EP2414629B1 - Two coil guidance system for tracking boreholes - Google Patents

Description

• The present invention relates to a method and apparatus for tracking and guiding the drilling of a borehole, and is applicable to tracking a borehole being drilled generally horizontally under an obstacle such as a river, a highway, a railroad, an airport runway or the like, where access to the ground above the borehole is difficult or perhaps not possible.
• Various well-known drilling techniques have been used in the placement of underground transmission lines, communication lines, pipelines, or the like through or beneath obstacles of various types. In order to traverse the obstacle, the borehole must be tunneled along a planned path underneath the obstacle from an entry point on the Earth's surface to a desired exit point. The borehole then may receive a casing which may be used, for example, as a pipeline or for receiving cables for use as power transmission lines, communication lines, or the like. In the drilling of such boreholes, it is important to maintain them on a carefully controlled track following a prescribed drilling proposal, for often the borehole must remain within a precisely defined right of way as it passes under the obstacle and its entry and exit points on opposite sides of the obstacle must often be at precisely defined locations. In order to do this, the driller must have an accurate determination of the lateral position of the borehole as it is being drilled, as well as the precise distance to the exit location so that appropriate adjustments to the inclination and direction of drilling can be made. Even if the radial distance to the exit location from the entry point of the borehole into the Earth is precisely known and the radial distance of the drill bit from its entry point into the Earth is precisely known, safety considerations alone give high priority to directly determining the relative location of the desired borehole exit point with respect to the drill bit location as the exit point is approached.
• Conventional directional drilling techniques used to drill such boreholes commonly use a steering tool which measures the borehole inclination, azimuth and tool roll angle at each station along the path where measurements are made. The borehole coordinates are computed and tabulated from these steering tool data as a function of the measured distance along the borehole, which may be referred to as the measured depth of the steering tool, or the "away" distance from a reference
  point such as the borehole entry point. However, these borehole coordinates suffer from serious cumulative effects caused by small errors in the inclination and azimuth determinations made at regularly spaced stations along the borehole, and the lateral errors generated by such conventional borehole surveying are intolerable.
• A number of prior electromagnetic systems present problems to the user since they require access to the land directly above the path to be followed by the borehole in order to permit placement of surface grids or other guidance systems. Often, however, access to this land is not available; furthermore, the placement of guidance systems of this kind can be extremely time consuming, and thus expensive. The Earth's magnetic field is usually utilized for determining azimuthal direction in such prior systems, but this creates additional problems because of the disturbances caused by nearby steel objects such as bridges, vehicular traffic and trains. Therefore, although steering tool inclinometers provide good inclination measurements, usually to a precision of 0.1 or 0.2 degrees, the standard steering tool azimuthal direction determination provided by the Earth Field magnetometers is inadequate.
• To avoid the need for placing guide wire grids on the earth's surface above the proposed borehole path, systems have been provided for guiding a drill in which a solenoid generates magnetic field signals that are measured at field sensors at the drill stem. One such system is illustrated in U.S. Pat. No. 5,513,710 to Kuckes , which discloses a drilling guidance method for drilling boreholes under rivers and other obstacles using a direct current powered solenoid and an industry standard MWD system.
• Patents Nos. 6,626,252 and 6,814,163, to Kuckes , similarly describe a drill tool which includes a three-axis magnetometer for detecting vector components of target magnetic fields produced by a large solenoid which incorporates a coil surrounding a large ferromagnetic core. The solenoid is connected to a reversible source of direct current of sufficient magnitude to provide a target magnetic field in the region of the proposed path of the drilling.
• In systems for guiding horizontal boreholes such as those described in the '710, the '252 and the '163 patents, the drilling equipment is placed at a location where a borehole is to be started, while the target solenoid is positioned at or near the area where the borehole is to exit the ground. The borehole entrance may be at or near one side of an obstacle, such as the bank of a river, with the borehole passing beneath the river to an exit beyond the opposite bank. Drilling the borehole is begun at the entrance site and conventional survey methods are used to guide the drill for a major part of the distance toward the exit location. As the borehole nears the desired exit site; for example, within about 100 meters, further guidance is by way of the target solenoid field. Whenever a survey is required, the drill is stopped and the sensor system in the drilling tool is activated to measure the x, y and z components of the total magnetic field in the region of the sensor. These measurements, together with downhole tool orientation measurements, are then used to determine the distance and direction from the drilling tool to the solenoid.
• Still another prior system is illustrated in U.S. Patent No. 6,466,020 to Kuckes et al , wherein a target magnetic field for borehole drilling guidance is generated by a loop/guide wire antenna system on the Earth's surface above the proposed path of the borehole. The loop is energized by a known current to generate a magnetic field that is measured by two single-axis electromagnetic field sensors that lie on an imaginary spherical surface at the known radial distance from a point such as the entry point of the borehole. These sensors are approximately perpendicular to the radius of this spherical segment and to each other. Measurement of the two perpendicular electromagnetic field components generated by the loop is made by the two sensors at a selected borehole site. The radial distance from the drill bit entry point to the selected measurement site is determined using standard integration techniques, as well as steering tool measurements of inclination from the Earth's gravity and azimuth from the Earth's magnetic field along the borehole together with the measured depth of the sensors. The electromagnetic field measurements by these two sensors together with this computed radial distance from the entry point are matched to theoretically computed values for field vectors on the imaginary spherical surface at the known radial distance of the measurement site to determine the location of the drill bit.
• A second embodiment of the system described in the '020 patent includes two electromagnetic field source loops, both near the punch-out location. A guide wire leg of each loop is positioned on the Earth's surface along the proposed path of the borehole to produce corresponding electromagnetic fields on that path. The loop geometry is designed so that there is a rapid variation in the x and y components of the fields generated near the portion of the borehole beyond the edge of the loops. The rapid variations in the x and y field components in this region are used find the radial distance from the drill bit to the proposed punch out location. The radial distance to the punch out point is usually a critical parameter.
• Although such prior systems are useful in various drilling guidance applications, it has been found that a continuing problem in horizontal drilling of boreholes is the accurate determination of the lateral position of the drill bit in a borehole and the direction of drilling when the drill bit is far from the point at which the borehole passes under an obstacle; for example when it is at a great distance from the shore of a river.
• United States Patent Publication No. US 3, 529, 682 A describes location detection and guidance systems for a burrowing device. European Patent Application Publication No. 0 622 524 A1 describes a system for measuring an excavation position of an underground excavator. However, neither of these publications discloses the use of differing coordinate systems to determine direction.
