Patent Publication Number: US-2020300064-A1

Title: Steering a Wellbore Using Stratigraphic Misfit Heat Maps

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of priority of U.S. Patent Application No. 62/820,191, filed Mar. 18, 2019, and entitled “Optimal Steering of a Wellbore Using Stratigraphic Misfit Heatmaps,” U.S. Patent Application No. 62/834,154, filed on Apr. 15, 2019, and entitled “Integrating Reference Data for Steering of a Wellbore Using Stratigraphic Misfit Heat Maps,” U.S. Patent Application No. 62/985,224, filed on Mar. 4, 2020, and entitled “Optimal Steering of a Wellbore Using Stratigraphic Misfit Heatmaps,” and U.S. Patent Application No. 62/844,488, filed on May 7, 2019, and entitled “Determining the Likelihood and Uncertainty of the Wellbore Being at a Particular Stratigraphic Vertical Depth,” each of which is hereby incorporated by reference herein. 
    
    
     BACKGROUND 
     Field of the Disclosure 
     The present disclosure provides systems and methods useful for optimally steering a wellbore into one or multiple geological target formations when one or multiple wells have already been drilled in the vicinity. The method can be executed with a programmed computer system in fully automated, semi-automated and manual modes. 
     Description of the Related Art 
     Drilling a borehole for the extraction of minerals has become an increasingly complicated operation due to the increased depth and complexity of many boreholes, including the complexity added by directional drilling. Drilling is an expensive operation and errors in drilling add to the cost and, in some cases, drilling errors may permanently lower the output of a well for years into the future. Conventional technologies and methods may not adequately address the complicated nature of drilling, and may not be capable of gathering and processing various information from downhole sensors and surface control systems in a timely manner, in order to improve drilling operations and minimize drilling errors. 
     The determination of the well trajectory from a downhole survey may involve various calculations that depend upon reference values and measured values. However, various internal and external factors may adversely affect the downhole survey and, in turn, the determination of the well trajectory. 
     A subject wellbore can be steered into one or multiple geological stratigraphic targets. The directional drilling process usually follows a spatial well plan, in which the position of the desired wellbore trajectory can be given in spatial coordinates. However, the desired geological target may not be exactly at the depth assumed when creating the well plan, due to unknown lateral variations and other uncertainties in geological stratigraphy. Therefore, the well plan may be updated based on new stratigraphic information from the wellbore, as it is being drilled. This stratigraphic information can be gained on one hand from Measurement While Drilling (MWD) and Logging While Drilling (LWD) sensor data, but could also include drilling dynamics data with information, for example, on the hardness of the rock. 
     SUMMARY 
     The systems and methods discussed herein may be used to help steer the drilling of a wellbore to a target. In one embodiment various parameters may be combined into a single characteristic function, both for the subject well and one or more offset wells. For every pair of subject well and offset well, a heat map can be computed to display the misfit between the characteristic functions of the subject and offset wells. The heat maps then enable the identification of paths (x(MD), y(MD)), parameterized by the measured depth (MD) along the subject well. These paths uniquely describe the vertical depth of the subject well relative to the geology (e.g., formation) at every offset well. Alternatively, the characteristic functions of the offset wells can be combined into a single characteristic function at the location of the subject wellbore. This combined characteristic function changes along the subject well with changes in the stratigraphy. The heat map may also be used to identify stratigraphic anomalies, such as structural faults, stringers and breccia. The identified paths may be used in updating the well plan with the latest data to steer the wellbore into the geological target(s) and keep the wellbore in the target zone. 
     In one embodiment, a method of geosteering a well is provided, with the method comprising taking measurements at a plurality measured depths (MD) along a borehole of a subject well being drilled; selecting an offset well log with measurements at a plurality of stratigraphic vertical depths (SVD); computing a misfit value (MD,SVD) for each of a plurality of pairs of measurements of the subject well MD and the offset well SVD; determining, responsive to the misfit value, a stratigraphic depth of the borehole of the subject well, SVD(MD); and drilling the borehole of the subject well responsive to the determined stratigraphic depth to a target stratigraphic depth. The misfit value may be computed using a plurality of offset well logs and may further comprise taking a plurality of measurements at a plurality of MDs along the borehole of the subject well; selecting a plurality of offset well logs, each having a plurality of measurements at true vertical depths (TVD); generating a mapping of TVD(SVD), wherein the mapping relates a stratigraphic vertical depth SVD to the corresponding TVD of all of the plurality of offset well logs; computing a combined Misfit value (MD,SVD) for each of the measurements at MD and offset measurements TVD(SVD); determining, responsive to the misfit value, the stratigraphic depth of the borehole of the subject well, SVD(MD), wherein the stratigraphic depth corresponds to a minimum cost related to the combined misfit value (MD,SVD); and updating a drill plan for the well and/or drilling the borehole of the subject well responsive to the determined stratigraphic depth, SVD(MD), to the target stratigraphic depth. The plurality of measurements from the multiple offset well logs may be combined into one reference log, and the methods may further comprise taking a plurality of measurements at measured depths MD along the borehole of the subject well; selecting a plurality of offset well logs with measurements at true vertical depths, TVD generating a mapping TVD(SVD), which relates a stratigraphic vertical depth, SVD, to the corresponding TVD of all offset well logs; combining the measurements of the offset well logs into a common reference well log organized by SVD; computing the Misfit value (MD,SVD) for each of a plurality of pairs of measurements at subject well MD and reference well log SVD; determining, responsive to the misfit value, the stratigraphic depth of the subject well, SVD(MD), by minimizing a cost related to the misfit (MD,SVD); and using the inferred stratigraphic depth, SVD(MD), to steer the subject well to a desired stratigraphic depth range. The measurements on the subject well log and offset well log(s) may include any one or more of gamma ray intensity, azimuthal gamma, resistivity, azimuthal resistivity, density, porosity, rate of progress, mechanical specific energy, rock compressive strength, rate of penetration, differential pressure, and weight on bit. In addition, a stochastic model may be used to characterize the misfit value. The misfit value (MD, SVD) may be displayed as a heat map, and may be displayed using a combination of MD, SVD, RSVD, TVD and Vertical Section on the X and Y axis. Moreover, the heat map display may include ancillary data, such as the well plan, the surveyed wellbore position, geological markers, seismic velocities, and/or other geophysical data. Multiple misfit heat maps for multiple measurement types or multiple offset well logs also may be displayed simultaneously, and may be displayed in 3D. In some embodiments, a cost function minimization of the cost function may be used and may be guided by the operator by specifying stratigraphic control points SVD(MD). In some implementations, an operator may manually interpret one or more paths of minimal misfit in the heat map to specify the SVD(MD) of the subject wellbore, while in other implementations, some or all of the steps are automatically performed by a computer system, and the computer system may be coupled to one or more control systems of a drilling rig and may automatically send one or more control signals to such rig control systems to adjust one or more drilling parameters or operations, such as to automatically adjust drilling to drill to the target geological zone. 
