RELATED APPLICATIONS 
     This application is a divisional application of U.S. patent application Ser. No. 12/816,250 filed Jun. 15, 2010, which is a continuation application of U.S. patent application Ser. No. 11/835,154 filed Aug. 7, 2007 and issued as U.S. Pat. No. 7,775,301 on Aug. 17, 2010, the disclosures of which are incorporated herein by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     The present application is generally related to steering tools for horizontal directional drilling and, more particularly, to a system and method using supplemental magnetic information in a steering tool type arrangement. 
     A boring tool is well-known as a steerable drill head that can carry sensors, transmitters and associated electronics. The boring tool is usually controlled through a drill string that is extendable from a drill rig. The drill string is most often formed of drill pipe sections, which may be referred to hereinafter as drill rods, that are selectively attachable with one another for purposes of advancing and retracting the drill string. Steering is often accomplished using a beveled face on the drill head. Advancing the drill string while rotating should result in the drill head traveling straight forward, whereas advancing the drill string with the bevel oriented at some fixed angle will result in deflecting the drill head in some direction. 
     One approach that has been taken by the prior art for purposes of monitoring the progress of a boring tool in the field of horizontal directional drilling, resides in what is commonly referred to as a “steering tool”. This term has come to describe an overall system which essentially predicts the position of the boring tool, as it is advanced through the ground using a drill string, such that the boring tool can be steered toward a desired target or along a planned drill path within the ground. Steering tool systems are considered as being distinct from other types of locating systems used in horizontal directional drilling at least for the reason that the position of the boring tool is monitored in a step-wise fashion as it progresses through the ground. For this reason, positional error can accumulate with increasing progress through the ground up to unacceptable levels. 
     Generally, in a steering tool system, pitch and yaw angles of the drill-head are measured in coordination with extension of the drill string. From this, the drill-head position coordinates are obtained by numerical integration. Nominal or measured drill rod lengths can serve as a step size during integration. While this method appears to be sound and might enable an experienced driller to use the steering tool successfully, there are a number of concerns with respect to its operation, as will be discussed immediately hereinafter. 
     With respect to the aforementioned positional error, it is noted that this error can be attributed, at least in part, to pitch and yaw measurement errors that accumulate during integration. This can often result in large position errors after only a few hundred feet of drilling. 
     Another concern arises with respect to underground disturbances of the earth&#39;s magnetic field, which can cause significant yaw measurement bias errors, potentially leading to very inaccurate position estimates. 
     Still another concern arises to the extent that steering effectiveness of a typical HDD drill bit depends on many factors including drill bit design, mud flow rate and soil conditions. For example, attempting to steer in wet and sandy soil with the tool in the 12 o&#39;clock roll position might become so ineffective that measured pitch does not provide correct vertical position changes. That is, the orientation of drill head, under such drilling conditions, does not necessarily reflect the direction of its travel. 
     One approach in dealing with the potential inaccuracy of the steering tool system is to confirm the position of the drill head independently. For example, the drill head can be fitted with a dipole transmitter. A walk over locator can then be used to receive the dipole field and independently locate the drill head. This approach is not always practical, for example, when drilling under a river, lake or freeway. In these situations, the operator might notice position errors too late during drilling and consequently might not have an opportunity to implement a drill-path correction. 
     The foregoing examples of the related art and limitations related therewith are intended to be illustrative and not exclusive. Other limitations of the related art will become apparent to those of skill in the art upon a reading of the specification and a study of the drawings. 
     SUMMARY 
     The following embodiments and aspects thereof are described and illustrated in conjunction with systems, tools and methods which are meant to be exemplary and illustrative, not limiting in scope. In various embodiments, one or more of the above-described problems have been reduced or eliminated, while other embodiments are directed to other improvements. 
     In general, a system and associated method are described in which a steering tool is movable by a drill string and steerable in a way that is intended to form an underground bore along an intended path, beginning from a starting position. 
     In one aspect, a sensing arrangement, forming one part of the steering tool, detects a pitch orientation and a yaw orientation of the steering tool at a series of spaced apart positions of the steering tool along the underground bore, each of which spaced apart positions is characterized by a measured extension of the drill string. At least one marker is positioned proximate to the intended path, for transmitting a rotating dipole field such that at least a portion of the intended path is exposed to the rotating dipole field. A receiver, forming another part of the steering tool, receives the rotating dipole field with the steering tool at a current one of the spaced apart positions to produce magnetic information. A processor is configured for using the detected pitch orientation, the detected yaw orientation and the measured extension of the drill string in conjunction with the magnetic information, corresponding to the current one of the positions of the steering tool, to determine a current location of the steering tool, relative to the starting position, with a given accuracy such that using only the detected pitch orientation, the detected yaw orientation and the measured extension of the drill string to determine the current position, without the magnetic information, would result in a reduced accuracy in the determination of the current location, as compared to the given accuracy. 
     In another aspect, a sensing arrangement is provided, forming one part of the steering tool, for detecting a pitch orientation and a yaw orientation of the steering tool. The steering tool is moved sequentially through a series of spaced apart positions along the underground bore. Each of the spaced apart positions is characterized by a measured extension of the drill string. At least one marker is arranged, proximate to the intended path, for transmitting a rotating dipole field such that at least a portion of the intended path is exposed to the rotating magnetic dipole. The dipole field is received using a receiver that forms another part of the steering tool, with the steering tool at a current one of the spaced apart positions on the portion of the intended path, to produce magnetic information. A processor is configured for using the detected pitch orientation, the detected yaw orientation and the measured extension of the drill string in conjunction with the magnetic information, corresponding to the current one of the positions of the steering tool, to determine a current location of the steering tool relative to the starting position with a given accuracy such that using only the detected pitch orientation, the detected yaw orientation and the measured extension of the drill string to determine the current location, without the magnetic information, results in a reduced accuracy in the determination of the current location, as compared to the given accuracy. 