• In one aspect, the invention provides an apparatus for borehole surveys, according to Claim 1.
• In another aspect, the invention provides a method for location determination according to Claim 6.
• In order that the invention will be more readily understood, embodiments thereof will now be described, by way of example only, in relation to the drawings, and in which:-
• FIG. 1 is a diagrammatic illustration of a drill guidance system embodying the invention for guiding the drilling of a horizontal borehole following a proposed path to a proposed punch out point, in which two electromagnetic guidance coils are deployed on the far side of an obstacle, illustrated as a river;
• FIG. 2 is a diagrammatic top plan view of a portion of the guidance system of FIG. 1;
• FIG. 3 is a diagrammatic illustration of exemplary downhole instrumentation and an uphole computer in accordance with an embodiment of the invention;
• FIG. 4 is a diagrammatic illustration of the electronics powering each of the two loops of the illustrated configuration of FIG. 1;
• FIG. 5 is a diagrammatic illustration of the current and clock signal controlling each of the loops of the configuration of FIG. 1;
• FIG. 6 is a diagrammatic illustration of the wire loop geometry together with representations of the symbols used for the relative position vectors of a loop and a downhole location P;
• FIG. 7 is a diagrammatic illustration showing the representation of quantities which enter into the theoretical computation of the electromagnetic field generated by a loop segment at the point P;
• FIG. 8 is a diagrammatic representation of the xyz unit vectors and of the bes unit vectors together with the direction of gravity; and
• FIG. 9 is a top view, looking vertically down showing the relationship of the a, e, and r unit vectors of the aer coordinate system and of the b, e, and s unit vectors of the bes coordinate system.
• Embodiments of the present invention are directed to an improved method and apparatus for providing guidance in drilling boreholes. Embodiments of the invention disclosed herein use two electromagnetic sources consisting of two laterally spaced-apart loops of wire carrying alternating current to provide target magnetic fields that are used to determine the location and direction of the drill head and to guide further drilling. At least one of the loops is significantly off to one side of the borehole being tracked, and both loops are laterally spaced from the proposed path. In one embodiment, two laterally spaced loops are on opposite sides of the proposed path and of the exit point of the borehole. Vector components of the electromagnetic fields are measured near the drill bit in the borehole by an array of magnetic field sensors, and accelerometers are provided to determine the direction of the Earth's gravitational field at the location of the magnetic field sensors. These electromagnetic field and accelerometer measurements are analyzed by mathematical techniques using the known loop geometry and electric currents to determine the drill bit location and the azimuthal direction of drilling. The drilling inclination direction is determined using standard accelerometer data. Alternatively, the location vector to the drilling point can be determined from the measured electromagnetic field vectors without the use of an accelerometer. The detailed geometry, power levels and frequency of operation and signal averaging time are subject to design criteria constraints including factors such as the size and location of the spaces available for positioning the loops and the consequent diameters of the loops and the number of turns of wire in each loop.
• The method, and the advantages, of the present system are illustrated by selecting an exemplary set of typical parameters for the wire loops used in embodiments of the invention. In this example, two source wire loops are provided, one on each side of the borehole path, with each loop being approximately 50 meters in diameter and the loops being laterally separated by 150 meters. Each loop is powered by 5 kilowatts to generate corresponding magnetic fields, which may be referred to as target fields. Measurement of the target magnetic fields generated by this current at a selected point, such as a field detector on a drill head 500 meters distant from the loops, readily provides a lateral location precision of 5 meters with respect to the proposed path and a few degrees precision in the drilling direction after a short period of signal averaging. Measurements made at a drilling location closer to the source loops will yield more precise results.
• A unique and novel feature of the present application is the utilization of two independent, laterally separated source loops to generate electromagnetic fields for use in determining the location and direction of field sensors, and more particularly for guiding borehole drilling. Such laterally displaced source loops give a much greater useful range away from the electromagnetic field source for better distant location resolution and for more accurately guiding the drilling of a borehole, than has heretofore been possible.
• One embodiment of the apparatus utilized in the method of the present invention is generally illustrated at 10 in FIG. 1, wherein a borehole 12 is drilled under an obstacle such as a river for use in the laying of pipeline, for example. The apparatus incorporates a drill head having a down-hole industry standard drilling motor 14 for driving a drilling bit under the control of equipment at a drill rig 16 at the Earth's surface. The crossing of the river 18 may entail drilling along a planned path 20 which may pass under the river at a depth of 30 meters, for example, to a planned "punch-out point", or exit location 22, which may be, for example, 1000 to 1500 meters away from a borehole entry point 24. The progress of the borehole may be tracked for part of the distance to the exit point by conventional survey equipment, but as the drill bit moves further away from the entry point 24, guidance of the drill bit is best carried out by the use of target magnetic fields generated by two wire loops 26 and 28. In this case the loops are illustrated as being located beyond the far bank 32 of the river 18, on the surface 34 of the Earth at the exit side of the river. It will be understood that the loops may be located on the near side of the river or other obstacle, or in some cases pairs of loops may be provided on both sides of the obstacle for precision guidance of the drill. The method is particularly important for drilling guidance while drilling under a river or other obstacle where it is impractical to deploy a guide wire system on the surface.
• The loops 26 and 28 are energized, as will be explained below, to produce target magnetic fields that are detected at selected measurement stations along the proposed path by a measurement while drilling (MWD) package 36 located on the drill head at or near the drilling motor 14 to provide the information needed to guide the drilling at each measurement station under the river and subsequently under the earth's surface 34 as drilling progresses toward the proposed exit location 22. As illustrated in the embodiment of FIG. 1, the drilling motor 14 is mounted on a drill stem 38 to drive a drill bit 40, in conventional manner. The MWD package 36 preferably includes a conventional three component accelerometer to measure the direction of gravity and a conventional three component magnetometer to measure alternating magnetic fields, and is mounted on the drill stem adjacent, and just above, the drilling motor.
• In an embodiment of the invention, the loop sources 26 and 28 are laterally spaced from each other, and are positioned by land surveying techniques at selected, known locations with respect to the proposed path 20 of the borehole being drilled. At least one of the loop sources is laterally displaced a significant distance from the proposed path 20. Both loops are displaced from the path. And advantageously the loops are on opposite sides of path 20, as illustrated in FIGS. 1 and 2. Each loop may consist of 5 turns of #12 wire, and may be 50 meters in diameter d. As illustrated in FIG. 2, the centers 50 and 51 of loops 26 and 28, respectively, are separated by approximately 150 meters, and in the illustrated embodiment are equally spaced by distances 52 and 53 on opposite sides of the path 20, with a line between the center points 50 and 51 being perpendicular to the proposed path 20.
