sub.2 of electrodes at A2 and B2, and each potential station measuring the potential differences $U_{M1N1}^{\left(2\right)}\mathrm{and}{U}_{M2N2}^{\left(2\right)}$ across electrode pairs M1-N1 and M2-N2, respectively, corresponding to current injection at A2 and B2; S5: based on the supply currents I.sub.1 and I.sub.2, and the spatial relationships between A1, B1, and point P, obtaining a first current density vector generated by A1 and B1 at P and a second current density vector generated by A2 and B2 at P; using $U_{M1N1}^{\left(1\right)}\mathrm{and}{U}_{M2N2}^{\left(1\right)}$ along with the electrode spacing to derive the first electric field intensity vector generated by A1 and B1 at point P; using $U_{M1N1}^{\left(2\right)}\mathrm{and}{U}_{M2N2}^{\left(2\right)}$ along with the electrode spacing to derive the second electric field intensity vector generated by A2 and B2 at point P; S6: performing vector decomposition of the first current density, the second current density, the first electric field vector, and the second electric field vector in a common coordinate system to compute the apparent resistivity tensor.

2. The method of claim 1, wherein the survey area is first divided into multiple sub-areas before executing step S1, and steps S1-S6 are performed within each sub-area to complete the tensor resistivity measurement across the entire survey area.

3. The method of claim 1, wherein prior to step S1, dividing the survey area into multiple sub-areas; executing steps S1-S2 in each sub-area to arrange potential measurement stations P and potential measurement electrodes M1, N1, M2, N2; wherein the power supply arrangement and selection in step S3 are replaced by: arranging k internal current injection points in each sub-area, and l external current injection points around the perimeter; one of the k internal points is selected as point O, and two of the l external points are selected as A1 and A2, respectively; both B1 and B2 are coincided with point O; executing steps S4-S6 within each sub-area to complete the apparent resistivity tensor.

4. The method of claim 1, wherein prior to step S1, dividing the survey area into multiple sub-areas; executing steps S1-S2 to deploy potential measurement stations and electrodes in each sub-area; wherein the power supply arrangement and selection in step S3 are replaced by: deploying not fewer than four external injection points around each sub-area; selecting four of the external injection points as A1, B1, A2, and B2, with a line between A1 and B1 intersecting with a line between A2 and B2; executing steps S4-S6 within each sub-area to complete the apparent resistivity tensor.

5. The method of claim 4, wherein the external current injection points are distributed in all directions around each sub-area.

6. The method of claim 5, wherein eight external injection points are arranged around each sub-area, distributed in the east, west, south, north, southwest, northwest, southeast, and northeast directions.

7. The method of claim 1, wherein in step S2, the potential electrodes N1 and N2 are co-located with the measurement station P.

8. The method of claim 1, wherein current injection points B1 and B2 are co-located at point O, and the supply station is arranged at O.

9. The method of claim 1, wherein the spacing between adjacent potential measurement stations is 2-3 times the electrode spacing, and the spacing between adjacent current injection points is 3-5 times that of adjacent potential measurement stations.

10. A randomly distributed tensor resistivity measurement system, comprising a plurality of current supply stations and potential measurement stations; wherein each current supply station comprises a supply station host, long-distance power cables, current injection electrodes, and a boost power supply; the supply station host comprises a first control module, first power module, first built-in battery, current measurement module, and first GPS module; the first control module is configured to control electrode channel selection, power parameter configuration, external power supply integration, internal power management, current measurement, GPS timing, and synchronization of current data into a time series stored at preset intervals; the current measurement module is configured to digitize the current values through an A/D converter; the first power module is configured to select specific supply channels and connects to the boost power supply; the first built-in battery is configured to supply long-term operating power; the first GPS module, with an external antenna, is configured to provide location and time data to the host; the host is configured to provide four power output ports connected via long-distance cables to current electrodes at points A1, B1, A2, and B2 for sequential power delivery underground; each potential measurement station comprises a station host, potential measurement cables, and electrodes; the host comprises a second control module, second power module, second built-in battery, potential measurement module, and second GPS module; the second control module is configured to control channel selection, measurement settings, power management, dual-channel acquisition, and GPS-synchronized time-stamped data storage; the potential measurement module is configured to acquire differential voltage from two orthogonal channels simultaneously; the second power module is configured to manage the internal battery condition and charging; the second built-in battery is configured to ensure long-term operation and essential standby functions; the second GPS module, with an external antenna, is configured to provide location and clock data to the host; the host comprises four measurement ports connected via cables to electrodes at M1, N1, M2, and N2 to receive potential difference signals between electrode pairs.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a flowchart illustrating the randomly distributed tensor resistivity measurement method of the present invention.