     In still another aspect, a sensing arrangement is provided, forming one part of the steering tool, for detecting a pitch orientation and a yaw orientation of the steering tool. The steering tool is moved sequentially through a series of spaced apart positions to form the underground bore. Each of the spaced apart positions is characterized by a measured extension of the drill string, a detected pitch orientation and a detected yaw orientation. At least one portion of the intended path is identified along which an enhanced accuracy of a determination of the current location of the steering tool is desired. One or more markers is arranged proximate to the portion of the intended path, each of which transmits a rotating dipole field such that at least the portion of the intended path is exposed to one or more rotating dipole fields. A receiver is provided, as part of the steering tool, for generating magnetic information responsive to the rotating dipole fields. A processor is configured for operating in a first mode using the detected pitch orientation, the detected yaw orientation and the measured extension of the drill string to determine a current location of the steering tool corresponding to any given one of the spaced apart positions with at least a given accuracy and for defaulting to a second mode using the detected pitch orientation, the detected yaw orientation, the measured extension of the drill string and the magnetic information, when the magnetic information is received, to determine the current location of the steering tool with an enhanced accuracy that is greater than the given accuracy. 
     In yet another aspect, a method for establishing a customized accuracy in determination of a position of the steering tool with respect to the intended path is described. A sensing arrangement, forming one part of the steering tool, detects a pitch orientation and a yaw orientation of the steering tool. The steering tool is moved sequentially through a series of spaced apart positions to form the underground bore. Each of the spaced apart positions is characterized by a measured extension of the drill string, a detected pitch orientation and a detected yaw orientation. One or more portions of the intended path are identified along which an enhanced accuracy of the determination of the current location of the steering tool is desired. One or more markers are arranged proximate to each one of the portions of the intended path where each of the markers transmits a rotating dipole field such that each one of the identified portions of the intended path is exposed to one or more rotating dipole fields. As a result of the transmission range of the rotating dipole field, more than just those portions of the intended path may be exposed to the rotating dipole field(s). A receiver is provided, as part of the steering tool, for generating magnetic information responsive to the rotating dipole fields. A processor is configured for operating in a first mode using the detected pitch orientation, the detected yaw orientation and the measured extension of the drill string to determine a current location of the steering tool corresponding to any given one of the spaced apart positions with at least a given accuracy and for operating in a second mode using the detected pitch orientation, the detected yaw orientation, the measured extension of the drill string and the magnetic information to determine the current location of the steering tool with an enhanced accuracy that is greater than the given accuracy, at least for the one or more portions of the intended path, to customize an overall position determination accuracy along the intended path. 
     In addition to the exemplary aspects and embodiments described above, further aspects and embodiments will become apparent by reference to the drawings and by study of the following descriptions. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Exemplary embodiments are illustrated in referenced figures of the drawings. It is intended that the embodiments and figures disclosed herein are to be illustrative rather than limiting. 
         FIG. 1  is a diagrammatic view, in elevation, of a system according to the present disclosure operating in a region. 
         FIG. 2  is a diagrammatic plan view of the system of  FIG. 1  in the region. 
         FIG. 3   a  is a block diagram which illustrates one embodiment of a steering tool that is useful in the system of  FIGS. 1 and 2 . 
         FIG. 3   b  is a diagrammatic view, in perspective, of a marker that is useful in the system of  FIGS. 1 and 2 . 
         FIG. 3   c  shows a coordinate system in which pitch and yaw are illustrated. 
         FIGS. 4 and 5  illustrate one embodiment of a setup technique that can be used in conjunction with the system of  FIGS. 1 and 2 . 
         FIG. 6  is a diagrammatic view, in elevation, of a drill path along which the steering tool is disposed, shown here to illustrate one embodiment of a technique for providing an initial solution estimate for the position of the steering tool. 
         FIG. 7   a  is a diagrammatic, further enlarged view, of a portion of  FIG. 6 , shown here to illustrate further details of the initial solution estimate technique. 
         FIG. 7   b  is a flow diagram which illustrates one possible embodiment of a technique for determining the position of the steering tool using a Kalman filter. 
         FIG. 8  is a plot of random distance error versus distance. 
         FIGS. 9   a  and  9   b  are plots of pitch angle and yaw angle, respectively, versus drill string length for use in a detailed simulation. 
         FIG. 10   a  is a plot in a simulation of estimated Y (lateral) steering tool position with respect to X position, employing a basic steering tool without the use of markers. 
         FIG. 10   b  is a plot in the simulation of estimated Z (elevational) steering tool position with respect to X position, employing a basic steering tool without the use of markers. 
         FIG. 10   c  is a plot, for the simulation of  FIGS. 10   a  and  10   b , of steering tool coordinate position error versus X position, which illustrates positional errors for the X, Y and Z axes without the use of markers. 
         FIG. 11   a  is a plot in a simulation of estimated Y (lateral) steering tool position with respect to X position, employing a steering tool in conjunction with one marker. 
         FIG. 11   b  is a plot in the simulation of estimated Z (elevational) steering tool position with respect to X position, employing the steering tool in conjunction with one marker. 
         FIG. 11   c  is a plot, for the simulation of  FIGS. 11   a  and  11   b , of steering tool coordinate position error versus X position, which illustrates positional errors for the X, Y and Z axes with the use of one marker. 
         FIG. 12   a  is a plot in a simulation of estimated Y (lateral) steering tool position with respect to X position, employing a steering tool in conjunction with two markers. 
         FIG. 12   b  is a plot in the simulation of estimated Z (elevational) steering tool position with respect to X position, employing the steering tool in conjunction with two markers. 
         FIG. 12   c  is a plot, for the simulation of  FIGS. 12   a  and  12   b , of steering tool coordinate position error versus X position, which illustrates positional errors for the X, Y and Z axes with the use of two markers. 
         FIG. 13   a  is a plot in a simulation of estimated Y (lateral) steering tool position with respect to X position, employing a steering tool in conjunction with three markers. 
         FIG. 13   b  is a plot in the simulation of estimated Z (elevational) steering tool position with respect to X position, employing the steering tool in conjunction with three markers. 
         FIG. 13   c  is a plot, for the simulation of  FIGS. 13   a  and  13   b , of steering tool coordinate position error versus X position, which illustrates positional errors for the X, Y and Z axes with the use of three markers. 
         FIGS. 14   a - c  are plots of position error estimates, available through the Kalman filter analysis, versus the X axis and directly compared with position error plots show in  FIG. 13   c  for the drill path of  FIGS. 13   a  and  13   b    
         FIG. 15  is a diagrammatic plan view of a drilling region for a concluding portion of an intended drill path, shown here to illustrate various aspects of arranging and moving markers along the drill path. 
         FIG. 16  is a diagrammatic plan view of the drilling region and drill path of  FIG. 16 , shown her to illustrate further aspects with respect to arranging and moving markers along the drill path. 
     