• The loops 26 and 28 are energized by a 5 kw motor/generator set 54 through suitable power control electronics 56 and power lines 58 and 60, respectively, as diagrammatically illustrated in FIG. 1. The control electronics simultaneously supply power to the two loops at different frequencies; for example, loop 26 may be energized by an alternating current having a frequency of 1 Hertz, while loop 28 is energized by alternating current at a frequency of 1.14 Hertz, to produce corresponding magnetic fields in the Earth in the region of the path 20. Alternatively, the loops may be separately energized by AC currents at the same frequency, in which case two sets of measurements are made sequentially, first with one loop powered and then with the second being powered. The important point is that independent measurements are made of the electromagnetic field components generated by each loop.
• In the illustrated embodiment of the invention, the magnetic field source loops 26 and 28 are positioned near the planned punch-out point 22 of the borehole, in this case on the far side of the obstacle under which the borehole is to be drilled, from the entry point of the borehole. It will be understood, however, that the source loops can be positioned at any location along, or even beyond, the path 20, wherever precise lateral control of the drill bit and thus of the direction of drilling, is needed. It is to be noted that the larger the lateral separation of the loops and the fewer the number of turns for a fixed length of wire and excitation power, the better. The choice of parameters, in the example given above, is based on the limitations imposed by land accessibility and other such factors in the region of the path 20 of the borehole, and may vary in accordance with specific applications.
Data Acquisition and Processing
• Typically, the direction of drilling a borehole, such as the illustrated borehole 24, is initially controlled by standard borehole surveying techniques in which a steering tool measures the borehole inclination, azimuth and tool roll angle at successive stations along the path of the borehole. The borehole coordinates are computed and tabulated from the steering tool data as a function of the measured distance along the borehole and directional signals are generated and sent to the steering tool to control further drilling. However, as described above, these borehole coordinates suffer from serious cumulative effects caused by small errors in the inclination and azimuth determinations made at regularly spaced stations along the borehole, and the lateral errors generated by such conventional borehole surveying can cause serious problems. In accordance with embodiments of the present invention, these lateral errors are overcome by utilizing the magnetic fields generated by laterally spaced loops 26 and 28 located where lateral drilling precision is needed.
• To obtain the data needed for accurate location of the drill head, which may include the drilling motor 14 and the drill bit 40, the loops 26 and 28 are electrically excited by alternating current after drilling has been stopped at a measurement station along the proposed borehole path. The resulting alternating electromagnetic fields are detected by the downhole MWD instrument package 36 on the drill string, and the resulting output data signals are transmitted uphole to the drilling apparatus 16, a few minutes of data are recorded and the data file is generated.
• As illustrated in FIG. 3, the MWD package 36 incorporates conventional magnetometers 60 for measuring the x, y and z vector components of the AC magnetic fields generated by the current flow in the loops 26 and 28 and the
  x, y and z components of the Earth's magnetic field. The AC magnetometer output signals are supplied through corresponding AC amplifiers 62, and the Earth's magnetic field vector signals are supplied directly, to respective inputs of multiplexer and telemetry circuitry 64 which is driven by a suitable downhole clock circuit 66. The MWD package also incorporates x, y and z accelerometers 68 for measuring the x, y and z components of the Earth's gravity and supplies corresponding output signals to inputs of multiplexer 64. The multiplexer output is connected by way of a suitable communications link such as a cable 70 to uphole communication equipment 72 which includes an instrument power supply and telemetry circuits 74. Equipment 72 also includes an uphole computer 76 which receives demultiplexed data signals from the telemetry circuitry 74 at Earth's field circuit 77, gravity circuit 78, and AC field circuit 80. These signals are recorded at corresponding data files 82, 84 and 86, respectively, and the computer then determines the direction of further drilling in processor 90 and sends suitable drill control signals downhole by way of controller circuit 92 and telemetry 74 to a downhole controller 94 for the drill motor 14.
• As illustrated in FIG. 4, the powering electronics 56 for the motor/generator set 54 includes an uphole clock 98 which generates a square wave clock signal 100, illustrated in FIG. 5(A), that drives an FET switching circuit 102 which is connected by way of respective lines 58 and 60 to supply an alternating current 104, illustrated in FIG. 5(B), to the loops 26 and 28. Electronics 56 also includes meters 106 for measuring the current in each loop.
• The first step for processing the recorded magnetic field data at processor 90 is the generation of a reference wave form by a reference clock 110 (FIG. 3) which is time synchronized with the clock 98 for the loop switching circuitry 102. Magnetic field data signals from the downhole magnetometers 60 that are stored in the AC field data files 86 are time averaged with respect to the reference wave form to produce two single-row data matrices of the three magnetic field components. This is done for magnetic field signals generated by the magnetometers in response to the magnetic fields H1 and H2 produced by the loops 26 and 28, respectively, either simultaneously when the loops are energized by different frequencies, or sequentially when they are energized by a single frequency. The location of the magnetic field sensor 60 with respect to the two loops, and thus the location of the drill with respect to the proposed path of the borehole, and the direction of drilling, can be determined by mathematically optimizing the match between measured and computed field components using mathematically well known methods, e.g. by using a least squares fitting method.
• Since the least squares method, using Taylor series expansions to linearize the fitting process, is so well suited to computerized analysis of these data a procedure based upon this method will be described. The first step is to precisely lay out the two electromagnetic source loops, or coils, 26 and 28 on the ground as shown in FIG. 6 for loop 26. As illustrated, the loop is defined by n straight line segments 120, which may be referred to generally as segments Ws(1)...Ws(n), with adjacent segments being joined at nodes 122 The loop, or coil, 26 has N turns, and when energized by the source 56 carries an electric current I. In the illustration, the positive current flows counterclockwise. A surveyor's wire file listing the three dimensional coordinates of the vectors Rw(i), Rw(i-1)...Rw(i-n) with respect to a reference origin point 124 for the wire element nodes 122 of the coils 26 and 28 is generated using standard land surveying techniques with customary away, elevation, right (aer) coordinate axes as illustrated in FIG. 6. Each node vector is defined by a vector Rw(i) from the origin 124 by its away, elevation and right (aer) coordinates. The elevation unit vector e defining the aer system points vertically upwards, as illustrated by axis 130. The away axis 132 is horizontal, usually chosen to be from the entry point toward the proposed borehole exit point., and the right unit vector axis 134 is also horizontal and points to the right of the away axis.
• The electromagnetic Law of Biot Savart can be used to compute the H1aer and H2aer (away, elevation, right) vector components of the electromagnetic fields H1 and H2 specified in the away, elevation, right coordinate system for any location P at which the target electromagnetic field is to be evaluated. The location P is defined by the vector Raer as shown in FIG. 7. This is done by breaking each loop into an ensemble of wire segments such as segment Ws(i) shown in FIG. 7. The field generated by the current I carried in each of the N wires of each segment is readily found by summing vectorally the expressions for H(i) generated by each of the n wire segments Ws(i) shown in FIG. 6 for all the wire segments constituting a loop.
• The field H(i) for a wire segment as shown in FIG. 7,using "MATLAB" notation conventions in the following expressions, is: $$\mathrm{H}\left(\mathrm{i}\right)=\left(\mathrm{N}*\mathrm{I}*\mathrm{Huv}\left(\mathrm{i}\right)/\left(4*\mathrm{pi}*\mathrm{b}\right)\right)*\left(\mathrm{cosB}\left(\mathrm{i}\right)-\mathrm{cosA}\left(\mathrm{i}\right)\right)$$