(2) FIG. 2 is a schematic diagram showing the relationship between current injection electrodes and potential measurement electrodes in tensor resistivity measurement.

(3) FIG. 3 is another schematic diagram showing an alternative layout of current injection and potential measurement electrodes for tensor resistivity measurement.

(4) FIG. 4 is a schematic diagram illustrating the layout effect of measurement points in a randomly distributed tensor resistivity measurement.

(5) FIG. 5 is a schematic diagram showing rolling measurements within tensor resistivity sub-survey areas.

(6) FIG. 6 is a schematic diagram of the current supply station host.

(7) FIG. 7 is a schematic diagram of the potential measurement station host.

(8) FIG. 8 is a numerical simulation comparison diagram between traditional scalar resistivity imaging results and imaging results obtained using the tensor resistivity method.

(9) FIG. 9 is a comparison diagram of imaging results using the tensor resistivity method with different current electrode lengths.

DETAILED DESCRIPTION OF THE EMBODIMENTS

(10) The invention will now be described in detail with reference to the accompanying drawings and preferred embodiments. The objectives and advantages of the invention will become more apparent. It should be understood that the specific embodiments described herein are merely illustrative and are not intended to limit the scope of the invention.

(11) In practical electrical exploration, the resistivity , which is the reciprocal of conductivity, is commonly used to describe the electrical properties of a medium. At the ground surface interface, the vertical components of the current density and electric field intensity in Ohm's Law become zero, and the resistivity tensor degenerates into a 22 matrix. This can be expressed and calculated by instrumentation as:

(12) $\stackrel{.fwdarw.}{E}=\stackrel{}{}.Math.\stackrel{.fwdarw.}{J}$

(13) In inhomogeneous or anisotropic media, the directions of the electric field intensity vector {right arrow over (E)} and the current density vector {right arrow over (J)} are no longer aligned, resulting in an angle between them. For configurations involving dual current injection and dual potential difference measurements, the electric field intensity vector {right arrow over (E)} and corresponding current density vector {right arrow over (J)} can be obtained by measuring the potential differences generated by two distinct current fields. The apparent resistivity tensor p is then calculated, and inversion methods are used to derive the true resistivity tensor of the subsurface medium.

(14) As illustrated in FIG. 1, this embodiment of the randomly distributed tensor resistivity measurement method includes two major components: (I) measurement point design and layout including the following steps S1-S3, and (II) data acquisition process and procedures including the following steps S4-S6.

(15) I. Measurement Point Design and Layout

(16) A key feature and advantage of the randomly distributed tensor resistivity method is that the current supply and potential measurement units are independently and separately designed. Each unit autonomously manages its own data acquisition, while the entire system enables coordinated, multi-station operation. Both the current supply and potential measurement stations adopt dual-channel, node-based acquisition designs using either three or four electrodes, allowing for flexible layout according to terrain conditions. The system imposes no specific restrictions on electrode pair positioning or orientation; however, uniform distribution within the survey area is encouraged for consistent subsurface coverage. The acquisition parameters, including potential electrode spacing, current electrode spacing, and point density, are determined based on the target exploration depth. The potential and current electrode points are then planned and laid out on a map. The specific measurement point design steps are as follows: S1: Based on surface conditions, a plurality of potential measurement stations P are randomly distributed across the survey area to ensure complete coverage. The spacing between adjacent potential measurement stations is set to 2-3 times the electrode spacing. S2: Four potential electrodes M1, N1, M2, and N2 are arranged around each potential measurement station P. A potential measurement unit is arranged at each station P, with its four terminals connected to the corresponding potential electrodes via potential measurement cables. The line connecting M1 and N1 intersects with the line connecting M2 and N2 (at a wide angle, not necessarily orthogonal). The distances between M1 and N1, and the distance between M2 and N2 are referred to as electrode spacing (not required to be equal, with the specific length determined by coordinate positioning). Specifically, =( 1/10 1/20) H, where H is the exploration depth. S3: Multiple current injection points are arranged across the survey area, with the spacing between adjacent current injection points being 3-5 times that between adjacent potential measurement stations P. Any four current injection points are selected within the survey area (specific injection configurations and selection methods are detailed later), designated as A1, B1, A2, and B2. The connecting lines A1-B1 and A2-B2 must intersect. Electrodes at A1, B1, A2, and B2 are connected to the current supply station via long-distance power cables.

(17) The relationship between current injection electrodes and potential measurement electrodes for tensor resistivity measurement is illustrated in FIG. 2.