    
    
     DETAILED DESCRIPTION 
     The following description is presented to enable one of ordinary skill in the art to make and use the invention and is provided in the context of a patent application and its requirements. Various modifications to the described embodiments will be readily apparent to those skilled in the art and the generic principles taught herein may be applied to other embodiments. Thus, the present invention is not intended to be limited to the embodiment shown, but is to be accorded the widest scope consistent with the principles and features described herein, including modifications and equivalents, as defined within the scope of the appended claims. It is noted that the drawings are not to scale and are diagrammatic in nature in a way that is thought to best illustrate features of interest. Descriptive terminology such as, for example, upper/lower, right/left, front/rear top/bottom, underside and the like has been adopted for purposes of enhancing the reader&#39;s understanding, with respect to the various views provided in the figures, and is in no way intended as being limiting. 
     Turning now to the figures, wherein like components are designated by like reference numbers whenever practical, attention is immediately directed to  FIGS. 1 and 2 , which illustrate an advanced steering tool system that is generally indicated by the reference number  10  and produced according to the present disclosure.  FIG. 1  is a diagrammatic elevation view of the system, whereas  FIG. 2  is a diagrammatic plan view of the system. System  10  includes a drill rig  18  having a carriage  20  received for movement along the length of an opposing pair of rails  22  which are, in turn, mounted on a frame  24 . A conventional arrangement (not shown) is provided for moving carriage  20  along rails  22 . A steering tool  26  includes an asymmetric face  28  and is attached to a drill string  30  which is composed of a plurality of drill pipe sections  32 . An intended path  40  of the steering tool includes positions that are designated as k and k+1. The steering tool is advanced from position k to k+1 by either a full or a fraction rod length. If very short drill pipe sections are used, the distance between positions k and k+1 could be greater than a rod length. By way of example, drill pipe sections have a rod length of two feet would be considered as very short. The steering tool is shown as having already passed through points  1  and  2 , where point  1  is the location at which the steering tool enters the ground at  42 , serving as the origin of the master coordinate system. While a Cartesian coordinate system is used as the basis for the master coordinate systems employed by the various embodiments disclosed herein, it is to be understood that this terminology is used in the specification and claims for descriptive purposes and that any suitable coordinate system may be used. 
     An x axis  44  extends from entry point  42  to a target location T that is on the intended path of the steering tool, as seen in  FIG. 1  and illustrated as a rectangle, while a y axis  46  extends to the left when facing in the forward direction along the x axis, as seen in  FIG. 2 . A z axis  48  extends upward, as seen in  FIG. 1 . Further descriptions will be provided at an appropriate point below with respect to establishing this coordinate system. 
     As the drilling operation proceeds, respective drill pipe sections, which may be referred to interchangeably as drill rods, are added to the drill string at the drill rig. For example, a most recently added drill rod  32   a  is shown on the drill rig in  FIG. 2 . An upper end  50  of drill rod  32   a  is held by a locking arrangement (not shown) which forms part of carriage  20  such that movement of the carriage in the direction indicated by an arrow  52  causes section  32   a  to move therewith, which pushes the drill string into the ground thereby advancing the boring operation. A clamping arrangement  54  is used to facilitate the addition of drill pipe sections to the drill string. The drilling operation is controlled by an operator (not shown) at a control console  60  which itself can include a telemetry section  62  connected with a telemetry antenna  64 , a display screen  66 , an input device such as a keyboard  68 , a processor  70 , and a plurality of control levers  72  which, for example, control movement of carriage  20 . 
     Turning now to  FIG. 3   a , an electromechanical block diagram is shown, illustrating one embodiment of steering tool  26  that is configured in accordance with the present disclosure. Steering tool  26  includes a slotted non-magnetic drill tool housing  100 . A triaxial magnetic field sensing arrangement  102  is positioned in housing  100 . For this purpose, a triaxial magnetometer or coil arrangement may be used depending on considerations such as, for example, space and accuracy. A triaxial accelerometer  104  is also located in the housing. Outputs from magnetic field sensing arrangement  102  and accelerometer  104  are provided to a processing section  106  having a microprocessor at least for use in determining a pitch orientation and a yaw heading of the steering tool. A dipole antenna and associated transmitter  108  are optionally located in the steering tool which can be used, responsive to the processing section, for telemetry purposes, for transferring encoded data such as roll, pitch, magnetometer readings and accelerometer readings to above ground locations such as, for example, telemetry receiver  62  ( FIG. 1 ) of console  60  via a dipole electromagnetic field  110  and for locating determinations such as, for example, determining a distance to the steering tool. For such locating determinations, dipole electromagnetic field  110  can be used in conjunction with a walkover locator, although this is not a requirement and is not practical in some cases, as discussed above. Generally, the dipole axis of the dipole antenna is oriented coaxially with an elongation axis of the steering tool in a manner which is well-known in the art. Of course, all of these functions are readily supported by processing section  106 , which reads appropriate inputs from the magnetometer and accelerometer, performs any necessary processing and then performs the actual encoding of information that is to be transmitted. 
     In another embodiment, processing section  106  is configured for communication with processor  70  ( FIG. 1 ) of console  60  using a wire-in-pipe approach wherein a conductor is provided in drill string  30  for transferring information above ground as described, for example, in commonly owned U.S. Pat. No. 6,223,826 entitled AUTO-EXTENDING/RETRACTING ELECTRICALLY ISOLATED CONDUCTORS IN A SEGMENTED DRILL STRING, which is incorporated by reference in its entirety. The conductor in the drill string is in electrical communication with a line  112  that is in electrical communication with processing section  106 . It is noted that this approach may also be used to provide power to a power supply  114  from above ground, as an alternative or supplemental to the use of batteries. 
     Still referring to  FIG. 3   a , regulated power supply  114 , which may be powered using batteries or through the aforedescribed wire-in-pipe arrangement, provides appropriate power to all of the components in the steering tool, as shown. It is noted that magnetic field sensor  102  can be used to measure the field generated by a rotating magnet as well as measuring the Earth&#39;s magnetic field. The later may be thought of as a constant, much like a DC component of an electrical signal. In this instance, the Earth&#39;s magnetic field may be used advantageously to determine a yaw heading. 
     Referring again to  FIGS. 1 and 2 , system  10  is illustrated having three markers  140   a - c , each of which includes a rotating magnet for generating a rotating dipole field. Markers  140   a  and  140   b  are arranged along a line that is generally orthogonal to the X axis, while marker  140   c  is offset toward drill rig  18 . A rotating dipole field can be generated either by a rotating magnet or by electromagnetic coils. Throughout this disclosure, the discussion may be framed in terms of a rotating magnet, but the described applications of magnets carry over to coils and wire loops with only minor modifications. As will be described in further detail, markers can be placed along the drill-path so that they are at least generally close to the target or other points of interest where high positioning accuracy is required, although one or two markers may provide sufficient accuracy for many drilling applications. That is, the marker signal should be receivable by the steering tool along a portion of the intended path including the target or other point(s) of interest. Aside from this consideration, the position of each marker can be arbitrary. Markers can be placed on the ground, on an elevated structure or even lowered within the ground. In each case, the marker can be at an arbitrary angular orientation. The rotation frequency (revolutions per second) of each magnet can be on the order of 1 Hz, but dipole field frequencies should be distinguishable if more than one marker/magnet is in use. A frequency difference of at least 0.5 Hz is considered to be acceptable for this purpose. Each magnet emits a rotating magnetic dipole field whose total flux is recorded by the steering tool magnetometer and subsequently converted to distance between magnet and tool. During rotation, the magnet of each marker emits a time dependent magnetic dipole field that is measured by the tri-axial magnetometer of the steering tool. As will be seen, a minimum value of the recorded total flux provides a distance between each marker and the steering tool. 