  

  $$\mathrm{Huv}=\mathrm{cross}\left(\mathrm{Ws}\left(\mathrm{i}\right),\mathrm{r}\left(\mathrm{i}\right)\right)/\mathrm{mag}(\mathrm{cross}\left(\mathrm{Ws}\left(\mathrm{i}\right),\mathrm{r}\left(\mathrm{i}\right)\right)$$

  

  b =mag(cross(Ws(i),r(i)) $$\begin{array}{l}\mathrm{cosB}\left(\mathrm{i}\right)=\mathrm{dot}\left(\mathrm{r}\left(\mathrm{i}\right),\mathrm{Ws}\left(\mathrm{i}\right)\right)/\left(\mathrm{mag}\left(\mathrm{r}\left(\mathrm{i}\right)\right)\ast \mathrm{mag}\left(\mathrm{Ws}\left(\mathrm{i}\right)\right)\right)\\ \mathrm{cosA}\left(\mathrm{i}\right)=\mathrm{dot}\left(\mathrm{r}\left(\mathrm{i}-1\right),\mathrm{Ws}\left(\mathrm{i}\right)\right)/\left(\mathrm{mag}\left(\mathrm{r}\left(\mathrm{i}-1\right)\right)\ast \mathrm{mag}\left(\mathrm{Ws}\left(\mathrm{i}\right)\right)\right)\end{array}$$

  

  The function mag denotes the conventional magnitude of a vector, e.g. mag(R) denotes sqrt(dot(R,R)). Thus the theoretical vectors Hlaerth and H2aerth for the electromagnetic field vectors H1 and H2 generated by each loop, as represented by their aer vector components, is readily done.
• To control the drilling of a borehole, the driller is given the proposed borehole path coordinates expressed in the aer coordinate system and the measured borehole coordinates found in the course of drilling are expressed in the same aer system. Usually, the coordinates of a borehole being drilled are obtained in the course of drilling using conventional borehole surveying techniques based upon integrating a large number of station measurements of gravity and of the Earth's magnetic field along the borehole. The difficulty is that, far away from the borehole entry point, a borehole survey generated in this way does not have sufficient precision due to the cumulative build-up of error along the borehole length. In accordance with an embodiment of the present invention, the driller is provided with the approximate survey file of the borehole, and this is improved upon using embodiments of the method and apparatus of this invention.
• To make an electromagnetic determination of the location and drilling azimuth, measurements of three components of the electromagnetic fields generated by the loops 26 and 28, and of gravity, are made by the borehole instrument 36 at a location P at a chosen depth, or away distance along the borehole. The instrument 36 sensing axes are defined by an xyz coordinate system 140 fixed to the downhole drilling tool. The z axis 142 of system 140, illustrated in FIG. 8, is along the axis of the borehole, with the positive z axis pointing in the direction of drilling. The x and y axes 144 and 146, respectively, are perpendicular to z and to each other, the xyz system forms a right handed system of coordinate axes.
• The analysis of the H1xyz and H2xyz data, i.e., the xyz vector components of the measurements of the down-hole electromagnetic fields produced by loops 26 and 28, respectively, is best analyzed by transforming them to a bes coordinate system 150, whose axes are shown in FIG. 9, and are shown together with those of the xyz system 140 in FIG. 8. The "b" axis of the bes system is the projection of the borehole drilling direction on to the horizontal plane. The "e" direction is vertically upwards, i.e., the direction "e" of the "bes" coordinate axes is
  the same as "e" direction of the surveyors "aer" system. The direction "s" is in the horizontal plane and points to the right of the borehole. These directions are defined by the accelerometer measurements. The direction of s is given by the vector cross product of the measured gravity vector g, g=[gx; gy; gz] and z, i.e., [0; 0; 1]; the unit vector s is found by taking this cross product and dividing by the magnitude of this vector, thus: $$\mathrm{s}=\mathrm{cross}\left(\mathrm{g},\mathrm{z}\right)/\mathrm{mag}\left(\mathrm{cross}\left(\mathrm{g},\mathrm{z}\right)\right)$$

  
• The elevation unit vector "e" is a unit vector pointing in the opposite direction as g, i.e. $$\mathrm{e}=-\mathrm{g}/\mathrm{mag}\left(\mathrm{g}\right)$$

  
• Finally, the projection of the borehole direction z in the horizontal plane is in the direction of the unit vector b given by $$\mathrm{b}=\mathrm{cross}\left(\mathrm{e},\mathrm{s}\right)$$

  
• The matrix to transform a vector represented by its xyz components to its representation using bes components is given by the 3 by 3 matrix: $$\mathrm{xyztobes}=\left[\mathrm{b}\prime ;\mathrm{e}\prime ;\mathrm{s}\prime \right]$$

  
• The measurements of the electromagnetic fields H1x, H1y, H1z generated by loop 1 are conveniently represented by the 3 by 1 matrix: $$\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{xyz}=\left[\mathrm{H}1\mathrm{x};\mathrm{H}1\mathrm{y};\mathrm{H}1\mathrm{z}\right]$$

  

  Similarly, the measurements H2x, H2y, H2z are represented by the 3 by 1 matrix: $$\mathrm{H}2\mathrm{xyz}=\left[\mathrm{H}2;\mathrm{H}2\mathrm{y};\mathrm{H}2\mathrm{z}\right]$$

  