     In some embodiments, the present disclosure includes a computer system, with the computer system comprising a processor; a memory coupled to the processor, the memory containing instructions executable by the processor for performing some or all of the following steps: taking measurements at a plurality measured depths (MD) along a borehole of a subject well being drilled; selecting an offset well log with measurements at a plurality of stratigraphic vertical depths (SVD); computing a misfit value (MD,SVD) for each of a plurality of pairs of measurements of the subject well MD and the offset well SVD; determining, responsive to the misfit value, a stratigraphic depth of the borehole of the subject well, SVD(MD); and sending a signal to one or more control systems of a drilling rig drilling the subject well to drill the borehole of the subject well responsive to the determined stratigraphic depth to a target stratigraphic depth. The misfit value may be computed by the system by using a plurality of offset well logs and the instructions further comprise instructions for: taking a plurality of measurements at a plurality of MDs along the borehole of the subject well; selecting a plurality of offset well logs, each having a plurality of measurements at true vertical depths (TVD); generating a mapping of TVD(SVD), wherein the mapping relates a stratigraphic vertical depth SVD to the corresponding TVD of all of the plurality of offset well logs; computing a combined Misfit value (MD,SVD) for each of the measurements at MD and offset measurements TVD(SVD); determining, responsive to the misfit value, the stratigraphic depth of the borehole of the subject well, SVD(MD), wherein the stratigraphic depth corresponds to a minimum cost related to the combined misfit value (MD,SVD); and sending one or more control signals to the one or more control systems to drill the borehole of the subject well responsive to the determined stratigraphic depth, SVD(MD), to the target stratigraphic depth. The computer system may further compromise instructions for: combining the plurality of measurements from the multiple offset well logs into one reference log; taking a plurality of measurements at measured depths MD along the borehole of the subject well; selecting a plurality of offset well logs with measurements at true vertical depths, TVD; generating a mapping TVD(SVD), which relates a stratigraphic vertical depth, SVD, to the corresponding TVD of all offset well logs; combining the measurements of the offset well logs into a common reference well log organized by SVD; computing the Misfit value (MD,SVD) for each of a plurality of pairs of measurements at subject well MD and reference well log SVD; determining, responsive to the misfit value, the stratigraphic depth of the subject well, SVD(MD), by minimizing a cost related to the Misfit(MD,SVD) using the inferred stratigraphic depth, SVD(MD), to steer the subject well to a desired stratigraphic depth range; and sending one or more control signals to the one or more control systems of the drilling rig to drill to the desired stratigraphic depth range. The measurements on the subject well log and offset well log include any one or more of gamma ray intensity, azimuthal gamma, resistivity, azimuthal resistivity, density, porosity, rate of progress, mechanical specific energy, rock compressive strength, rate of penetration, differential pressure, and weight on bit. A stochastic model may be used by the computer system to characterize the misfit value. The collection of misfit values (MD, SVD) may be displayed as a heat map, with the display being located either or both at the drilling site for the subject well or at a remote location from the drilling site. The misfit value (MD,SVD) may be displayed using a combination of MD, SVD, RSVD, TVD and Vertical Section on the X and Y axis. In addition, at least a portion of the heat map display may include ancillary data, such as some or all of the well plan, the surveyed wellbore position, geological markers, seismic velocities, and/or other geophysical data, are displayed on at least a portion of the display of the heat map. A plurality of misfit heat maps for either or both of a plurality of measurement types or a plurality of offset well logs also may be displayed simultaneously, and/or may be displayed in 3D. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       For a more complete understanding of the present invention and its features and advantages, reference is now made to the following description, taken in conjunction with the accompanying drawings, in which: 
         FIG. 1  is a depiction of a drilling system for drilling a borehole; 
         FIG. 2  is a depiction of a drilling environment including the drilling system for drilling a borehole; 
         FIG. 3  is a depiction of a borehole generated in the drilling environment; 
         FIG. 4  is a depiction of a drilling architecture including the drilling environment; 
         FIG. 5  is a depiction of rig control systems included in the drilling system; 
         FIG. 6  is a depiction of algorithm modules used by the rig control systems; 
         FIG. 7  is a depiction of a steering control process used by the rig control systems; 
         FIG. 8  is a depiction of a graphical user interface provided by the rig control systems; 
         FIG. 9  is a depiction of a guidance control loop performed by the rig control systems; 
         FIG. 10  is a depiction of a controller usable by the rig control systems; 
         FIG. 11A  is a depiction of a conventional display of a vertical section of a wellbore, showing a gamma ray log and the wellbore and geology against true vertical depth. 