     Turning now to  FIG. 3   b , one embodiment of marker  140  is diagrammatically illustrated. It is noted that aforedescribed markers  140   a - c  may be of this design as well as any additional markers used hereinafter. In this embodiment, each marker  140  can include a drive motor  142  having an output shaft  144  which directly spins a magnet  146  having a north pole, which is visible. Motor  142  is electrically driven by a motor controller  148  to provide stable rotation of the magnet. The motor can rotate the magnet slowly, for example, at about 1 revolution per second (1 Hz), as indicated by arrow  150 , thereby emitting a rotating magnetic dipole field  152  (only partially shown). It should be appreciated that a relatively wide range of rotational speeds may employed, for example, from approximately 0.5 Hz to 600 Hz. In one embodiment, a proportional-integral-derivative (PID) controller can be used to drive motor  142  with user selectable rotational velocity. It is noted that such PIDs are commercially available. A benefit associated with using lower rotational velocity resides in a decreased influence by local magnetic objects such as, for example, rebar. If a higher rotational velocity is desired loop antennas can be used to create the rotational field. Further, the rotational velocity can be varied so that the fields from various markers are distinguishable when simultaneously rotating. A suitable power supply can be used, as will be recognized by one having ordinary skill in the art, such as for example a battery and voltage regulator, which have not been shown. It should be appreciated that there is no need for an encoder, since the specific angle of the magnet, corresponding to a particular measurement position, is not involved in making the determinations that are described below. Further, orientation sensors and a telemetry section in marker  140  are not needed. As will be seen, variation in rotation rate of magnet  146  will introduce associated positional error. Hence, a desire to increase measurement accuracy is associated with increasing the rotational stability of magnet  146 . 
     Still referring to  FIG. 3   b , while the axis of rotation of magnet  146  is illustrated as being vertical, this is not a requirement. The axis of rotation can be horizontal or at some arbitrary tilted orientation. Moreover, positioning of the marker for field use does not require orienting the marker in any particular way. This remarkable degree of flexibility and ease of positioning these markers is one of the benefits of the system and method taught herein. 
     Most conventional applications of the steering tool function rely on a nominal value for drill rod length when integrating pitch and yaw to determine position. In accordance with the present disclosure, however, pitch and yaw can be measured more than once along each drill rod such that the distance between successive steering tool measurement positions can be less than the nominal length of one drill rod. This is particularly the case when the length of the drill rod is exceptionally long such as, for example, thirty feet. For this purpose, a laser distance meter, a potentiometer, an ultrasonic arrangement or some other standard distance measurement device can be mounted on the drill rig. An ultrasonic arrangement will be described immediately hereinafter. 
     Referring again to  FIGS. 1 and 2 , a drill string measuring arrangement includes a stationary ultrasonic transmitter  202  positioned on drill frame  18  and an ultrasonic receiver  204  with an air temperature sensor  206  ( FIG. 2 ) positioned on carriage  20 . It should be noted that the positions of the ultrasonic transmitter and receiver may be interchanged with no effect on measurement capabilities. Transmitter  202  and receiver  204  are each coupled to processor  70  or a separate dedicated processor (not shown). In a manner well known in the art, transmitter  202  emits an ultrasonic wave  208  that is picked up at receiver  204  such that the distance between the receiver and the transmitter may be determined to within a fraction of an inch by processor  70  using time delay and temperature measurements. By monitoring movements of carriage  20 , in which drill string  30  is either pushed into or pulled out of the ground, and clamping arrangement  54 , processor  70  can accurately track the length of drill string  30  throughout a drilling operation. While it is convenient to perform measurements in the context of the length of the drill rods, with measurement positions corresponding to the ends of the drill rods, it should be appreciated that this is not a requirement and the ultrasonic arrangement can provide the total length of the drill string at any given moment in time. Further, the length according to the number of drill rods multiplied by nominal rod length can be correlated to the length that is determined by ultrasonic measurement. 
     Referring to  FIG. 1 , control console  60 , in this embodiment, serves as a base station to communicate with steering tool  26 , to monitor its power supply, to receive and process steering tool data and to send commands to the steering tool, if so desired. Determined drill-path positions and estimated position errors can be displayed on display screen  66  for monitoring by the system operator. This functionality may also be extended to a remote base station configuration, for example, by using telemetry section  64  to transmit information  210  to a remote base station  212  for display on a screen  214 . 
     Measured Quantities 
     The steering method requires measurement of the following variables: 
     Tool pitch and yaw angles φ, β 
     Distances D i  between N M  magnets and the steering tool (i=1, . . . N M ) 
     Magnet positions (X M     i   , Y M     i   , Z M     i   ), (i=1, . . . N M ) 
     Initial tool position (X 1 , Y 1 , Z 1 ) 
     Rod length increments Δs k+1  (k=1, 2, 3, . . . ) 
     Referring to  FIGS. 1 and 2 , pitch and yaw are measured and magnet-to-tool distances are determined at a series of tool positions including the initial tool position. Point  1 , which is additionally denoted by the reference number  42 , designates the position of drill begin. The steering tool is currently located at a measurement position k and is intended to proceed to position k+1. These positions can correspond to the end points of a drill rod or to intermediate points along the length of each drill rod. As discussed above, intermediate points may be needed, for example, when an exceptionally long drill rod is used such as, for example, 30 feet. Higher accuracy will generally be provided through the use of relatively more measurement positions. In some cases, the drill rod length may be sufficiently short that the number of drill rods may provide a sufficiently accurate value as to the length of the drill string. The latter situation may also be characterized by drill rods having a tolerance in their average length that is reasonably close to a nominal value. In some embodiments, there may be no correspondence between the drill rod length and the measurement positions, for example, where a measurement system, such as is employed by system  10 , is capable of measuring and monitoring an overall length of the drill string. For purposes of simplicity of description, it will be assumed that the drill rod length is used in the remainder of this description to establish the measurement positions. It is noted that measurements at each measurement position may be performed on-the-fly while pushing and/or rotating the drill string; however, enhanced accuracy can be achieved by stopping movement of the steering tool at each of the measurement positions during the measurements. A rod length increment Δs k+1  is defined as the arc-length between tool measurement positions (X k , Y k , Z k ) and (X k+1 , Y k+1 , Z k+1 ). The setup of this coordinate system is described immediately hereinafter. 
     Set-Up of Steering System 
     Referring to  FIG. 3   c , in conjunction with  FIGS. 1 and 2 , the origin and directions of the X, Y, Z-coordinate system can be specified in relation to drill begin point  1  and target T. The location where drilling begins is a convenient choice for the origin and the direction from this position to the projection of the target onto a level plane through the origin defines the X-coordinate axis. The Z-coordinate axis is positive upward and the Y-coordinate axis completes a right-handed system. If desired, a different right-handed Cartesian coordinate system or any suitable coordinate system may be used. In the present example, the formulation constrains the X-axis to be level. As noted above, yaw orientation is designated as β measured from the X axis in a level X, Y plane, whereas pitch orientation is designated as φ measured vertically from the yawed tool position in the X, Y plane as represented by a dashed line in the X, Y plane.  FIG. 3   c  defines pitch and yaw as Euler angles that require a particular sequence of yaw and pitch rotations in order to rotate the steering tool from a hypothetical position along the X axis into its illustrated position. 
     Magnet Position Measurements 
     The use of an Electronic Distance Measurement device (EDM) is currently the quickest and most accurate method of defining the X-coordinate axis and measuring magnet position coordinates. However, using an EDM for this purpose requires the presence of a surveyor at the HDD job site, which may sometimes be difficult to arrange. Accordingly, any suitable method may be used. 
     As an alternative to an EDM, a laser distance measurement device can be used. Devices of this kind are commercially available with a maximum range of about 650 feet and a distance measurement accuracy of ⅛ of an inch; the Leica Disto™ laser distance meter is an example of such a device. The device is placed at the position of drill-begin and pointed at the target to obtain the distance between these two positions. For short range measurements, the device can be handheld, but for larger distances it should be fixedly mounted to focus reliably. When an EDM, laser distance measurement device or similar device is used to determine the magnet positions, the accuracy of the device itself can be used as the magnet position error in the context of the discussions below. 
     Referring to  FIGS. 4 and 5 , one embodiment of a setup technique is illustrated.  FIG. 4  illustrates a diagrammatic plan view of steering tool  26  positioned ahead of drill begin point  1  with target T arranged along the X axis and a marker M1 that is offset from the X axis. The target is located at coordinates X t , Y t , Z t .  FIG. 5  illustrates a diagrammatic elevational view of steering tool  26  positioned ahead of drill begin point  1  on a surface  230  of the ground. In one embodiment, a laser distance meter (LDM) can be used having a tilt sensor so that horizontal and vertical distances X t , Z t  to the target can either be calculated or are directly provided by the LDM. The relative position (ΔX, ΔY, ΔZ) between the target and a marker, M1, located near the target can also be measured using the LDM, with a measuring tape or in any other suitable manner. Marker position coordinates can be obtained by adding position increments to target coordinates, as follows:
 