  And the representation of the H1 and H2 measurements in the bes coordinate system is given by the matrix products: $$\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{bes}=\mathrm{xyztobes}*\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{xyz}\mathrm{and}\mathrm{H}2\mathrm{bes}=\mathrm{xyztobes}*\mathrm{H}2\mathrm{xyz}$$

  
• These 3 by 1 matrix representations of the measurements of H1 and H2 are to be compared to theoretical values of the corresponding quantities H1th and H2th, which are conveniently calculated using the Law of Biot Savart with respect to the "aer" coordinate system used by the land surveyor. The matrix representation of H1th and H2th with respect to the aer coordinate axes will be written as: $$\begin{array}{l}\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{aerth}=\left[\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{ath};\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{eth};\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{rth}\right]\mathrm{and}\\ \mathrm{H}2\mathrm{aerth}=\left[\mathrm{H}2\mathrm{ath},\mathrm{H}2\mathrm{eth},\mathrm{H}2\mathrm{rth}\right]\end{array}$$

  
• To compare the theoretical and measured field values, represent the H1th and H2th vectors in the bes coordinate system. The matrix to transform vectors from the aer system to the bes system is given by: $$\mathrm{aertobes}=\left[\cos \left(\mathrm{Aab}\right)0\sin \left(\mathrm{Aab}\right);010;-\mathrm{<figure-callout\; id="0"\; label="sin\; Aab"\; filenames="imgf0005.png"\; state="\{\{state\}\}">sin</figure-callout>}\left(\mathrm{Aab}\right)\left(\right)0\cos \left(\mathrm{Aab}\right)\right]$$

  

  The representation of the theoretical values H1th and H2th calculated with respect to a surface aer system can be converted to a down-hole bes system by the expressions: $$\begin{array}{cc}\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{besth}=\mathrm{aertobes}*\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{aerth}& \mathrm{H}2\mathrm{besth}=\mathrm{aertobes}*\mathrm{H}2\mathrm{aerth}\end{array}$$

  
• The mathematical problem is thus reduced to finding the value of the vector R from the reference origin to the drill bit represented by the 3 by 1 matrix Raer and of the azimuthal angle Aab (FIG. 9) between the away axis and the horizontal projection of the drilling direction. Thus, four numbers are to be determined to give the best fit of H1besth and H2besth to the 6 measurements of the H1bes and H2bes. To do this, start with an approximate value of R which is called Raer0, and an approximate value of the drilling azimuth which is called Aab0, and then form from them a 4 by 1 matrix RA0 i.e.: $$\mathrm{<figure-callout\; id="0"\; label="RA"\; filenames="imgf0005.png"\; state="\{\{state\}\}">RA</figure-callout>}0=\left[\mathrm{<figure-callout\; id="0"\; label="Ra"\; filenames="imgf0005.png"\; state="\{\{state\}\}">Ra</figure-callout>}0;\mathrm{<figure-callout\; id="0"\; label="Re"\; filenames="imgf0005.png"\; state="\{\{state\}\}">Re</figure-callout>}0;\mathrm{<figure-callout\; id="0"\; label="Rr"\; filenames="imgf0005.png"\; state="\{\{state\}\}">Rr</figure-callout>}0;\mathrm{<figure-callout\; id="0"\; label="Aab"\; filenames="imgf0005.png"\; state="\{\{state\}\}">Aab</figure-callout>}0\right]$$

  
• The values of the elements of RA0 can usually obtained from the driller's borehole survey, but if a survey is not available, an educated guess is usually sufficient. Then, together with RA0, define an associated 4 by 1 difference matrix: $$\mathrm{dRA}=\left[\mathrm{dRa};\mathrm{dRe};\mathrm{dRe},\mathrm{dAab}\right]$$

  

  The intent is to make the elements of $$\mathrm{RA}=\mathrm{<figure-callout\; id="0"\; label="RA"\; filenames="imgf0005.png"\; state="\{\{state\}\}">RA</figure-callout>}0+\mathrm{dRA}$$

  

  such that the 6 values of H1besth and H2besth evaluated at the 4 element values of RA match the 6 electromagnetic field measurement values contained in H1bes and H2bes. Call the values of H1besth and H2besth, evaluated at RA0, H1besth0 and H2besth0 and form a 6 by 1 matrix H12besth0, i.e.: $$\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{besth}0=\left[\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{<figure-callout\; id="0"\; label="bth"\; filenames="imgf0005.png"\; state="\{\{state\}\}">bth</figure-callout>}0;\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{<figure-callout\; id="0"\; label="eth"\; filenames="imgf0005.png"\; state="\{\{state\}\}">eth</figure-callout>}0;\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{<figure-callout\; id="0"\; label="sth"\; filenames="imgf0005.png"\; state="\{\{state\}\}">sth</figure-callout>}0;\mathrm{H}2\mathrm{<figure-callout\; id="0"\; label="bth"\; filenames="imgf0005.png"\; state="\{\{state\}\}">bth</figure-callout>}0;\mathrm{H}2\mathrm{<figure-callout\; id="0"\; label="eth"\; filenames="imgf0005.png"\; state="\{\{state\}\}">eth</figure-callout>}0;\mathrm{H}2\mathrm{<figure-callout\; id="0"\; label="sth"\; filenames="imgf0005.png"\; state="\{\{state\}\}">sth</figure-callout>}0\right]$$

  
• The corresponding theoretically computed field values computed at RA, which at the moment is unknown, can be written as: $$\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{besth}=\left[\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{bth};\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{eth};\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{sth};\mathrm{H}2\mathrm{bth};\mathrm{H}2\mathrm{eth};\mathrm{H}2\mathrm{sth}\right]$$

  

  Now Taylor expand H12besth0 about the region RA0, and call that value H12besth i.e. write the matrix equation which represents 6 linear equations with 4 unknowns, i.e.: $$\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{besth}=\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{<figure-callout\; id="0"\; label="besth"\; filenames="imgf0005.png"\; state="\{\{state\}\}">besth</figure-callout>}0+\mathrm{<figure-callout\; id="12"\; label="dH"\; filenames="imgf0001.png"\; state="\{\{state\}\}">dH</figure-callout>}12\mathrm{besdRAb}*\mathrm{dRA}.$$

  