         FIG. 11B  is a depiction of a display of information (like that included in  FIG. 11A ) for a wellbore that is displayed with measured depth on the x-axis and stratigraphic vertical depth on the y-axis. 
         FIG. 12  is a depiction of a simple form of stratigraphic misfit heat map in accordance with the present disclosure. 
         FIG. 13  is a depiction of a 3D geomodel in accordance with the present disclosure. 
         FIG. 14  illustrates a heat map in accordance with the present disclosure. 
         FIG. 15  illustrates an alternative type of heat map in accordance with the present disclosure. 
         FIG. 16  illustrates a comparison of a well borehole interpretation using an automated heat map analysis versus a well borehole interpretation using a conventional technique. 
         FIG. 17  is a 3D plot of a least squares estimation for linear segments; 
         FIG. 18  is a 3D plot of a semi-automated interpretation using way points; 
         FIG. 19  is a 3D plot of a manual interpretation; and 
         FIG. 20  is a depiction of a user interface for displaying multiple 3D plots. 
     
    
    
     DESCRIPTION OF PARTICULAR EMBODIMENT(S) 
     In the following description, details are set forth by way of example to facilitate discussion of the disclosed subject matter. It should be apparent to a person of ordinary skill in the field, however, that the disclosed embodiments are exemplary and not exhaustive of all possible embodiments. 
     Throughout this disclosure, a hyphenated form of a reference numeral refers to a specific instance of an element and the un-hyphenated form of the reference numeral refers to the element generically or collectively. Thus, as an example (not shown in the drawings), device “ 12 - 1 ” refers to an instance of a device class, which may be referred to collectively as devices “ 12 ” and any one of which may be referred to generically as a device “ 12 ”. In the figures and the description, like numerals are intended to represent like elements. 
     Drilling a well typically involves a substantial amount of human decision-making during the drilling process. For example, geologists and drilling engineers use their knowledge, experience, and the available information to make decisions on how to plan the drilling operation, how to accomplish the drilling plan, and how to handle issues that arise during drilling. However, even the best geologists and drilling engineers perform some guesswork due to the unique nature of each borehole. Furthermore, a directional human driller performing the drilling may have drilled other boreholes in the same region and so may have some similar experience. However, during drilling operations, a multitude of input information and other factors may affect a drilling decision being made by a human operator or specialist, such that the amount of information may overwhelm the cognitive ability of the human to properly consider and factor into the drilling decision. Furthermore, the quality or the error involved with the drilling decision may improve with larger amounts of input data being considered, for example, such as formation data from a large number of offset wells. For these reasons, human specialists may be unable to achieve optimal drilling decisions, particularly when such drilling decisions are made under time constraints, such as during drilling operations when continuation of drilling is dependent on the drilling decision and, thus, the entire drilling rig waits idly for the next drilling decision. Furthermore, human decision-making for drilling decisions can result in expensive mistakes, because drilling errors can add significant cost to drilling operations. In some cases, drilling errors may permanently lower the output of a well, resulting in substantial long term economic losses due to the lost output of the well. 
     Referring now to the drawings, Referring to  FIG. 1 , a drilling system  100  is illustrated in one embodiment as a top drive system. As shown, the drilling system  100  includes a derrick  132  on the surface  104  of the earth and is used to drill a borehole  106  into the earth. Typically, drilling system  100  is used at a location corresponding to a geographic formation  102  in the earth that is known. 
     In  FIG. 1 , derrick  132  includes a crown block  134  to which a traveling block  136  is coupled via a drilling line  138 . In drilling system  100 , a top drive  140  is coupled to traveling block  136  and may provide rotational force for drilling. A saver sub  142  may sit between the top drive  140  and a drill pipe  144  that is part of a drill string  146 . Top drive  140  may rotate drill string  146  via the saver sub  142 , which in turn may rotate a drill bit  148  of a bottom hole assembly (BHA)  149  in borehole  106  passing through formation  102 . Also visible in drilling system  100  is a rotary table  162  that may be fitted with a master bushing  164  to hold drill string  146  when not rotating. 
     A mud pump  152  may direct a fluid mixture  153  (e.g., a mud mixture) from a mud pit  154  into drill string  146 . Mud pit  154  is shown schematically as a container, but it is noted that various receptacles, tanks, pits, or other containers may be used. Mud  153  may flow from mud pump  152  into a discharge line  156  that is coupled to a rotary hose  158  by a standpipe  160 . Rotary hose  158  may then be coupled to top drive  140 , which includes a passage for mud  153  to flow into borehole  106  via drill string  146  from where mud  153  may emerge at drill bit  148 . Mud  153  may lubricate drill bit  148  during drilling and, due to the pressure supplied by mud pump  152 , mud  153  may return via borehole  106  to surface  104 . 