 X   M1   =X   t   +ΔX   (1)
 
 Y   M1   =ΔY   (2)
 
 Z   M1   =Z   t   +ΔZ   (3)
 
     The foregoing procedure can be repeated for any number of markers that are arranged proximate to the target. 
     In another embodiment, the position of each marker can be measured directly, for example using an EDM, with no need to measure the location of the target, so long as some other position has been provided that establishes the X axis from point  1  of drill begin. For example, a marker M2 may be arranged along the X axis. As will be further described, location accuracy along the X axis can be customized based on the arrangement of markers therealong. The need for enhanced accuracy for some portion of the path of the steering tool can be established, for example, based on the presence of a known inground obstacle  232 . 
     Reference Yaw Angle 
     Continuing to refer to  FIGS. 4 and 5 , a reference yaw angle β ref  is defined as the yaw angle of the steering tool, measured by the steering tool, with its elongation axis aligned with the X-direction. In the present example, the reference yaw angle is measured as a compass orientation from magnetic north, based on the Earth&#39;s magnetic field. Since steering tool yaw has previously been defined as positive for a counterclockwise rotation the particular reference yaw angle β ref  shown in  FIG. 4  is negative. Accordingly, in order to measure yaw accurately without interference from the magnetic influence of the drill rig, the steering tool can be placed on a level ground a sufficient distance ahead of the drill; 30 feet is usually adequate. The elongation axis of the steering tool is at least approximately on or at least parallel to the X-axis. Yaw angle β m , measured as a compass heading during steering, is subsequently replaced by β=β m −β ref . 
     Steering Procedure Formulation 
     Nomenclature 
     c A =pitch and yaw error covariance matrix 
     c e =empirical coefficient 
     c M =magnet position error covariance matrix 
     D=distance between marker and steering tool 
     F=continuous state equations matrix 
     H=observation coefficient vector 
     N M =number of markers 
     P=error covariance matrix= 
     Q=continuous process noise covariance parameter matrix 
     Q k =discrete process noise covariance matrix 
     R=observation covariance scalar 
     {right arrow over (r)}=vector of magnet position measurement error 
     s=arc-length along drill-rod axis 
     ν D =distance measurement noise 
     ν M =magnet position measurement noise 
     {right arrow over (x)}=state variables vector 
     X, Y, Z=global coordinates 
     X k , Y k , Z k =steering tool position coordinates 
     z=measurement scalar 
     β=yaw angle 
     δX, δY, δZ=position state variables 
     δX M , δY M , δZ M =magnet position increments 
     δβ, δφ=yaw and pitch angle increments 
     Δs=rod length increment 
     φ=pitch angle 
     Φ k =discrete state equation transition matrix 
     σ=standard deviation 
     σ 2 =variance, square of standard deviation 
     Subscripts 
     bias=bias error 
     D=distance 
     ex=exact value 
     i=i-th magnet 
     k=k-th position on drill path 
     M=magnet 
     m=measured 
     ref=reference 
     1=initial tool position (drill begin at k=1) 
     Superscripts 
     
       
         
           
             
               
                 ( 
                 
                     
                 
                 ) 
               
               . 
             
             = 
             
               ⅆ 
               
                 ⅆ 
                 s 
               
             
           
         
       
     
     ( ) − =indicates last available estimate 
     ( )′=transpose 
     ( )*=nominal drill path 
     {right arrow over ({circumflex over (x)}=state variables vector estimate 
     Tracking Equations 
     The method is based on two types of equations, referred to as steering tool process equations and distance measurement equations. The former are a set of ordinary differential equations describing how tool position (X, Y, Z) changes along the drill-path as a function of measured pitch φ and yaw β and shown as equations 4. 
     
       
         
           
             
               
                 
                   
                     { 
                     
                       
                         
                           
                             X 
                             . 
                           
                         
                       
                       
                         
                           
                             Y 
                             . 
                           
                         
                       
                       
                         
                           
                             Z 
                             . 
                           
                         
                       
                     
                     } 
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ϕ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             β 
                           
                         
                       
                       
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ϕ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             β 
                           
                         
                       
                       
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ϕ 
                           
                         
                       
                     
                     } 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     The over-dot indicates that derivatives of position coordinates are to be taken with respect to arc-length s along the axis of the drill rod. Pitch and yaw angles are illustrated in  FIG. 3   c . Accordingly, the premise of a conventional steering tool resides in a numerical integration of equations 4 with respect to arc length s of the drill string. Unfortunately, as discussed above, this technique readily produces potentially serious positional errors in and by itself. 
     The aforementioned distance measurement equations are of the form:
 
 D   2 =( X   M   −X ) 2 +( Y   M   −Y ) 2 +( Z   M   −Z ) 2   (5)
 
     The distance measurement equations express distance D between the center of a rotating magnet of a marker and the center of tri-axial steering tool magnetometer  102  (see  FIG. 1 ) in terms of tool position (X, Y, Z) and magnet position (X M , Y M , Z M ). Accordingly, N M  of such equations can be written for a system, corresponding to the total number of markers. 
     The origin of the global X, Y, Z-coordinate system in which tool position will be tracked can be chosen to coincide with the location of drill begin (point  1  in  FIGS. 1 and 2 ).
 
 X   1 =0  Y   1 =0  Z   1 =0  (6)
 
     Equations (4), (5) and (6) represent an initial value problem that can be solved for steering tool position coordinates. 
     Nonlinear Solution Procedures 
     The foregoing initial value problem can be solved using either a nonlinear solution procedure, such as the method of nonlinear least squares, the SIMPLEX method, or can be based on Kalman filtering. The latter will be discussed in detail beginning at an appropriate point below. Initially, however, an application of the SIMPLEX method will be described where the description is limited to the derivation of the nonlinear algebraic equations that are to be solved at each drill-path position. Details of the solver itself are well-known and considered as within the skill of one having ordinary skill in the art in view of this overall disclosure. 
     The present technique and other solution methods can replace the derivatives {dot over (X)}, {dot over (Y)}, Ż in equations (4) with finite differences that are here written as: 
     
       
         
           
             
               
                 
                   
                     X 
                     . 
                   
                   = 
                   
                     
                       
                         X 
                         
                           k 
                           + 
                           1 
                         
                       
                       - 
                       
                         X 
                         k 
                       
                     
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         s 
                         
                           k 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     Y 
                     . 
                   
                   = 
                   
                     
                       
                         Y 
                         
                           k 
                           + 
                           1 
                         
                       
                       - 
                       
                         Y 
                         k 
                       
                     
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         s 
                         
                           k 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
             
               
                 
                   
                     Z 
                     . 
                   
                   = 
                   
                     
                       
                         Z 
                         
                           k 
                           + 
                           1 
                         
                       
                       - 
                       
                         Z 
                         k 
                       
                     
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         s 
                         
                           k 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Resulting algebraic equations read:
 
 f   1   =X   k+1   −X   k   −Δs   k+1  cos φ k  cos β k =0  (10)
 
 f   2   =Y   k+1   −Y   k   −Δs   k+1  cos φ k  sin β k =0  (11)
 
 f   3   =Z   k+1   −Z   k   −Δs   k+1  cos φ k =0  (12)
 
     The distance measurement equations (5) provide additional N M  equations written as:
 
 f   4     i     =D   k+1,i   2 −( X   k+1   −X   M     i   ) 2 −( Y   k+1   −Y   M     i   ) 2 −( Z   k+1   −Z   M     i   ) 2 =0  (13)
 
     Starting with the known initial values (Equations 6) at drill begin, the coordinates of subsequent positions along the drill path can be obtained by solving the above set of nonlinear algebraic equations (10-13) for each new tool position. The coordinates of position k+1 are calculated iteratively beginning with some assumed initial solution estimate that is sufficiently close to the actual location to assure convergence to the correct position. One suitable estimate will be described immediately hereinafter. 
     Referring to  FIGS. 6 and 7   a , the X, Z plane is illustrated with a drill path  240  formed therein and in a direction  242  using a plurality of drill rods  32 , at least some of which have been designated by reference numbers.  FIG. 7   a  is an enlarged view within a dashed circle  244  of  FIG. 6 . An initial solution estimate is given by a point on what may be referred to as a nominal drill-path  246  that can be found by linear extrapolation of the previously predicted/last determined position to a predicted position  248 . The linear extrapolation is based on equations 4 and a given incremental movement Δs k+1  of the steering tool from a k th  position where: 
     
       
         
           
             
               
                 
                   
                     { 
                     
                       
                         
                           
                             X 
                             
                               k 
                               + 
                               1 
                             
                             * 
                           
                         
                       
                       
                         
                           
                             Y 
                             
                               k 
                               + 
                               1 
                             
                             * 
                           
                         
                       
                       
                         
                           
                             Z 
                             
                               k 
                               + 
                               1 
                             
                             * 
                           
                         
                       
                     
                     } 
                   
                   = 
                   
                     
                       { 
                       
                         
                           
                             
                               X 
                               k 
                             
                           
                         
                         
                           
                             
                               Y 
                               k 
                             
                           
                         