  The quantity dH12besdRAb is a 6 by 4 matrix with 24 "derivative" elements, i.e., $$\mathrm{<figure-callout\; id="12"\; label="dH"\; filenames="imgf0001.png"\; state="\{\{state\}\}">dH</figure-callout>}12\mathrm{besdRAb}=\left[\mathrm{<figure-callout\; id="1"\; label="dH"\; filenames="imgf0005.png"\; state="\{\{state\}\}">dH</figure-callout>}1\mathrm{<figure-callout\; id="1"\; label="bthdRa\; dH"\; filenames="imgf0005.png"\; state="\{\{state\}\}">bthdRa</figure-callout>}\mathrm{dH}1\mathrm{<figure-callout\; id="1"\; label="bthdRe\; dH"\; filenames="imgf0005.png"\; state="\{\{state\}\}">bthdRe</figure-callout>}\mathrm{dH}1\mathrm{<figure-callout\; id="1"\; label="bthdRr\; dH"\; filenames="imgf0005.png"\; state="\{\{state\}\}">bthdRr</figure-callout>}\mathrm{dH}1\mathrm{bthdAab};\mathrm{<figure-callout\; id="1"\; label="dH"\; filenames="imgf0005.png"\; state="\{\{state\}\}">dH</figure-callout>}1\mathrm{<figure-callout\; id="1"\; label="ethdRa\; dH"\; filenames="imgf0005.png"\; state="\{\{state\}\}">ethdRa</figure-callout>}\mathrm{dH}1\mathrm{<figure-callout\; id="1"\; label="bethdRe\; dH"\; filenames="imgf0005.png"\; state="\{\{state\}\}">bethdRe</figure-callout>}\mathrm{dH}1\mathrm{<figure-callout\; id="1"\; label="ethdRr\; dH"\; filenames="imgf0005.png"\; state="\{\{state\}\}">ethdRr</figure-callout>}\mathrm{dH}1\mathrm{bthdAab};\mathrm{<figure-callout\; id="1"\; label="dH"\; filenames="imgf0005.png"\; state="\{\{state\}\}">dH</figure-callout>}1\mathrm{<figure-callout\; id="1"\; label="sthdRa\; dH"\; filenames="imgf0005.png"\; state="\{\{state\}\}">sthdRa</figure-callout>}\mathrm{dH}1\mathrm{<figure-callout\; id="1"\; label="eshdRe\; dH"\; filenames="imgf0005.png"\; state="\{\{state\}\}">eshdRe</figure-callout>}\mathrm{dH}1\mathrm{<figure-callout\; id="1"\; label="sthdRr\; dH"\; filenames="imgf0005.png"\; state="\{\{state\}\}">sthdRr</figure-callout>}\mathrm{dH}1\mathrm{bthdAab};\mathrm{dH}2\mathrm{<figure-callout\; id="12"\; label="bthdRa\; dH"\; filenames="imgf0001.png"\; state="\{\{state\}\}">bthdRa</figure-callout>}\mathrm{dH}12\mathrm{bthdRe}\mathrm{dH}2\mathrm{bthdRr}\mathrm{dH}2\mathrm{bthdAab};\mathrm{dH}2\mathrm{ethdRa}\mathrm{dH}2\mathrm{ethdRe}\mathrm{dH}2\mathrm{ethdRr}\mathrm{dH}2\mathrm{ethdAab};\mathrm{dH}2\mathrm{sthdRa}\mathrm{dH}2\mathrm{sthdRe}\mathrm{dH}2\mathrm{sthdRr}\mathrm{dH}2\mathrm{shdAab}\right]$$

  
• Each of the derivatives in this relation are to be read as calculus partial derivatives evaluated at the element values contained in RA0; e.g., dH1bthdRa in the notation of calculus would be written as dH1bth/dRa. The derivative dH1bthdRa is the change in H1bth per unit change in the parameter Ra near the RA0 while holding Re0, Rr0 and Aab0 constant. In a computer program it can be computed, for example by evaluating H1bth at [Ra0; Re0; Rr0; Aab0] and at [Ra0+0.01; Re0; Rr0; ;Aab0] (where "0.01" is any appropriately small number). The first value H1bth is subtracted from the second and this difference is divided by 0.01, following the basic definition of differentiation.
• To proceed, equate the Taylor series expansion (Eq.16) of H12besth about the 4 by 1 matrix RA0 to the measurement matrix H12besmeas: $$\begin{array}{l}\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{besmeas}=\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{besth}=\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{<figure-callout\; id="0"\; label="besth"\; filenames="imgf0005.png"\; state="\{\{state\}\}">besth</figure-callout>}0+\mathrm{<figure-callout\; id="12"\; label="dH"\; filenames="imgf0001.png"\; state="\{\{state\}\}">dH</figure-callout>}12\mathrm{besdRA}*\mathrm{<figure-callout\; id="12"\; label="dRA\; H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">dRA</figure-callout>}& \\ \mathrm{H}\mathrm{}12\mathrm{besmeas}=\left[\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{bes};\mathrm{H}2\mathrm{bes}\right]\end{array}$$

  

  This over determined equation is readily solved for dRA in the least squares sense using the computer program "MATLAB" operation of left matrix division as $$\mathrm{dRA}=\mathrm{<figure-callout\; id="12"\; label="dH"\; filenames="imgf0001.png"\; state="\{\{state\}\}">dH</figure-callout>}12\mathrm{<figure-callout\; id="12"\; label="besdRA\; \backslash \; H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">besdRA</figure-callout>}\backslash \left(\mathrm{H}\right)\left(\mathrm{}12\mathrm{besmeas}-\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{<figure-callout\; id="0"\; label="besth"\; filenames="imgf0005.png"\; state="\{\{state\}\}">besth</figure-callout>}0\right)$$

  
• The improved value for the 4 by 1 matrix RA , which contains the element values wanted is: $$\mathrm{RA}=\mathrm{<figure-callout\; id="0"\; label="RA"\; filenames="imgf0005.png"\; state="\{\{state\}\}">RA</figure-callout>}0+\mathrm{dRA}$$

  

  This procedure is repeated iteratively using this value of RA for a new RA0. The procedure quickly minimizes the quantity Error2 given by the inner matrix product below of the last iteration; i.e.: $$\mathrm{Error}2=\left(\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{bes}-\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{besth}\right)\text{'}*\left(\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{bes}-\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{besth}\right)$$

  