(18) To facilitate field deployment and improve operational efficiency, the current injection points B1 and B2 may be co-located at a single point, designated as point O, where the current supply station is arranged. For similar reasons, the potential measurement electrodes N1 and N2 may be co-located with the potential measurement station P. The layout shown in FIG. 3 is more conducive to practical field implementation.

(19) In one exemplary embodiment, a mixed layout scheme for current injection and potential measurement electrode deployment is illustrated in FIG. 4, demonstrating the effectiveness of the arrangement.

(20) In practice, to achieve inversion imaging at greater exploration depths, the distance between current injection points and potential measurement stations should increase proportionally with the target depth. Due to limitations on the number of available current and potential measurement instruments, it is typically impractical to cover the entire survey area with instruments in a single deployment. To achieve comprehensive and relatively uniform imaging coverage of the survey area, the area is first subdivided into multiple sub-survey zones. Current injection points and potential measurement stations are arranged and deployed within each sub-zone. After completing the measurements in one sub-zone, the system rolls forward sequentially to the next sub-zone and repeats steps S1-S2, continuing until data acquisition is complete in the final sub-zone, thereby finalizing the data collection for the entire survey area.

(21) During actual field operations, the deployment process includes both paper-based planning and on-site validation. Once the measurement locations are finalized, GPS or similar surveying equipment is used to record the coordinates and identifiers of each electrode pair. These data support subsequent optimization of the acquisition plan and configuration of instrument parameters during field implementation.

(22) Furthermore, various flexible approaches are available for the deployment and powering strategies of current injection points in Step S3:

(23) 1. Quasi-Equidistant Grid Node Powering Mode

(24) A grid of designed current injection points is used to form near-orthogonal powering combinations (e.g., BB1-BB2 and B1-BB12 in FIG. 5). The advantage of this mode is that the system can advance sequentially along orthogonal directions defined by the grid layout, and the power cables remain relatively short, which is beneficial for field deployment. However, due to the short spacing between injection points, mutual electromagnetic interference is more likely during measurements. To mitigate this, higher supply voltage is applied to increase current magnitude, or the dipole length is increased by using every few points as power electrodes.

(25) 2. Peripheral Illumination Imaging Powering Mode

(26) In this mode, l external current injection points are arranged around each sub-survey area. From l external current injection pints, any four points are selected and designated as A1, A2, B1, and B2, which serves as the current supply station combination (the lines A1-B1 and A2-B2 intersect at point O). As illustrated in FIG. 5, the survey area is divided into four square sub-areas, with small red dots representing current injection points. Taking sub-area 1 at lower-left quadrant as an example, it includes 8 peripheral long-distance current injection points and 25 potential measurement stations. The 8 peripheral injection points are labeled as AA1 through AA3, AA5, AA7, and AA9 through AA11, and are distributed around the east, west, south, north, southwest, northwest, southeast, and northeast directions. For example, AA5 and AA7 can be selected as A1 and B1, and AA2 and AA10 as A2 and B2. Alternatively, AA9 and AA2 can serve as A1 and B1, and AA2 and AA11 as A2 and B2. For any selected power combination, all potential measurement stations (both M1-N1 and M2-N2 channels) in the sub-area perform simultaneous, parallel measurements. This powering mode minimizes field workload.

(27) 3. Hybrid Near-Far Powering Mode

(28) As shown in FIG. 5, the entire survey area is divided into four square sub-survey areas. Sub-area 1 includes 36 internal current electrodes, 8 peripheral long-distance injection points, and 25 potential measurement stations. In this mode, combinations of long-distance illumination points (AA-type) and internal short-distance points (BB-type) are used for powering. Specifically, k internal current injection points are arranged within each sub-area, and l external points around the perimeter. One internal point is selected and labeled as point O. Two external points are selected and labeled A1 and A2, with B1 and B2 coinciding at O. In the configuration illustrated in FIG. 5, a combination such as AA5, BB34 and AA2, BB34 is used. Numerous AA-BB combinations are available, enabling multi-directional illumination of the underground target, improving measurement density and resolution. For example, the system can begin with (AA1:BB1) and (AA2:BB1), then sequentially move through (AA1:BB2) and (AA2:BB2) up to (AA1:BB61) and (AA2:BB61). Next, based on combinatorial patterns, configurations like (AA2:BB1) and (AA3:BB1) are selected. This iterative sequence continues until all combinations of AA- and BB-type points are exhausted, thereby completing the powering phase. During this process, the two powering channels are activated sequentially, while all 25 potential measurement stations simultaneously collect voltage data for each supply event. After completing the measurement for sub-area 1, all supply and potential measurement stations are relocated to sub-area 2. The powering and measurement process is then repeated. The same procedure is followed sequentially in sub-areas 3 and 4 until all data acquisition tasks for the full survey area are completed.