                         
                           
                             
                               Z 
                               k 
                             
                           
                         
                       
                       } 
                     
                     + 
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         s 
                         
                           k 
                           + 
                           1 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ϕ 
                                   k 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   β 
                                   k 
                                 
                               
                             
                           
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ϕ 
                                   k 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   β 
                                   k 
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ϕ 
                                   k 
                                 
                               
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     Where predicted positions are indicated in equations 14 using an asterisk ( )*. It should be appreciated that the position of the steering tool is characterized as predicted or estimated since the location is not identified in an affirmative manner such as is the case, for example, when a walk-over locater is used. The use of a steering tool differs at least for the reason that the position of the steering tool is estimated or predicted based on its previous positions. Thus, the actual position of the steering tool, for a sufficiently long drill path, can be significantly different than the position that is determined by a steering tool technique, as a result of accumulating error, if this error is not managed appropriately. 
     Application of the SIMPLEX method requires definition of a function that is to be minimized during the solution procedure. An example of such a function that is suitable in the present application reads: 
     
       
         
           
             
               
                 
                   F 
                   = 
                   
                     
                       ∑ 
                       
                         p 
                         = 
                         1 
                       
                       
                         3 
                         + 
                         
                           N 
                           M 
                         
                       
                     
                     ⁢ 
                     
                       f 
                       p 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     As noted above, it is considered that one having ordinary skill can conclude the solution procedure under SIMPLEX in view of the foregoing. 
     Kalman Filter Solution 
     In another embodiment, a method is described for solving the tracking equations employing Kalman filtering. The filter minimizes the position error caused by measurement uncertainties in a least square sense. The filter determines position coordinates as well as position error estimates. 
     The three tool position coordinates (X, Y, Z) are chosen as the main system parameters. Increments (δX, δY, δZ) of these parameters are referred to as state variables. The solution method can be characterized as a predictor-corrector technique. Assuming all drill-path variables are known at a last determined position and a drill string increment is known, the current or next-determined position on the drill path can be approximated by linear extrapolation, as described above with respect to  FIGS. 6 and 7   a . This is the predictor step that gives a point on nominal drill path  246 . The Kalman filter, in turn, performs a corrector step in which state variables are calculated and added to the nominal drill path. 
     Initial tool position coordinates (X 1 , Y 1 , Z 1 ) are assumed and corresponding error variances (σ X     1     2 , σ Y     1     2 , σ Z     1     2 ) are known. For example, at (X 1 , Y 1 , Z 1 ), which is the origin of the coordinate system, the error variances are zero. If (X 1 , Y 1 , Z 1 ) is not the origin, the error variances are based on the accuracy of measurement from the origin. The tracking procedure starts from this initial position and proceeds along the drill path, as follows: 
     As is illustrated in  FIGS. 6 and 7   a , the last known drill path position (X k , Y k , Z k ) is extrapolated linearly to obtain an approximate or estimated tool position, previously introduced as nominal drill path position (X k+1 *, Y k+1 *, Z k+1 *). 
     The filter determines state variables (δX k+1 , δY k+1 , δZ k+1 ) and standard deviations of position error (σ X     k+1   , σ Y     k+1   , σ Z     k+1   ). 
     State variables are added to the nominal drill path position to find the new tool position (X k+1 , Y k+1 , Z k+1 ). 
                     {           X     k   +   1                 Y     k   +   1                 Z     k   +   1             }     =       {           X     k   +   1     *               Y     k   +   1     *               Z     k   +   1     *           }     +     {           δ   ⁢           ⁢     X     k   +   1                   δ   ⁢           ⁢     Y     k   +   1                   δ   ⁢           ⁢     Z     k   +   1               }               (   16   )               
Measurement Errors
 
     The Kalman filter takes the following random measurement errors into account which must therefore be known before tracking begins. 
     Tool pitch and yaw angle errors σ φ , σ β   
     Distance error σ D    
     Magnet position errors (σ X     M   , σ Y     M   , σ Z     M   ) 
     Initial tool position errors (σ X     1   , σ Y     1   , σ Z     1   ) 
     Error values are empirical and depend on the type of instrumentation used. Note that the effect of drill rod length measuring error is not part of the analysis since arc-length along the axis of the drill rod is used as an independent variable. 
     Knowing initial tool position errors (σ X     1   , σ Y     1   , σ Z     1   ), the corresponding error covariance matrix P 1  is given as: 
     
       
         
           
             
               
                 
                   
                     P 
                     1 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             σ 
                             
                               X 
                               1 
                             
                             2 
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             σ 
                             
                               Y 
                               1 
                             
                             2 
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             σ 
                             
                               Z 
                               1 
                             
                             2 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Adding the latter to equations (4) to (6) completes the formulation of the initial value problem to be solved by Kalman filtering. 
     Linearized Tracking Equations 
     In addition to various measured quantities that are summarized above, the Kalman filter solution uses input of the following parameters. 
     Φ k  discrete state equation transition matrix 
     Q k  discrete process noise covariance matrix 
     z measurement scalar 
     H observation coefficient vector 
     R observation error covariance scalar 
     The above parameters are derived by linearizing the steering tool process equations and distance measurement equations about the nominal drill path position. The resulting two sets of linear equations are the so-called state equations and the observation equations. They are summarized below. 
     The state variables are defined as position increments.
 
 {right arrow over (x)}= (δ X,δY,δZ )′  (18a)
 
 {dot over ({right arrow over (x)}= (δ {dot over (X)},δ{dot over (Y)},δŻ )′  (18b)
 
     The state equations governing state variables read
 
 {right arrow over (x)}   k+1 =Φ k   {right arrow over (x)}   k   +Δs   k+1   G   k   {right arrow over (u)}   k   (19)
 
     Where Δs k+1 G k {right arrow over (u)} k  represents pitch and yaw measurement noise. It is noted that, hereinafter, subscripts may be dropped for purposes of clarity. Accordingly: 
     
       
         
           
             
               
                 
                   Φ 
                   = 
                   I 
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     Q 
                     = 
                     
                       cov 
                       ⁡ 
                       
                         ( 
                         
                           
                             ( 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               s 
                             
                             ) 
                           
                           ⁢ 
                           G 
                           ⁢ 
                           
                             u 
                             → 
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   
                     u 
                     → 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ϕ 
                         
                         , 
                         
                           δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           β 
                         
                       
                       ) 
                     
                     ′ 
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   G 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               - 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ϕ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             β 
                           
                         
                         
                           
                             
                               - 
                               cos 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ϕ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             β 
                           
                         
                       
                       
                         
                           
                             
                               - 
                               sin 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ϕ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             β 
                           
                         
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ϕ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             β 
                           
                         
                       
                       
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ϕ 
                           
                         
                         
                           0 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     The discrete noise covariance matrix Q k  becomes: 
     
       
         
           
             
               
                 
                   
                     c 
                     A 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             σ 
                             ϕ 
                             2 
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             σ 
                             β 
                             2 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   Q 
                   = 
                   
                     
                       
                         
                           c 
                           e 
                         
                         ⁡ 
                         
                           ( 
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             s 
                           
                           ) 
                         
                       
                       2 
                     
                     ⁢ 
                     G 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       c 
                       A 
                     
                     ⁢ 
                     
                       G 
                       ′ 
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     Note that the empirical coefficient c e  has been added to equation (25) in order to account for pitch and yaw bias errors. It has unit value if pitch and yaw measurement errors are entirely random. 
     The observation equation of a rotating magnet reads:
 
 z=H{right arrow over (x)}+ν   D +ν M   (27)
 
 R =cov(ν D +ν M )  (28)
 