  Thus the value of the drill bit location Raer and the azimuthal drilling direction with respect to the a axis; i.e., the angle Aab, have been determined from electromagnetic and accelerometer measurements.
• Alternatively, the location vector Raer to the drilling point can be determined from the measured vectors H1xyz and H2xyz without the use of an accelerometer. This can be done by comparing scalar vector invariants of the measured electromagnetic field vectors to corresponding computed values. To demonstrate this method, consider the magnitude of H1, the magnitude of H2 and the magnitude of the projection of (H1-H2) upon a plane perpendicular to (H1+H2). The approximate inverse cube decrease in the field magnitudes H1 and H2 with the distance away from each source loop makes these quantities good for determining the away component Ra and the right side component Rr. For two similar laterally separated loops lying on approximately the same plane the H1 and H2 vectors are vertical at the Earth's surface; however, the direction of these vectors diverge direction with depth into the Earth. Thus the projection of the difference vector H1-H2 onto the plane perpendicular to the direction of H1+H2; i.e., the near vertical direction, can give a good measure of the elevation component Re. These three parameters are not only good for this idealized configuration but also for most cases of practical interest.
• Thus, the three quantities H1mag, H2mag and H12mag, defined below, can be computed from the H1 and H2 measurements and from theoretical computations; i.e.: $$\begin{array}{l}\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{mag}=\mathrm{<figure-callout\; id="1"\; label="sqrt\; dot\; H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">sqrt</figure-callout>}\left(\mathrm{dot}\left(\mathrm{H}\right)\right)\left(\left(\mathrm{}1,\mathrm{H}1\right)\right)\\ \mathrm{H}2\mathrm{mag}=\mathrm{sqrt}\left(\mathrm{dot}\left(\mathrm{H}2,\mathrm{H}2\right)\right)\\ \mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{mag}=\mathrm{sqrt}\left(\mathrm{dot}\left(\mathrm{H}12\mathrm{p},\mathrm{H}12\mathrm{<figure-callout\; id="12"\; label="p\; H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">p</figure-callout>}\right)\right)& \\ \mathrm{H}\mathrm{}12\mathrm{p}=\left(\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1-\mathrm{H}2\right)-\mathrm{<figure-callout\; id="1"\; label="dot\; H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">dot</figure-callout>}\left(\left(\mathrm{H}\right)\right)\left(\left(\mathrm{}1-\mathrm{H}2\right),\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{plusuv}\right)*\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{<figure-callout\; id="12"\; label="plusuv\; H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">plusuv</figure-callout>}& \\ \mathrm{H}\mathrm{}12\mathrm{plus}=\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1+\mathrm{H}2\\ \mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{plusuv}=\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{plus}/\mathrm{<figure-callout\; id="12"\; label="sqrt\; dot\; H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">sqrt</figure-callout>}\left(\mathrm{dot}\left(\mathrm{H}\right)\right)\left(\left(\mathrm{}12\mathrm{plus},\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{plus}\right)\right)\end{array}$$

  
• From the three quantities H1mag, H2mag and H12mag, form the 3 by 1 field measurement matrix H12perpmeas; i.e.: $$\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{perpmeas}=\left[\mathrm{<figure-callout\; id="1"\; label="H"\; filenames="imgf0005.png"\; state="\{\{state\}\}">H</figure-callout>}1\mathrm{mag};\mathrm{H}2\mathrm{mag};\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{mag}\right];$$

  

  A first guess for the aer representation of the sensor location vector Raer0 is written as a 3 by 1 matrix, as follows: $$\mathrm{<figure-callout\; id="0"\; label="Raer"\; filenames="imgf0005.png"\; state="\{\{state\}\}">Raer</figure-callout>}0=\left[\mathrm{<figure-callout\; id="0"\; label="Ra"\; filenames="imgf0005.png"\; state="\{\{state\}\}">Ra</figure-callout>}0;\mathrm{<figure-callout\; id="0"\; label="Re"\; filenames="imgf0005.png"\; state="\{\{state\}\}">Re</figure-callout>}0;\mathrm{<figure-callout\; id="0"\; label="Rr"\; filenames="imgf0005.png"\; state="\{\{state\}\}">Rr</figure-callout>}0\right];$$

  

  The vector Raer to the best fit sensor location is written asRaer0 plus a small correction 3 by 1 correction matrix dRaer; i.e.: $$\mathrm{Raer}=\mathrm{<figure-callout\; id="0"\; label="Raer"\; filenames="imgf0005.png"\; state="\{\{state\}\}">Raer</figure-callout>}0+\mathrm{dRaer}=\mathrm{<figure-callout\; id="0"\; label="Raer"\; filenames="imgf0005.png"\; state="\{\{state\}\}">Raer</figure-callout>}0+\left[\mathrm{dRa};\mathrm{dRe};\mathrm{dRr}\right];$$

  
• Using the same procedure as defined by FIG. 7 and (Eq.1), compute H1th0 and H2th0 at the presumed location Raer0. Then compute theoretical values of the matrix elements of H12perpth0; i.e., the theoretical matrix following the steps in (Eq.23), To evaluate the matrix H12perpth at the true value of Raer, use a Taylor series approach using a derivative matrix dH12perpdRaer; i.e.: $$\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{perpth}=\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{<figure-callout\; id="0"\; label="perpth"\; filenames="imgf0005.png"\; state="\{\{state\}\}">perpth</figure-callout>}0+\mathrm{<figure-callout\; id="12"\; label="dH"\; filenames="imgf0001.png"\; state="\{\{state\}\}">dH</figure-callout>}12\mathrm{perpdRaer}*\mathrm{dRaer}$$

  

  The derivative matrix dH12perpdRaer is similar to the derivative matrix of (Eq.17); i.e.: $$\mathrm{<figure-callout\; id="12"\; label="dH"\; filenames="imgf0001.png"\; state="\{\{state\}\}">dH</figure-callout>}12\mathrm{perpdRaer}=\left[\mathrm{<figure-callout\; id="1"\; label="dH"\; filenames="imgf0005.png"\; state="\{\{state\}\}">dH</figure-callout>}1\mathrm{<figure-callout\; id="1"\; label="magdRa\; dH"\; filenames="imgf0005.png"\; state="\{\{state\}\}">magdRa</figure-callout>}\mathrm{dH}1\mathrm{<figure-callout\; id="1"\; label="magdRe\; dH"\; filenames="imgf0005.png"\; state="\{\{state\}\}">magdRe</figure-callout>}\mathrm{dH}1\mathrm{magdRr};\mathrm{dH}2\mathrm{magdRa}\mathrm{dH}2\mathrm{magdRe}\mathrm{dH}2\mathrm{magdRr};\mathrm{<figure-callout\; id="12"\; label="dH"\; filenames="imgf0001.png"\; state="\{\{state\}\}">dH</figure-callout>}12\mathrm{<figure-callout\; id="12"\; label="magdRa\; dH"\; filenames="imgf0001.png"\; state="\{\{state\}\}">magdRa</figure-callout>}\mathrm{dH}12\mathrm{<figure-callout\; id="12"\; label="magdRe\; dH"\; filenames="imgf0001.png"\; state="\{\{state\}\}">magdRe</figure-callout>}\mathrm{dH}12\mathrm{magdRr};\right]$$