(29) By deploying the current injection points beyond the periphery of each sub-survey area, not only is multi-directional illumination imaging enabled, but the electrode spacing for current injection is also extended. This increases the effective exploration depth and ensures that near-, mid-, and far-depth regions each have sufficient coverage from measurement points.

(30) Therefore, the measurement point design and deployment in this embodiment may alternatively be structured as follows:

(31) Prior to executing step S1, the entire survey area is divided into multiple sub-survey areas. Within each sub-survey area, the complete randomly distributed tensor resistivity measurement method is implemented. After completing the tensor resistivity measurement in each sub-area, the process is considered complete for the entire survey area.

(32) Alternatively, prior to executing step S1, the survey area is divided into multiple sub-survey areas. Then, within each sub-area, steps S1 and S2 are carried out to deploy potential measurement stations P and corresponding potential electrodes M1, N1, M2, and N2;

(33) The current electrode arrangement and selection in step S3 is modified as follows: k internal current injection points are deployed within each sub-survey area, and l external current injection points are arranged around the perimeter. One of the internal injection points is selected as point O. Two external injection points are selected and designated as A1 and A2, such that B1 and B2 are both co-located with point O.

(34) Alternatively, prior to executing step S1, the entire survey area is divided into multiple sub-survey areas. Within each sub-area, steps S1 and S2 are performed to deploy potential measurement stations P and corresponding potential electrodes M1, N1, M2, and N2;

(35) The current electrode arrangement and selection in step S3 is modified as follows: not fewer than four external current injection points are deployed around the perimeter of each sub-survey area. From these, any four are selected and designated as A1, B1, A2, and B2, such that the line connecting A1 and B1 intersects the line connecting A2 and B2.

(36) II. Data Acquisition Process and Acquisition Steps

(37) The operation of the current supply stations and the movement of measurement stations is carried out independently and separately from the potential measurement stations. No communication exists between the two systems; each function continuously according to preset time intervals until its assigned measurement task is complete, at which point it powers down automatically. The current injection points are first deployed based on the selected powering mode described earlier. Each supply station is powered on, performs a self-check (including GPS signal verification), and is configured with operational parameters. Upon successful initialization, the coordinates of the current injection points A1B1 and A2B2 are recorded using GPS or other surveying tools. Once safety is confirmed, the supply process is initiated. The instrument system automatically logs time, current magnitude, and other relevant data. After each measurement cycle, the station is relocated to the next designated injection point, and the process is repeated.

(38) The specific steps of the current supply measurement process are as follows: S4: The current supply station first energizes the electrodes at current injection points A1 and B1. Simultaneously, all potential measurement stations P measure voltage differences across their connected electrodes. The supply station records the supply current I.sub.1 at A1 and B1, and each potential station records the potential differences

(39) $U_{M1N1}^{\left(1\right)}\mathrm{and}{U}_{M2N2}^{\left(1\right)}$ across electrode pairs M1-N1 and M2-N2, respectively, corresponding to current injection at A1 and B1.

(40) Next, the supply station energizes the electrodes at points A2 and B2. All potential measurement stations perform simultaneous measurements. The supply station records the supply current I.sub.2 at A2 and B2, and each potential station records the potential differences

(41) $U_{M1N1}^{\left(2\right)}\mathrm{and}{U}_{M2N2}^{\left(2\right)}$ across electrode pairs M1-N1 and M2-N2, respectively, corresponding to current injection at A2 and B2. S5: Based on the supply currents I.sub.1 and I.sub.2, and the spatial relationships between A1, B1, and point P, a first current density vector generated by A1 and B1 at P and a second current density vector generated by A2 and B2 at P are obtained. Using

(42) $U_{M1N1}^{\left(1\right)}\mathrm{and}{U}_{M2N2}^{\left(1\right)}$ along with the electrode spacing, the first electric field intensity vector generated by A1 and B1 at point P is derived. Using

(43) $U_{M1N1}^{\left(2\right)}\mathrm{and}{U}_{M2N2}^{\left(2\right)}$ along with the electrode spacing, the second electric field intensity vector generated by A2 and B2 at point P is derived; S6: The first and second current density vectors and the first and second electric field intensity vectors are resolved within a common coordinate system to compute the apparent resistivity tensor.