     In drilling system  100 , drilling equipment (see also  FIG. 5 ) is used to perform the drilling of borehole  106 , such as top drive  140  (or rotary drive equipment) that couples to drill string  146  and BHA  149  and is configured to rotate drill string  146  and apply pressure to drill bit  148 . Drilling system  100  may include control systems such as a WOB/differential pressure control system  522 , a positional/rotary control system  524 , a fluid circulation control system  526 , and a sensor system  528 , as further described below with respect to  FIG. 5 . The control systems may be used to monitor and change drilling rig settings, such as the WOB or differential pressure to alter the ROP or the radial orientation of the toolface, change the flow rate of drilling mud, and perform other operations. Sensor system  528  may be for obtaining sensor data about the drilling operation and drilling system  100 , including the downhole equipment. For example, sensor system  528  may include MWD or logging while drilling (LWD) tools for acquiring information, such as toolface and formation logging information, that may be saved for later retrieval, transmitted with or without a delay using any of various communication means (e.g., wireless, wireline, or mud pulse telemetry), or otherwise transferred to steering control system  168 . As used herein, an MWD tool is enabled to communicate downhole measurements without substantial delay to the surface  104 , such as using mud pulse telemetry, while a LWD tool is equipped with an internal memory that stores measurements when downhole and can be used to download a stored log of measurements when the LWD tool is at the surface  104 . The internal memory in the LWD tool may be a removable memory, such as a universal serial bus (USB) memory device or another removable memory device. It is noted that certain downhole tools may have both MWD and LWD capabilities. Such information acquired by sensor system  528  may include information related to hole depth, bit depth, inclination angle, azimuth angle, true vertical depth, gamma count, standpipe pressure, mud flow rate, rotary rotations per minute (RPM), bit speed, ROP, WOB, among other information. It is noted that all or part of sensor system  528  may be incorporated into a control system, or in another component of the drilling equipment. As drilling system  100  can be configured in many different implementations, it is noted that different control systems and subsystems may be used. 
     Sensing, detection, measurement, evaluation, storage, alarm, and other functionality may be incorporated into a downhole tool  166  or BHA  149  or elsewhere along drill string  146  to provide donwhole surveys of borehole  106 . Accordingly, downhole tool  166  may be an MWD tool or a LWD tool or both, and may accordingly utilize connectivity to the surface  104 , local storage, or both. In different implementations, gamma radiation sensors, magnetometers, accelerometers, and other types of sensors may be used for the downhole surveys. Although downhole tool  166  is shown in singular in drilling system  100 , it is noted that multiple instances (not shown) of downhole tool  166  may be located at one or more locations along drill string  146 . 
     In some embodiments, formation detection and evaluation functionality may be provided via a steering control system  168  on the surface  104 . Steering control system  168  may be located in proximity to derrick  132  or may be included with drilling system  100 . In other embodiments, steering control system  168  may be remote from the actual location of borehole  106  (see also  FIG. 4 ). For example, steering control system  168  may be a stand-alone system or may be incorporated into other systems included with drilling system  100 . 
     In operation, steering control system  168  may be accessible via a communication network (see also  FIG. 10 ), and may accordingly receive formation information via the communication network. In some embodiments, steering control system  168  may use the evaluation functionality to provide corrective measures, such as a convergence plan to overcome an error in the well trajectory of borehole  106  with respect to a reference, or a planned well trajectory. The convergence plans or other corrective measures may depend on a determination of the well trajectory, and therefore, may be improved in accuracy using surface steering, as disclosed herein. 
     In particular embodiments, at least a portion of steering control system  168  may be located in downhole tool  166  (not shown). In some embodiments, steering control system  168  may communicate with a separate controller (not shown) located in downhole tool  166 . In particular, steering control system  168  may receive and process measurements received from downhole surveys, and may perform the calculations described herein for surface steering using the downhole surveys and other information referenced herein. 
     In drilling system  100 , to aid in the drilling process, data is collected from borehole  106 , such as from sensors in BHA  149 , downhole tool  166 , or both. The collected data may include the geological characteristics of formation  102  in which borehole  106  was formed, the attributes of drilling system  100 , including BHA  149 , and drilling information such as weight-on-bit (WOB), drilling speed, and other information pertinent to the formation of borehole  106 . The drilling information may be associated with a particular depth or another identifiable marker to index collected data. For example, the collected data for borehole  106  may capture drilling information indicating that drilling of the well from 1,000 feet to 1,200 feet occurred at a first rate of penetration (ROP) through a first rock layer with a first WOB, while drilling from 1,200 feet to 1,500 feet occurred at a second ROP through a second rock layer with a second WOB (see also  FIG. 2 ). In some applications, the collected data may be used to virtually recreate the drilling process that created borehole  106  in formation  102 , such as by displaying a computer simulation of the drilling process. The accuracy with which the drilling process can be recreated depends on a level of detail and accuracy of the collected data, including collected data from a downhole survey of the well trajectory. 
     The collected data may be stored in a database that is accessible via a communication network for example. In some embodiments, the database storing the collected data for borehole  106  may be located locally at drilling system  100 , at a drilling hub that supports a plurality of drilling systems  100  in a region, or at a database server accessible over the communication network that provides access to the database (see also  FIG. 4 ). At drilling system  100 , the collected data may be stored at the surface  104  or downhole in drill string  146 , such as in a memory device included with BHA  149  (see also  FIG. 10 ). Alternatively, at least a portion of the collected data may be stored on a removable storage medium, such as using steering control system  168  or BHA  149 , that is later coupled to the database in order to transfer the collected data to the database, which may be manually performed at certain intervals, for example. 
     In  FIG. 1 , steering control system  168  is located at or near the surface  104  where borehole  106  is being drilled. Steering control system  168  may be coupled to equipment used in drilling system  100  and may also be coupled to the database, whether the database is physically located locally, regionally, or centrally (see also  FIGS. 4 and 5 ). Accordingly, steering control system  168  may collect and record various inputs, such as measurement data from a magnetometer and an accelerometer that may also be included with BHA  149 . 