  
• As in (Eq. 17), each of the elements of dH12perpdRaer is a partial derivative; e.g., dH1magdRa is the partial derivative of H1mag with respect to Ra, etc., and can be computed in the same manner as outlined for the elements of the matrix of (Eq. 17). The three linear equations implicit in (Eq. 26) are solved for the three unknowns in dRaer, after equating H12perpth to the corresponding 3 by 1 measurement matrix H12perpmeas The solution for dRaer is found using the MATLAB "\" operation, as: $$\mathrm{dRaer}=\mathrm{<figure-callout\; id="12"\; label="dH"\; filenames="imgf0001.png"\; state="\{\{state\}\}">dH</figure-callout>}12\mathrm{<figure-callout\; id="12"\; label="perpdRaer\; \backslash \; H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">perpdRaer</figure-callout>}\backslash \left(\mathrm{H}\right)\left(\mathrm{}12\mathrm{perpmeas}-\mathrm{<figure-callout\; id="12"\; label="H"\; filenames="imgf0001.png"\; state="\{\{state\}\}">H</figure-callout>}12\mathrm{<figure-callout\; id="0"\; label="perpth"\; filenames="imgf0005.png"\; state="\{\{state\}\}">perpth</figure-callout>}0\right)$$

  

  The improved value for the 3 by 1 matrix Raer is: $$\mathrm{Raer}=\mathrm{<figure-callout\; id="0"\; label="Raer"\; filenames="imgf0005.png"\; state="\{\{state\}\}">Raer</figure-callout>}0+\mathrm{dRaer}$$

  

  This procedure is repeated iteratively using this value of Raer for a new Raer0 to minimize the error implicit with the linear nature of a Taylor expansion. The final Raer matrix has the three components of the drill bit location vector Raer required to locate the drill bit with respect to the proposed path. From this, the driller can determine the appropriate drill control signals needed to progress along the path to the desired exit point.
• Although the present invention has been described in terms of a preferred embodiment, it will be understood that variations and modifications may be made in the above-described system without departing from the scope thereof as defined in the following claims.

Claims (13)

1. Apparatus for borehole surveys, comprising:

   a downhole tool suitable for being located in a borehole in the Earth, said tool having an axis and including a magnetometer (60) for measuring x, y and z vector components of magnetic fields at the tool in a first coordinate system;

   an electromagnetic excitation system (54, 56) including two electrical loop conductors (26, 28) at known locations, laterally spaced from each other and from a proposed path (20) of said borehole (12), and carrying known alternating electrical currents producing corresponding alternating electromagnetic fields in the region of said magnetometer;

   computer circuitry (76);

   a telemetry link (70) for transmitting measurement data signals corresponding to the respective measured vectors of magnetic fields generated by each of said loops from said magnetometer to said computer circuitry; and

   circuits (90) in said computer circuitry responsive to said measurement signals for determining the location of the borehole sensors in a second coordinate system,

   wherein:

   said two electrical loop conductors are configured to carry first and second alternating currents, respectively, said currents producing corresponding first and second electromagnetic fields in the region of said downhole tool in said borehole; and

   said electrical loop conductors are configured to be spaced on opposite sides of said proposed path of said borehole.
2. The apparatus of Claim 1, wherein said downhole tool further includes sensors (68) for measuring the Earth's gravity vector and for determining the orientation of the downhole tool, said telemetry link being configured to transmit measurement data signals corresponding to said gravity vector and orientation measurement to said computer for determination of the direction of the borehole with respect to said proposed path.
3. The apparatus of Claim 1 or Claim 2, wherein said downhole tool is a measurement while drilling instrument.
4. The apparatus of any preceding claim, wherein the electrical current supplied to each of said loops is an alternating current of a different frequency.
5. The apparatus of Claim 1, wherein each of said loop conductors comprises multiple turns of wire.
6. A method for location determination, comprising:

   positioning first and second current-conducting loops (26, 28) at known locations laterally spaced from each other;

   energizing said loops with known electrical currents to produce corresponding first and second electromagnetic fields;

   positioning a magnetic field vector sensor (60) to detect x, y and z vector components of said first and second fields in a first coordinate system; and

   determining from said vector components the location of said sensor with respect to said loops in a second coordinate system;

   wherein:

   positioning said loops includes positioning the loops on opposite sides of a proposed path (20) to be followed by said sensor and laterally spaced from said proposed path; and

   determining the location of said sensor includes locating said sensor with respect to said path.
7. The method of Claim 6, wherein energizing said loops includes supplying alternating current of a known amplitude and frequency to one loop at a time.
8. The method of Claim 6 or Claim 7, wherein energizing said loops includes simultaneously supplying alternating currents of known amplitudes and different known frequencies to each loop.
9. The method of any of Claims 6 to 8, further including:

   detecting, at the location of said sensor, the Earth's gravity vectors and the orientation vectors of the sensor; and

   determining from said vectors the direction of the sensor with respect to the locations of said loops.
10. the method of any of Claims 6 to 9, wherein determining the location of the sensor includes:

    time averaging magnetic field vector data obtained for each of said loops by the sensor;

    calculating theoretical location vectors for the sensor; and

    optimizing the match between theoretical location values and measured magnetic field vector data to determine the location of the sensor.
11. The method of any of Claims 6 to 10, further including:

    determining a proposed path for a borehole; and

    positioning said magnetic field vector sensor at a measurement location in said borehole for drill guidance, and wherein:

    said first and second current-conducting loops are positioned at known, laterally separated locations with respect to said proposed path;

    energizing said loops includes causing said first and second loops to generate corresponding first and second magnetic fields, each of said fields having an alternating polarity;

    measuring, at said measurement location, said x, y and z vector components of said first and second magnetic fields; and

    determining, from said measured vector components of said first and second fields, the location of the sensor with respect to the locations of the magnetic loop sources.
12. The method of Claim 11, wherein determining the location of the sensor includes:

    time averaging magnetic field vector data obtained for each of said loops by the sensor;

    calculating theoretical location vectors for the sensor; and

    optimizing the match between theoretical location values and measured magnetic field vector data to determine the location of the sensor with respect to said loops and said proposed path.
13. The method of Claim 12, wherein determining the location of the sensor further includes determining the lateral distance of said measuring location from the proposed path and the direction of said borehole with respect to the direction of the proposed path.

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