     Where the term ν D  represents distance measurement noise and the term ν M  represents magnet position measurement noise. The term H will be described at an appropriate point below. The symbol z, seen in equation (27) is a difference between measured distance D and calculated distance D* from a marker to the nominal drill path position, given as:
 
 z=D−D*   (29)
 
 D*   2 =( X*−X   M ) 2 +( Y*−Y   M ) 2 +( Z*−Z   M ) 2   (30)
 
     The first term H on the right hand side of equation (27) is the observation coefficient vector, written as: 
     
       
         
           
             
               
                 
                   H 
                   = 
                   
                     ( 
                     
                       
                         
                           
                             X 
                             * 
                           
                           - 
                           
                             X 
                             M 
                           
                         
                         
                           D 
                           * 
                         
                       
                       , 
                       
                         
                           
                             Y 
                             * 
                           
                           - 
                           
                             Y 
                             M 
                           
                         
                         
                           D 
                           * 
                         
                       
                       , 
                       
                         
                           
                             Z 
                             * 
                           
                           - 
                           
                             Z 
                             M 
                           
                         
                         
                           D 
                           * 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
     
     The following form of the observation covariance scalar R is used in the steering tool method: 
                   R   =       σ   D   2     +     H   ⁢           ⁢     c   M     ⁢     H   ′                 (   32   )                 c   M     =     [           σ     X   M     2         0       0           0         σ     Y   M     2         0           0       0         σ     Z   M     2           ]             (   33   )               
Projection of State Variables and Estimation Errors
 
     An estimate of the state vector at the next steering tool position k+1 is denoted by {right arrow over ({circumflex over (x)} and its error covariance matrix is P −  where the superscript ( ) −  indicates the last available estimate. Before the filter is applied at the new tool position, set
 
 {right arrow over ({circumflex over (x)}={ 0}  (34)
 
     The error covariance matrix P k  is projected to the new position using
 
 P   k+1   − =Φ k   P   k Φ k   ′+Q   k   (35)
 
Kalman Filter Loop
 
     The filter loop is executed once for each marker, resulting in a flexible arrangement that is able to process any number of markers in use by the steering tool system. 
     The classical, well documented version of the filter loop is chosen as a basis for the current steering tool embodiment. It consists of three steps: 
     Kalman Gain:
 
 K=P   −   H ′( HP   −   H′+R ) −1   (36)
 
     State Variables:
 
 {right arrow over ({circumflex over (x)}={right arrow over ({circumflex over (x)}   −   +K ( z−H{right arrow over ({circumflex over (x)}   − )  (37)
 
     Error Covariance Matrix:
 
 P =( I−KH ) P   −   (38)
 
Position Coordinate Errors
 
     Having completed the filter analysis at a new position, its coordinates are given by equation (16). Corresponding one-sigma position errors follow from:
 
σ X =√{square root over ( P   11 )}  (39)
 
σ Y =√{square root over ( P   22 )}  (40)
 
σ Z =√{square root over ( P   33 )}  (41)
 
       FIG. 7   b  is a flow diagram, generally indicated by the reference number  260 , which illustrates one embodiment of a Kalman filter implementation according to the descriptions above. At  262 , the nominal position of the steering tool at k+1 is determined using equation 14. At  264 , the error covariance matrix is projected to position k+1 using equation 35. The state vector is initialized at  266 . Beginning with step  270 , a loop is entered using magnetic measurements associated with one marker. The distance D* between a point on the nominal drill path and the marker is determined per equation 30. The observation coefficient vector H in turn is calculated using equation 31. Equation 32 provides the observation covariance scalar R. At  272 , the Kalman filter is executed using equations 36-38. At  274 , a determination is made as to whether magnetic information is available that is associated with another marker. If so, execution returns to step  270 . If magnetic information from all markers has been processed, step  276  establishes the final coordinates of the current position of the steering tool based on equation 16 and can associate a position error estimate with these coordinates, based on equations 39-41. 
     Numerical Simulations 
     Several numerical simulations were performed to estimate positions of the steering tool assisted by up to three rotating magnets. In all cases the steering tool was tracked, moving along a drill-path defined by: 
     
       
         
           
             
               
                 
                   0 
                   ≤ 
                   
                     X 
                     ex 
                   
                   ≤ 
                   
                     300 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ft 
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
             
               
                 
                   
                     Y 
                     ex 
                   
                   = 
                   
                     15 
                     ⁢ 
                     
                       sin 
                       ⁡ 
                       
                         ( 
                         
                           
                             π 
                             300 
                           
                           ⁢ 
                           
                             X 
                             ex 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
             
               
                 
                   
                     Z 
                     ex 
                   
                   = 
                   
                     
                       - 
                       2 
                     
                     ⁢ 
                     
                       Y 
                       ex 
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     Note that drilling starts at the origin of the global coordinate system. The steering tool reaches a maximum depth of 30 feet and yaws to the side with a maximum lateral displacement of 15 feet before it reaches the target 300 feet out. The above coordinates are exactly known coordinates from which values for pitch, yaw and tool to magnet distances were derived. 
     Table 1 summarizes random and bias errors that were added to these exact values to generate “measured” data. 
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 Errors for Generating “Measured” Simulation Data 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Pitch Error 
                 σ φ  = 0.25 deg 
               
               
                   
                   
                 φ bias  = 0.25 deg 
               
               
                   
                 Yaw Error 
                 σ β  = 0.50 deg 
               
               
                   
                   
                 β bias  = 0.50 deg 
               
               
                   
                 Drill Rod Length Error 
                 σ Δs  = 0.01 ft 
               
               
                   
                 Distance Error 
                 D bias  = 0.02 ft 
               
               
                   
                   
                 (See also, FIG. 8) 
               
               
                   
               
             
          
         
       
     
       FIG. 8  sets forth random distance error σ D  in feet, plotted against distance D in feet. It is noted that errors were chosen based on empirical measurements with specific pitch and yaw sensors as well as with rotating magnets. Table 2 summarizes the random errors used as input for the filter. Note that the rod length increment error is used only for generating measured data; it is not used by the filter. 
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 2 
               
               
                   
               
               
                 Random Errors Used in Kalman Filter 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Pitch Error 
                 σ φ  = 0.5 deg 
               
               
                   
                 Yaw Error 
                 σ β  = 1 deg 
               
               
                   
                 Distance Error 
                 σ D  (see FIG. 8) 
               
               
                   
                 Magnet Position Errors 
                 σ X     M    = σ Y     M    = σ Z     M    = 0.02 ft 
               
               
                   
                 Initial Position Error 
                 σ X     1    = σ Y     1    = σ Z     1    = 0 
               
               
                   
               
             
          
         
       
     