(44) The apparent resistivity tensor is computed using the following formula:

(45) 0$\stackrel{\_}{}=\left[\begin{array}{cc}{}_{xx}& {}_{xy}\\ {}_{yx}& {}_{yy}\end{array}\right]=\frac{1}{j_x^{\left(1\right)}{j}_y^{\left(2\right)}-j_y^{\left(1\right)}{j}_x^{\left(2\right)}}[\begin{array}{cc}{E}_x^{\left(1\right)}{j}_y^{\left(2\right)}-E_x^2{j}_y^{\left(1\right)}& E_x^{\left(2\right)}{j}_x^{\left(1\right)}-E_x^{\left(1\right)}{j}_x^{\left(2\right)}\\ E_y^{\left(1\right)}{j}_y^{\left(2\right)}-E_y^{\left(2\right)}{j}_y^{\left(1\right)}& E_y^{\left(2\right)}{j}_x^{\left(1\right)}-E_y^{\left(1\right)}{j}_x^{\left(2\right)}\end{array}$$j_X^{\left(1\right)}=\frac{I_1}{2}\left[\frac{x_p-x_{A1}}{.Math.r_{A1P}.Math.}-\frac{x_p-x_{B1}}{{.Math.r_{B1P}.Math.}^3}\right]$$j_Y^{\left(1\right)}=\frac{I_1}{2}\left[\frac{y_p-y_{A1}}{{.Math.r_{A1P}.Math.}^3}-\frac{y_p-y_{B1}}{{.Math.r_{B1P}.Math.}^3}\right]$$j_X^{\left(2\right)}=\frac{I_2}{2}\left[\frac{x_p-x_{A2}}{{.Math.r_{A2P}.Math.}^3}-\frac{x_p-x_{B2}}{{.Math.r_{B2P}.Math.}^3}\right]$$j_Y^{\left(2\right)}=\frac{I_2}{2}\left[\frac{y_p-y_{A2}}{{.Math.r_{A2P}.Math.}^3}-\frac{y_p-y_{B2}}{{.Math.r_{B2P}.Math.}^3}\right]$$E_x^{\left(1\right)}=E_{M1N1}^{x\left(1\right)}+E_{M2N2}^{x\left(1\right)}=\frac{U_{M1N1}^{\left(1\right)}}{|r_{M1N1}{|}^2}.Math.\left(x_{N1}-x_{M1}\right)+\frac{U_{M2N2}^{\left(1\right)}}{|r_{M2N2}{|}^2}.Math.\left(x_{N2}-x_{M2}\right)$$E_y^{\left(1\right)}=E_{M1N1}^{y\left(1\right)}+E_{M2N2}^{y\left(1\right)}=\frac{U_{M1N1}^{\left(1\right)}}{|r_{M1N1}{|}^2}.Math.\left(y_{N1}-y_{M1}\right)+\frac{U_{M2N2}^{\left(1\right)}}{|r_{M2N2}{|}^2}.Math.\left(y_{N2}-y_{M2}\right)$$E_x^{\left(2\right)}=E_{M1N1}^{x\left(2\right)}+E_{M2N2}^{x\left(2\right)}=\frac{U_{M1N1}^{\left(2\right)}}{|r_{M1N1}{|}^2}.Math.\left(x_{N1}-x_{M1}\right)+\frac{U_{M2N2}^{\left(2\right)}}{|r_{M2N2}{|}^2}.Math.\left(x_{N2}-x_{M2}\right)$$E_y^{\left(2\right)}=E_{M1N1}^{y\left(2\right)}+E_{M2N2}^{y\left(2\right)}=\frac{U_{M1N1}^{\left(2\right)}}{|r_{M1N1}{|}^2}.Math.\left(y_{N1}-y_{M1}\right)+\frac{U_{M2N2}^{\left(2\right)}}{|r_{M2N2}{|}^2}.Math.\left(y_{N2}-y_{M2}\right)$$\left|r_{A1P}\right|=\sqrt{{\left(x_p-x_{A1}\right)}^2+{\left(y_p-y_{A1}\right)}^2}\left|r_{B1P}\right|=\sqrt{{\left(x_p-x_{B1}\right)}^2+{\left(y_p-y_{B1}\right)}^2}\left|r_{A2P}\right|=\sqrt{{\left(x_p-x_{A2}\right)}^2+{\left(y_p-y_{A2}\right)}^2}$$\left|r_{B2P}\right|=\sqrt{{\left(x_p-x_{B2}\right)}^2+{\left(y_p-y_{B2}\right)}^2}$$\left|r_{M1N1}\right|={\left(x_{N1}-x_{M1}\right)}^2+{\left(y_{N1}-y_{M1}\right)}^2$$\left|r_{M2N2}\right|={\left(x_{N2}-x_{M2}\right)}^2+{\left(y_{N2}-y_{M2}\right)}^2$