     Steering control system  168  may further be used as a surface steerable system, along with the database, as described above. The surface steerable system may enable an operator to plan and control drilling operations while drilling is being performed. The surface steerable system may itself also be used to perform certain drilling operations, such as controlling certain control systems that, in turn, control the actual equipment in drilling system  100  (see also  FIG. 5 ). The control of drilling equipment and drilling operations by steering control system  168  may be manual, manual-assisted, semi-automatic, or automatic, in different embodiments. 
     Manual control may involve direct control of the drilling rig equipment, albeit with certain safety limits to prevent unsafe or undesired actions or collisions of different equipment. To enable manual-assisted control, steering control system  168  may present various information, such as using a graphical user interface (GUI) displayed on a display device (see  FIG. 8 ), to a human operator, and may provide controls that enable the human operator to perform a control operation. The information presented to the user may include live measurements and feedback from the drilling rig and steering control system  168 , or the drilling rig itself, and may further include limits and safety-related elements to prevent unwanted actions or equipment states, in response to a manual control command entered by the user using the GUI. 
     To implement semi-automatic control, steering control system  168  may itself propose or indicate to the user, such as via the GUI, that a certain control operation, or a sequence of control operations, should be performed at a given time. Then, steering control system  168  may enable the user to imitate the indicated control operation or sequence of control operations, such that once manually started, the indicated control operation or sequence of control operations is automatically completed. The limits and safety features mentioned above for manual control would still apply for semi-automatic control. It is noted that steering control system  168  may execute semi-automatic control using a secondary processor, such as an embedded controller that executes under a real-time operating system (RTOS), that is under the control and command of steering control system  168 . To implement automatic control, the step of manual starting the indicated control operation or sequence of operations is eliminated, and steering control system  168  may proceed with only a passive notification to the user of the actions taken. 
     In order to implement various control operations, steering control system  168  may perform (or may cause to be performed) various input operations, processing operations, and output operations. The input operations performed by steering control system  168  may result in measurements or other input information being made available for use in any subsequent operations, such as processing or output operations. The input operations may accordingly provide the input information, including feedback from the drilling process itself, to steering control system  168 . The processing operations performed by steering control system  168  may be any processing operation associated with surface steering, as disclosed herein. The output operations performed by steering control system  168  may involve generating output information for use by external entities, or for output to a user, such as in the form of updated elements in the GUI, for example. The output information may include at least some of the input information, enabling steering control system  168  to distribute information among various entities and processors. 
     In particular, the operations performed by steering control system  168  may include operations such as receiving drilling data representing a drill path, receiving other drilling parameters, calculating a drilling solution for the drill path based on the received data and other available data (e.g., rig characteristics), implementing the drilling solution at the drilling rig, monitoring the drilling process to gauge whether the drilling process is within a defined margin of error of the drill path, and calculating corrections for the drilling process if the drilling process is outside of the margin of error. 
     Accordingly, steering control system  168  may receive input information either before drilling, during drilling, or after drilling of borehole  106 . The input information may comprise measurements from one or more sensors, as well as survey information collected while drilling borehole  106 . The input information may also include a well plan, a regional formation history, drilling engineer parameters, downhole tool face/inclination information, downhole tool gamma/resistivity information, economic parameters, reliability parameters, among various other parameters. Some of the input information, such as the regional formation history, may be available from a drilling hub  410 , which may have respective access to a regional drilling database (DB)  412  (see  FIG. 4 ). Other input information may be accessed or uploaded from other sources to steering control system  168 . For example, a web interface may be used to interact directly with steering control system  168  to upload the well plan or drilling parameters. 
     As noted, the input information may be provided to steering control system  168 . After processing by steering control system  168 , steering control system  168  may generate control information that may be output to drilling rig  210  (e.g., to rig controls  520  that control drilling equipment  530 , see also  FIGS. 2 and 5 ). Drilling rig  210  may provide feedback information using rig controls  520  to steering control system  168 . The feedback information may then serve as input information to steering control system  168 , thereby enabling steering control system  168  to perform feedback loop control and validation. Accordingly, steering control system  168  may be configured to modify its output information to the drilling rig, in order to achieve the desired results, which are indicated in the feedback information. The output information generated by steering control system  168  may include indications to modify one or more drilling parameters, the direction of drilling, the drilling mode, among others. In certain operational modes, such as semi-automatic or automatic, steering control system  168  may generate output information indicative of instructions to rig controls  520  to enable automatic drilling using the latest location of BHA  149 . Therefore, an improved accuracy in the determination of the location of BHA  149  may be provided using steering control system  168 , along with the methods and operations for surface steering disclosed herein. 
     Referring now to  FIG. 2 , a drilling environment  200  is depicted schematically and is not drawn to scale or perspective. In particular, drilling environment  200  may illustrate additional details with respect to formation  102  below the surface  104  in drilling system  100  shown in  FIG. 1 . In  FIG. 2 , drilling rig  210  may represent various equipment discussed above with respect to drilling system  100  in  FIG. 1  that is located at the surface  104 . 
     In drilling environment  200 , it may be assumed that a drilling plan (also referred to as a well plan) has been formulated to drill borehole  106  extending into the ground to a true vertical depth (TVD)  266  and penetrating several subterranean strata layers. Borehole  106  is shown in  FIG. 2  extending through strata layers  268 - 1  and  270 - 1 , while terminating in strata layer  272 - 1 . Accordingly, as shown, borehole  106  does not extend or reach underlying strata layers  274 - 1  and  276 - 1 . A target area  280  specified in the drilling plan may be located in strata layer  272 - 1  as shown in  FIG. 2 . Target area  280  may represent a desired endpoint of borehole  106 , such as a hydrocarbon producing area indicated by strata layer  272 - 1 . It is noted that target area  280  may be of any shape and size, and may be defined using various different methods and information in different embodiments. In some instances, target area  280  may be specified in the drilling plan using subsurface coordinates, or references to certain markers, that indicate where borehole  106  is to be terminated. In other instances, target area may be specified in the drilling plan using a depth range within which borehole  106  is to remain. For example, the depth range may correspond to strata layer  272 - 1 . In other examples, target area  280  may extend as far as can be realistically drilled. For example, when borehole  106  is specified to have a horizontal section with a goal to extend into strata layer  172  as far as possible, target area  280  may be defined as strata layer  272 - 1  itself and drilling may continue until some other physical limit is reached, such as a property boundary or a physical limitation to the length of the drill string. 