       FIGS. 9   a  and  9   b  are plots against drill string length, in feet, which compare exact with “measured” pitch and yaw angles, respectively, used in all the simulations described below. Exact pitch and yaw values are shown by dotted lines, while measured pitch and yaw values are shown by solid lines. Increments between adjacent measurement positions along the drill-path were approximately three feet. 
     Estimated steering tool positions and position errors are illustrated by  FIGS. 10   a - c , as an application of the basic steering tool function without the use of markers. It is noted that, in subsequent figures, an increasing number of magnets is added to the system to demonstrate the improvements that are provided through the use of markers. Illustrated position errors are shown as the differences between estimated and exact values. Since “measured” values for pitch and yaw contain bias as well as random components, lateral and vertical position errors are also biased.  FIG. 10   a  is a diagrammatic plan view of the estimated drill path, designated by the reference number  300 , whereas  FIG. 10   b  is an elevational view of the estimated drill path, designated by the reference number  302 .  FIG. 10   c  illustrates the X coordinate positional error as a solid line  310 , the Y coordinate positional error as a dashed line  312  and the Z coordinate positional error as a dotted line  314 . In the present example, without the use of magnets, it can be seen that there is a continuously accumulating Y coordinate error, which increases to about three feet upon reaching X=300 feet, the X axis coordinate of target T. The Z coordinate error is over one foot. 
     Referring collectively to  FIGS. 11   a - c , simulations are now presented including the use of markers. One marker  320  is used at a location of X=300 ft, Y=−5 ft, and Z=5 ft.  FIG. 11   a  is a diagrammatic plan view of the estimated drill path, designated by the reference number  322 , whereas  FIG. 11   b  is an elevational view of the estimated drill path, designated by the reference number  324 .  FIG. 11   c  illustrates the X coordinate positional error as a solid line  326 , the Y coordinate positional error as a dashed line  328 , and the Z coordinate positional error as a dotted line  330 . In the present example, with the use of only one magnet near target T, it can be seen that the Y coordinate error is dramatically reduced to just over one foot upon reaching the target X coordinate at 300 feet. 
     Referring collectively to  FIGS. 12   a - c , a second marker  340  is added at a location of X=305 ft, Y=0 ft and Z=5 ft.  FIG. 12   a  is a diagrammatic plan view of the estimated drill path, designated by the reference number  342 , whereas  FIG. 12   b  is an elevational view of the estimated drill path, designated by the reference number  344 .  FIG. 12   c  illustrates the X coordinate positional error as a solid line  346 , the Y coordinate positional error as a dashed line  348 , and the Z coordinate positional error as a dotted line  350 . In the present example, with the use of two magnets near target T, it can be seen that the Y coordinate error is still further reduced to a relatively small fraction of one foot upon reaching the target X coordinate at 300 feet. Moreover, the X and Z coordinate errors are likewise reduced to a small fraction of one foot upon reaching the target X coordinate at 300 feet. 
     Referring collectively to  FIGS. 13   a - c , a third marker  360  is added at a location of X=300 ft, Y=5 ft and Z=5 ft.  FIG. 13   a  is a diagrammatic plan view of the estimated drill path, designated by the reference number  362 , whereas  FIG. 13   b  is an elevational view of the estimated drill path, designated by the reference number  364 .  FIG. 12   c  illustrates the X coordinate positional error as a solid line  366 , the Y coordinate positional error as a dashed line  368 , and the Z coordinate positional error as a dotted line  370 . In the present example, with the use of three magnets near target T, it can be seen that the X, Y and Z coordinate errors are reduced to a very small fraction of one foot upon reaching the target X coordinate at 300 feet. In view of the foregoing, the use of two or three markers proximate to a point of interest on the drill path (such as the target) enables a high precision guidance of the steering tool to a target at least 300 feet out from the point of drill begin, or enables high precision steering relative to some point of interest along the drill path at least 300 feet out. 
     It should be appreciated that, in the aforedescribed numerical simulations, errors defined as the difference between estimated and exact positions can be calculated, since exact drill-path coordinates are known. This type of error can not be calculated during actual drill-head tracking. Accordingly, a different type of error estimate is used for actual drilling. The Kalman filter analysis provides such an error estimate in the form of standard deviations of position coordinates. In this regard,  FIGS. 14   a - c  illustrate the two types of position errors for the drill-path of  FIGS. 13   a - b  with three markers placed near the target. The solid lines denote the +1 sigma position error provided by the Kalman filter analysis, whereas the dashed lines represent the corresponding −1 sigma errors. For comparison, the position errors of  FIG. 13   c , defined as the difference between estimated and exact positions, are also shown in  FIGS. 14   a - c . As seen, position errors expressed in terms of standard deviations vary smoothly along the drill-path since they are based on a statistical measure. In contrast, estimated positions and, hence, the errors shown as dotted lines in  FIGS. 14   a - c  are based on one set of partly random measurements resulting in an irregular distribution of position errors. Repeating the Kalman filter analysis with a different set of random measurements would produce different error distributions of this type. Numerical simulations were performed with c e =16. 
     Attention is now directed to  FIGS. 15 and 16  for purposes of describing additional aspects of the present disclosure.  FIG. 15  illustrates a plan view of a drilling region  400  having a concluding section of an intended drill path  402  defined therein. Further, a first inground obstacle  404  and a second inground obstacle  406  are shown in relation to intended path  402 . As can be seen, intended path  402  has been specifically designed to avoid inground obstacles  404  and  406 . Such path design can be based on any knowledge of inground features that should be avoided and can include a reliance on any suitable resource including but not limited to utility surveys, available design drawings and exploratory excavations. Moreover, inground obstacles  404  and  406  are intended to represent any type of feature within the ground that should be avoided. 
     Still referring to  FIGS. 14 and 15 , an exemplary plurality of markers  140   a - e  is distributed along intended path  402  such that markers  140   a  and  140   b  are in the vicinity of obstacle  404 , marker  104   c  is in the vicinity of obstacle  406 , and markers  140   d  and  140   e  are in the vicinity of target T. It should be appreciated that orientation of the markers is arbitrary so long as the steering tool, on the intended path and proximate to some inground feature of interest, is capable of receiving at least the magnetic field that is emanated by the markers in its general vicinity. As seen above, with each marker that is added proximate to target T, there is a corresponding increase in steering tool accuracy. That is, the steering tool tracks the intended path with proportionally increasing accuracy. Placement of markers proximate to points of interest, as illustrated, likewise produces a corresponding increase in accuracy along any portion of the intended path that is exposed to the magnetic field that is emanated by that marker. In this way, an enhanced steering accuracy, of a selective degree, can be provided at any desired point or points along the intended path. Accordingly, a highly advantageous customized steering accuracy is provided along the intended path. In this regard, as discussed above, the described technique readily accommodates receiving signals from any number of markers at any given point along the intended path or receiving no marker signals for some portions of the path, such as might be the case at a point  410  midway between markers  140   c  and  140   d  of the present example. 
     Even though the present example illustrates the use of five markers, fewer markers may actually be necessary since the markers can be moved along the intended drill path responsive to the progression of the steering tool. For example, after the steering tool passes obstacle  404 , marker  140   a  can be moved to the position of marker  140   c . At a suitable time, marker  140   b  can be moved to the position of marker  140   d . Once the steering tool passes obstacle  406 , marker  140   a  can then be moved to the illustrated position of marker  140   e . Accordingly, long drill runs can be made with as few as one or two markers. 
     Applicants consider that sweeping advantages are provided over the state-of-the-art with respect to steering tool systems and methods. While there are systems in the prior art that use rotating magnet signals, it should be apparent from the detailed descriptions above that providing the capability to use rotating magnet signals in the context of a steering tool system is neither trivial nor obvious. In this regard, Applicants are unaware of any prior art use of a rotating magnet signal in the context of a steering tool system and, particularly, with such flexibility and ease of use where the rotating magnet field markers can not only be arbitrarily placed, but arbitrarily oriented. 
     While a number of exemplary aspects and embodiments have been discussed above, those of skill in the art will recognize certain modifications, permutations, additions and sub-combinations thereof. It is therefore intended that the following appended claims and claims hereafter introduced are interpreted to include all such modifications, permutations, additions and sub-combinations as are within their true spirit and scope.
A steering tool is movable by a drill string to form an underground bore along an intended path. A sensing arrangement of the steering tool detects its pitch and yaw orientations at a series of spaced apart positions along the bore, each position is characterized by a measured extension of the drill string. The steering tool further includes a receiver. At least one marker is positioned proximate to the intended path, for transmitting a rotating dipole field to expose a portion of the intended path to the field for reception by the receiver. The detected pitch orientation, the detected yaw orientation and the measured extension of the drill string are used in conjunction with magnetic information from the receiver to locate the steering tool. The steering tool may automatically use the magnetic information when it is available. A customized overall position determination accuracy can be provided along the intended path.