(46) Where: (x.sub.A1, y.sub.A1), (x.sub.B1, y.sub.B1), (x.sub.A2, y.sub.A2), (x.sub.B2, y.sub.B2) are coordinates of current electrodes A1, B1, A2, and B2, respectively; (x.sub.M1, y.sub.M1), (x.sub.N1, y.sub.N1), (x.sub.M2, y.sub.M2), (x.sub.N2, y.sub.N2) are coordinates of potential measurement electrodes; (x.sub.P, y.sub.P) is the coordinate of potential measurement station P;

(47) $j_X^{\left(1\right)}\mathrm{and}{j}_Y^{\left(1\right)}$
are the components of the first current density vector along the x and y axes, respectively;

(48) $j_X^{\left(2\right)}\mathrm{and}{j}_Y^{\left(2\right)}$
are the components of the second current density vector along the x and y axes, respectively;

(49) $E_x^{\left(1\right)}\mathrm{and}{E}_y^{\left(1\right)}$
are the components of the first electric field intensity vector along the x and y axes, respectively;

(50) $E_x^{\left(2\right)}\mathrm{and}{E}_y^{\left(2\right)}$
are the components of the second electric field intensity vector along the x and y axes, respectively; |r.sub.A1P|, |r.sub.B1P|, |r.sub.A2P|, |r.sub.B2P|, |r.sub.M1N1|, |r.sub.M2N2| are the components of the distance between both points, respectively.

(51) The randomly distributed tensor resistivity measurement system described in the present invention includes multiple independent current supply stations and potential measurement stations. Each operates autonomously, using a built-in GPS clock to provide unified timing. Measurement data are recorded and stored sequentially according to GPS timestamps, and are later synchronized by time during post-processing to extract current or potential difference data corresponding to each measurement station at specific times.

(52) In this embodiment, the current supply station includes a supply station host, four long-distance power cables, four to ten stainless steel or copper electrodes, a boost power supply, and either a generator or battery unit.

(53) As illustrated in FIG. 6, the supply station host includes a first control module, a first power module, a first built-in battery, a current measurement module, and a first GPS module. The first control module serves as the core of the current supply station and is responsible for managing current electrode channel selection, power parameter configuration, external power source integration, internal power management, real-time current measurement, and synchronization of supply current data with GPS timing signals (time-stamping). It generates time-series current data and segments it into binary data files according to preset time intervals for storage. The file header contains information such as start and end times of current supply, supply station ID, channel ID, and associated electrode position coordinates.

(54) The current measurement module, controlled by the first control module, uses an analog-to-digital conversion (A/D) circuit to measure and digitize the current values, which are then recorded and stored by the control system.

(55) Under the control of the first control module, the first power module selects specific channels (A1, B1 or A2, B2) and connects them to the external boost power supply through the power interface to receive high-voltage DC input. The first power module also monitors the status of the built-in battery and manages the charging process.

(56) The first built-in battery provides the necessary energy for long-term operation of the supply station and ensures minimal energy consumption for the system clock, memory, and essential chip operations during shutdown or standby.

(57) The first GPS module consists of an external antenna and GPS control module, governed by the first control module. After successful system startup and self-check, it continuously transmits GPS codes to the host, providing real-time position and timing data. The host records these timestamps alongside the current measurements to support synchronized data processing.

(58) The supply station host is designed with multiple functional interfaces: 1. Four power output ports for connecting to distant electrodes at A1, B1, A2, and B2 via long-distance cables to sequentially supply current underground. 2. A power input interface connected to the boost power supply for high-power, high-voltage DC input. 3. A charging port for recharging the internal battery. 4. A communication interface for exporting time-series supply current data recorded during field operations. This interface is also preconfigured to accommodate an external wireless communication module for future upgrades to support remote data transmission and telemetry control.

(59) The boost power supply is a high-power DC output device with inverter and step-up capabilities. Its power output and configuration are determined according to the depth of the exploration. It may be powered either by connecting to a battery or battery pack for DC-to-DC step-up conversion, or through a generator using 220V AC step-up followed by rectification to DC.

(60) The selection between a generator, battery, or battery pack is based on the required exploration depth (power needs) and surface terrain accessibility. Batteries and portable packs offer ease of mobility but are limited in power and duration. Generators provide a broader range of output power and runtime but are bulkier and more difficult to transport in rough terrain. The choice can be made flexibly depending on the surface conditions and exploration requirements.

(61) As shown in FIG. 7, the potential measurement station includes a station host, two to four potential measurement cables, and three to four non-polarizable electrodes.

(62) The potential measurement station host includes a second control module, a second power module, a second built-in battery, a potential measurement module, and a second GPS module. The second control module functions as the core of the potential measurement station, managing channel selection for differential voltage measurements, measurement parameter configuration, internal power management, dual-channel voltage acquisition and storage, and synchronization of voltage data with GPS timing (time-stamping). It generates time-series potential difference data, segmented and stored as binary files at preset intervals. File headers include measurement start and end times, station ID, channel ID, and associated electrode coordinates.