     Also visible in  FIG. 2  is a fault line  278  that has resulted in a subterranean discontinuity in the fault structure. Specifically, strata layers  268 ,  270 ,  272 ,  274 , and  276  have portions on either side of fault line  278 . On one side of fault line  278 , where borehole  106  is located, strata layers  268 - 1 ,  270 - 1 ,  272 - 1 ,  274 - 1 , and  276 - 1  are unshifted by fault line  278 . On the other side of fault line  278 , strata layers  268 - 2 ,  270 - 3 ,  272 - 3 ,  274 - 3 , and  276 - 3  are shifted downwards by fault line  278 . 
     Current drilling operations frequently include directional drilling to reach a target, such as target area  280 . The use of directional drilling has been found to generally increase an overall amount of production volume per well, but also may lead to significantly higher production rates per well, which are both economically desirable. As shown in  FIG. 2 , directional drilling may be used to drill the horizontal portion of borehole  106 , which increases an exposed length of borehole  106  within strata layer  272 - 1 , and which may accordingly be beneficial for hydrocarbon extraction from strata layer  272 - 1 . Directional drilling may also be used alter an angle of borehole  106  to accommodate subterranean faults, such as indicated by fault line  278  in  FIG. 2 . Other benefits that may be achieved using directional drilling include sidetracking off of an existing well to reach a different target area or a missed target area, drilling around abandoned drilling equipment, drilling into otherwise inaccessible or difficult to reach locations (e.g., under populated areas or bodies of water), providing a relief well for an existing well, and increasing the capacity of a well by branching off and having multiple boreholes extending in different directions or at different vertical positions for the same well. Directional drilling is often not limited to a straight horizontal borehole  106 , but may involve staying within a strata layer that varies in depth and thickness as illustrated by strata layer  172 . As such, directional drilling may involve multiple vertical adjustments that complicate the trajectory of borehole  106 . 
     Referring now to  FIG. 3 , one embodiment of a portion of borehole  106  is shown in further detail. Using directional drilling for horizontal drilling may introduce certain challenges or difficulties that may not be observed during vertical drilling of borehole  106 . For example, a horizontal portion  318  of borehole  106  may be started from a vertical portion  310 . In order to make the transition from vertical to horizontal, a curve may be defined that specifies a so-called “build up” section  316 . Build up section  316  may begin at a kick off point  312  in vertical portion  310  and may end at a begin point  314  of horizontal portion  318 . The change in inclination in build up section  316  per measured length drilled is referred to herein as a “build rate” and may be defined in degrees per one hundred feet drilled. For example, the build rate may have a value of 6°/100 ft., indicating that there is a six degree change in inclination for every one hundred feet drilled. The build rate for a particular build up section may remain relatively constant or may vary. 
     The build rate used for any given build up section may depend on various factors, such as properties of the formation (i.e., strata layers) through which borehole  106  is to be drilled, the trajectory of borehole  106 , the particular pipe and drill collars/BHA components used (e.g., length, diameter, flexibility, strength, mud motor bend setting, and drill bit), the mud type and flow rate, the specified horizontal displacement, stabilization, and inclination, among other factors. An overly aggressive built rate can cause problems such as severe doglegs (e.g., sharp changes in direction in the borehole) that may make it difficult or impossible to run casing or perform other operations in borehole  106 . Depending on the severity of any mistakes made during directional drilling, borehole  106  may be enlarged or drill bit  146  may be backed out of a portion of borehole  106  and redrilled along a different path. Such mistakes may be undesirable due to the additional time and expense involved. However, if the built rate is too cautious, additional overall time may be added to the drilling process, because directional drilling generally involves a lower ROP than straight drilling. Furthermore, directional drilling for a curve is more complicated than vertical drilling and the possibility of drilling errors increases with directional drilling (e.g., overshoot and undershoot that may occur while trying to keep drill bit  148  on the planned trajectory). 
     Two modes of drilling, referred to herein as “rotating” and “sliding”, are commonly used to form borehole  106 . Rotating, also called “rotary drilling”, uses top drive  140  or rotary table  162  to rotate drill string  146 . Rotating may be used when drilling occurs along a straight trajectory, such as for vertical portion  310  of borehole  106 . Sliding, also called “steering” or “directional drilling” as noted above, typically uses a mud motor located downhole at BHA  149 . The mud motor may have an adjustable bent housing and is not powered by rotation of the drill string. Instead, the mud motor uses hydraulic power derived from the pressurized drilling mud that circulates along borehole  106  to and from the surface  104  to directionally drill borehole  106  in build up section  316 . 
     Thus, sliding is used in order to control the direction of the well trajectory during directional drilling. A method to perform a slide may include the following operations. First, during vertical or straight drilling, the rotation of drill string  146  is stopped. Based on feedback from measuring equipment, such as from downhole tool  166 , adjustments may be made to drill string  146 , such as using top drive  140  to apply various combinations of torque, WOB, and vibration, among other adjustments. The adjustments may continue until a tool face is confirmed that indicates a direction of the bend of the mud motor is oriented to a direction of a desired deviation (i.e., build rate) of borehole  106 . Once the desired orientation of the mud motor is attained, WOB to the drill bit is increased, which causes the drill bit to move in the desired direction of deviation. Once sufficient distance and angle have been built up in the curved trajectory, a transition back to rotating mode can be accomplished by rotating the drill string again. The rotation of the drill string after sliding may neutralize the directional deviation caused by the bend in the mud motor due to the continuous rotation around a centerline of borehole  106 . 