(63) Controlled by the second control module, the potential measurement module simultaneously acquires differential voltage data from two orthogonally arranged channels, which are then recorded and stored by the control circuit.

(64) The second power module, under the control of the second control module, monitors the status of the built-in battery and manages the charging process.

(65) The second built-in battery provides long-term operating power for the potential measurement station, and maintains minimum energy consumption for system clock, memory retention, and essential chip operation during shutdown or standby.

(66) The second GPS module, composed of an external antenna and GPS unit, is controlled by the second control module. After a successful power-on self-test, it continuously transmits GPS codes to the control module, providing real-time position and timing data. This data is recorded along with voltage measurements to generate synchronized timestamps for post-processing.

(67) The potential measurement station host is equipped with several functional interfaces: 1. Four potential measurement ports connect to remote electrodes at M1, N1, M2, and N2 via measurement cables, receiving potential difference data between electrode pairs. 2. A charging port for recharging the internal battery. 3. A communication port for exporting field-recorded, time-series potential difference data during post-processing. The communication interface is preconfigured to support a wireless communication module for future upgrades, enabling remote data transmission and telemetry control.

(68) To perform inversion imaging of the subsurface within the survey area, the data from all current supply stations and potential measurement stations must first be collected. Using processing software, the four-channel voltage measurement data from each potential measurement station is extracted and synchronized with the corresponding current supply data from the supply stations based on timestamp alignment. By integrating the station IDs and spatial coordinates, the apparent resistivity tensor for each current-potential station combination is computed. These tensors are then used directly for inversion imaging, or alternatively, rotational invariants of the apparent resistivity tensors may be extracted and used for inversion to generate the final resistivity model of the subsurface.

(69) Rotational invariants are typically extracted using the following two methods:

(70) 1. Based on a Combination of Three Independent Rotational Invariants

(71) There are three independent rotational invariants (Determinant, RMS, and Trace) associated with the elements of the apparent resistivity tensor matrix. From these three rotational invariants, an infinite number of invariant sets can be derived and combined to characterize and evaluate the electrical properties of subsurface structures.

(72) (1) Square Root of the Determinant:

(73) ${}_{}=\sqrt{}=\sqrt{\left({}_{\mathrm{xx}}{}_{\mathrm{yy}}-{}_{\mathrm{xy}}{}_{\mathrm{yx}}\right)}$
(2) {square root over (2)} Times the Root Mean Square (RMS):

(74) ${}_{\mathrm{ssq}}=\sqrt{\frac{\left({}_{\mathrm{xx}}^2+{}_{\mathrm{xy}}^2+{}_{\mathrm{yx}}^2+{}_{\mathrm{yy}}^2\right)}{2}}=\sqrt{2}.Math.\mathrm{RMS}$
(3) Average of the Trace:

(75) ${}_{\mathrm{tr}\mathrm{ace}}=\frac{{}_{\mathrm{xx}}+{}_{\mathrm{yy}}}{2}$
2. WAL Rotational Invariants

(76) By decomposing the apparent resistivity tensor, it can be expressed as a combination of rotational invariants and variable components:

(77) $\stackrel{}{}={.Math.}_1.Math.\left[\begin{array}{cc}\cos 2& \sin 2\\ \sin 2& -\cos 2\end{array}\right]+{.Math.}_2.Math.\left[\begin{array}{cc}\cos 2& \sin 2\\ -\sin 2& \cos 2\end{array}\right]\text{Where:}{.Math.}_1=\frac{1}{2}\sqrt{{\left({}_{\mathrm{xy}}+{}_{\mathrm{yx}}\right)}^2+{\left({}_{\mathrm{yy}}-{}_{\mathrm{xx}}\right)}^2}{.Math.}_2=\frac{1}{2}\sqrt{{\left({}_{\mathrm{yx}}-{}_{\mathrm{xy}}\right)}^2+{\left({}_{\mathrm{yy}}+{}_{\mathrm{xx}}\right)}^2}\sin 2=\frac{{}_{\mathrm{xx}}+{}_{\mathrm{yx}}}{\sqrt{{\left({}_{\mathrm{yx}}+{}_{\mathrm{xy}}\right)}^2+{\left({}_{\mathrm{yy}}-{}_{\mathrm{xx}}\right)}^2}}\sin 2=\frac{{}_{\mathrm{xy}}-{}_{\mathrm{yx}}}{\sqrt{{\left({}_{\mathrm{yx}}-{}_{\mathrm{xy}}\right)}^2+{\left({}_{\mathrm{yy}}+{}_{\mathrm{xx}}\right)}^2}}$

(78) Where, .sub.1, .sub.2, and sin 2 are invariants. Referring to the WAL rotational invariants used in magnetotelluric impedance tensor analysis (which includes 7 parameters). The resistivity tensor of DC resistivity can reference and retain the three rotational invariants I.sub.1, I.sub.3, and I.sub.5, which can be used for evaluating ID, 2D, and 3D subsurface structure conditions.