     Referring now to  FIG. 4 , a drilling architecture  400  is illustrated in diagram form. As shown, drilling architecture  400  depicts a hierarchical arrangement of drilling hubs  410  and a central command  414 , to support the operation of a plurality of drilling rigs  210  in different regions  402 . Specifically, as described above with respect to  FIGS. 1 and 2 , drilling rig  210  includes steering control system  168  that is enabled to perform various drilling control operations locally to drilling rig  210 . When steering control system  168  is enabled with network connectivity, certain control operations or processing may be requested or queried by steering control system  168  from a remote processing resource. As shown in  FIG. 4 , drilling hubs  410  represent a remote processing resource for steering control system  168  located at respective regions  402 , while central command  414  may represent a remote processing resource for both drilling hub  410  and steering control system  168 . 
     Specifically, in a region  401 - 1 , a drilling hub  410 - 1  may serve as a remote processing resource for drilling rigs  210  located in region  401 - 1 , which may vary in number and are not limited to the exemplary schematic illustration of  FIG. 4 . Additionally, drilling hub  410 - 1  may have access to a regional drilling DB  412 - 1 , which may be local to drilling hub  410 - 1 . Additionally, in a region  401 - 2 , a drilling hub  410 - 2  may serve as a remote processing resource for drilling rigs  210  located in region  401 - 2 , which may vary in number and are not limited to the exemplary schematic illustration of  FIG. 4 . Additionally, drilling hub  410 - 2  may have access to a regional drilling DB  412 - 2 , which may be local to drilling hub  410 - 2 . 
     In  FIG. 4 , respective regions  402  may exhibit the same or similar geological formations. Thus, reference wells, or offset wells, may exist in a vicinity of a given drilling rig  210  in region  402 , or where a new well is planned in region  402 . Furthermore, multiple drilling rigs  210  may be actively drilling concurrently in region  402 , and may be in different stages of drilling through the depths of formation strata layers at region  402 . Thus, for any given well being drilled by drilling rig  210  in a region  402 , survey data from the reference wells or offset wells may be used to create the well plan, and may be used for surface steering, as disclosed herein. In some implementations, survey data or reference data from a plurality of reference wells may be used to improve drilling performance, such as by reducing an error in estimating TVD or a position of BHA  149  relative to one or more strata layers, as will be described in further detail herein. Additionally, survey data from recently drilled wells, or wells still currently being drilled, including the same well, may be used for reducing an error in estimating TVD or a position of BHA  149  relative to one or more strata layers. 
     Also shown in  FIG. 4  is central command  414 , which has access to central drilling DB  416 , and may be located at a centralized command center that is in communication with drilling hubs  410  and drilling rigs  210  in various regions  402 . The centralized command center may have the ability to monitor drilling and equipment activity at any one or more drilling rigs  210 . In some embodiments, central command  414  and drilling hubs  412  may be operated by a commercial operator of drilling rigs  210  as a service to customers who have hired the commercial operator to drill wells and provide other drilling-related services. 
     In  FIG. 4 , it is particularly noted that central drilling DB  416  may be a central repository that is accessible to drilling hubs  410  and drilling rigs  210 . Accordingly, central drilling DB  416  may store information for various drilling rigs  210  in different regions  402 . In some embodiments, central drilling DB  416  may serve as a backup for at least one regional drilling DB  412 , or may otherwise redundantly store information that is also stored on at least one regional drilling DB  412 . In turn, regional drilling DB  412  may serve as a backup or redundant storage for at least one drilling rig  210  in region  402 . For example, regional drilling DB  412  may store information collected by steering control system  168  from drilling rig  210 . 
     In some embodiments, the formulation of a drilling plan for drilling rig  210  may include processing and analyzing the collected data in regional drilling DB  412  to create a more effective drilling plan. Furthermore, once the drilling has begun, the collected data may be used in conjunction with current data from drilling rig  210  to improve drilling decisions. As noted, the functionality of steering control system  168  may be provided at drilling rig  210 , or may be provided, at least in part, at a remote processing resource, such as drilling hub  410  or central command  414 . 
     As noted, steering control system  168  may provide functionality as a surface steerable system for controlling drilling rig  210 . Steering control system  168  may have access to regional drilling DB  412  and central drilling DB  416  to provide the surface steerable system functionality. As will be described in greater detail below, steering control system  168  may be used to plan and control drilling operations based on input information, including feedback from the drilling process itself. Steering control system  168  may be used to perform operations such as receiving drilling data representing a drill trajectory and other drilling parameters, calculating a drilling solution for the drill trajectory based on the received data and other available data (e.g., rig characteristics), implementing the drilling solution at drilling rig  210 , monitoring the drilling process to gauge whether the drilling process is within a margin of error that is defined for the drill trajectory, or calculating corrections for the drilling process if the drilling process is outside of the margin of error. 