(79) (1) 1D Rotational Invariant

(80) $I_1={.Math.}_2=\frac{1}{2}\sqrt{{\left({}_{\mathrm{yx}}-{}_{\mathrm{xy}}\right)}^2+{\left({}_{\mathrm{yy}}+{}_{\mathrm{xx}}\right)}^2}$
(2) 2D Rotational Invariant

(81) 0$I_3=\frac{{.Math.}_1}{{.Math.}_2}=\sqrt{\frac{{\left({}_{\mathrm{xy}}+{}_{\mathrm{yx}}\right)}^2+{\left({}_{\mathrm{yy}}-{}_{\mathrm{xx}}\right)}^2}{{\left({}_{\mathrm{yx}}-{}_{\mathrm{xy}}\right)}^2+{\left({}_{\mathrm{yy}}+{}_{\mathrm{xx}}\right)}^2}}$
(3) 3D Rotational Invariant

(82) $I_5=\sin 2=\frac{{}_{\mathrm{xy}}-{}_{\mathrm{yx}}}{\sqrt{{\left({}_{\mathrm{yx}}-{}_{\mathrm{xy}}\right)}^2+{\left({}_{\mathrm{yy}}+{}_{\mathrm{xx}}\right)}^2}}$

(83) These invariants are typically used in combination for inversion and visualization. By eliminating directional dependencies inherent in conventional resistivity measurements, they improve map clarity, enable multi-parameter cross-validation, and enhance interpretation accuracy.

(84) To validate the beneficial effects of the method proposed in this invention, numerical simulations and imaging were performed using both the conventional resistivity method and the tensor resistivity method of the present application. The results are shown in FIG. 8. In FIG. 8, (e) illustrates the model setup, which includes a square underground wall structure, with 1313 electrodes evenly distributed along the X and Y directions around it. In the conventional resistivity method, measurements were taken by selecting current injection and potential difference measurement points along either the X or Y direction, followed by inversion imaging. This yielded four different imaging results. The .sub.xx in (a) represents the imaging results of X-direction current and X-direction potential measurement. The .sub.xy in (b) represents the imaging results of X-direction current and Y-direction potential measurement. The .sub.yx in (c) represents the imaging results of Y-direction current and X-direction potential measurement. The .sub.yy in (d) represents the imaging results of Y-direction current and Y-direction potential measurement. As evident from the imaging results, the conventional scalar resistivity method suffers from pronounced directional dependence. For example, X-direction current with X-direction potential measurement (pxx) fails to clearly image wall boundaries along the Y direction and introduces false anomalies outside the Y boundaries. Similarly, Y-direction current with Y-direction potential measurement (yy) fails to resolve X-direction boundaries and produces false anomalies on the X periphery. The pxy and pyx configurations result in even more complex image distortions, boundary loss, and artificial anomalies. The (f) in FIG. 8 shows the imaging result obtained using the tensor resistivity method. The rotational invariants of the apparent resistivity tensor were extracted and used for inversion. The result shows a strong match with the true wall model, eliminates directional deficiencies, and effectively suppresses directional artifacts.

(85) FIG. 9 presents a comparison of numerical simulations using different current injection strategies. The (a) in FIG. 9 is a model setup with five small target bodies placed in the survey area, with grid-based electrode arrangement identical to that shown in FIG. 7. The (b) in FIG. 9 is an imaging result using the tensor resistivity method with current injected at single grid-node spacing. The five targets are not clearly resolved. The (c) in FIG. 9 is an imaging result using six-node spacing. The targets remain poorly resolved. The (d) in FIG. 9 is an imaging result with current injected from electrodes placed at twelve-node spacing (near the model boundary). All five targets are clearly resolved. Additional simulations with even greater spacing demonstrated that increasing the distance between current electrodesthrough the use of external current injection points or combined external and internal injection schemessignificantly enhances imaging performance.

(86) It will be understood by those skilled in the art that the above-described embodiments are merely preferred examples and are not intended to limit the scope of the invention. Although the invention has been described in detail with reference to these embodiments, modifications or equivalent replacements of the technical features disclosed herein are still possible. Any changes made within the spirit and scope of the invention shall be considered within the protection scope of the present application.