     Referring now to  FIG. 5 , an example of rig control systems  500  is illustrated in schematic form. It is noted that rig control systems  500  may include fewer or more elements than shown in  FIG. 5  in different embodiments. As shown, rig control systems  500  includes steering control system  168  and drilling rig  210 . Specifically, steering control system  168  is shown with logical functionality including an autodriller  510 , a bit guidance  512 , and an autoslide  514 . Drilling rig  210  is hierarchically shown including rig controls  520 , which provide secure control logic and processing capability, along with drilling equipment  530 , which represents the physical equipment used for drilling at drilling rig  210 . As shown, rig controls  520  include WOB/differential pressure control system  522 , positional/rotary control system  524 , fluid circulation control system  526 , and sensor system  528 , while drilling equipment  530  includes a draw works/snub  532 , top drive  140 , a mud pumping  536 , and an MWD/wireline  538 . 
     Steering control system  168  represent an instance of a processor having an accessible memory storing instructions executable by the processor, such as an instance of controller  1000  shown in  FIG. 10 . Also, WOB/differential pressure control system  522 , positional/rotary control system  524 , and fluid circulation control system  526  may each represent an instance of a processor having an accessible memory storing instructions executable by the processor, such as an instance of controller  1000  shown in  FIG. 10 , but for example, in a configuration as a programmable logic controller (PLC) that may not include a user interface but may be used as an embedded controller. Accordingly, it is noted that each of the systems included in rig controls  520  may be a separate controller, such as a PLC, and may autonomously operate, at least to a degree. Steering control system  168  may represent hardware that executes instructions to implement a surface steerable system that provides feedback and automation capability to an operator, such as a driller. For example, steering control system  168  may cause autodriller  510 , bit guidance  512  (also referred to as a bit guidance system (BGS)), and autoslide  514  (among others, not shown) to be activated and executed at an appropriate time during drilling. In particular implementations, steering control system  168  may be enabled to provide a user interface during drilling, such as the user interface  850  depicted and described below with respect to  FIG. 8 . Accordingly, steering control system  168  may interface with rig controls  520  to facilitate manual, assisted manual, semi-automatic, and automatic operation of drilling equipment  530  included in drilling rig  210 . It is noted that rig controls  520  may also accordingly be enabled for manual or user-controlled operation of drilling, and may include certain levels of automation with respect to drilling equipment  530 . 
     In rig control systems  500  of  FIG. 5 , WOB/differential pressure control system  522  may be interfaced with draw works/snubbing unit  532  to control WOB of drill string  146 . Positional/rotary control system  524  may be interfaced with top drive  140  to control rotation of drill string  146 . Fluid circulation control system  526  may be interfaced with mud pumping  536  to control mud flow and may also receive and decode mud telemetry signals. Sensor system  528  may be interfaced with MWD/wireline  538 , which may represent various BHA sensors and instrumentation equipment, among other sensors that may be downhole or at the surface. 
     In rig control systems  500 , autodriller  510  may represent an automated rotary drilling system and may be used for controlling rotary drilling. Accordingly, autodriller  510  may enable automate operation of rig controls  520  during rotary drilling, as indicated in the well plan. Bit guidance  512  may represent an automated control system to monitor and control performance and operation drilling bit  148 . 
     In rig control systems  500 , autoslide  514  may represent an automated slide drilling system and may be used for controlling slide drilling. Accordingly, autoslide  514  may enable automate operation of rig controls  520  during a slide, and may return control to steering control system  168  for rotary drilling at an appropriate time, as indicated in the well plan. In particular implementations, autoslide  514  may be enabled to provide a user interface during slide drilling to specifically monitor and control the slide. For example, autoslide  514  may rely on bit guidance  512  for orienting a tool face and on autodriller  510  to set WOB or control rotation or vibration of drill string  146 . 
       FIG. 6  illustrates one embodiment of control algorithm modules  600  used with steering control system  168 . The control algorithm modules  600  of  FIG. 6  include: a slide control executor  650  that is responsible for managing the execution of the slide control algorithms; a slide control configuration provider  652  that is responsible for validating, maintaining, and providing configuration parameters for the other software modules; a BHA &amp; pipe specification provider  654  that is responsible for managing and providing details of BHA  149  and drill string  146  characteristics; a borehole geometry model  656  that is responsible for keeping track of the borehole geometry and providing a representation to other software modules; a top drive orientation impact model  658  that is responsible for modeling the impact that changes to the angular orientation of top drive  140  have had on the tool face control; a top drive oscillator impact model  660  that is responsible for modeling the impact that oscillations of top drive  140  has had on the tool face control; an ROP impact model  662  that is responsible for modeling the effect on the tool face control of a change in ROP or a corresponding ROP set point; a WOB impact model  664  that is responsible for modeling the effect on the tool face control of a change in WOB or a corresponding WOB set point; a differential pressure impact model  666  that is responsible for modeling the effect on the tool face control of a change in differential pressure (DP) or a corresponding DP set point; a torque model  668  that is responsible for modeling the comprehensive representation of torque for surface, downhole, break over, and reactive torque, modeling impact of those torque values on tool face control, and determining torque operational thresholds; a tool face control evaluator  672  that is responsible for evaluating all factors impacting tool face control and whether adjustments need to be projected, determining whether re-alignment off-bottom is indicated, and determining off-bottom tool face operational threshold windows; a tool face projection  670  that is responsible for projecting tool face behavior for top drive  140 , the top drive oscillator, and auto driller adjustments; a top drive adjustment calculator  674  that is responsible for calculating top drive adjustments resultant to tool face projections; an oscillator adjustment calculator  676  that is responsible for calculating oscillator adjustments resultant to tool face projections; and an autodriller adjustment calculator  678  that is responsible for calculating adjustments to autodriller  510  resultant to tool face projections. 
       FIG. 7  illustrates one embodiment of a steering control process  700  for determining an optimal corrective action for drilling. Steering control process  700  may be used for rotary drilling or slide drilling in different embodiments. 
     Steering control process  700  in  FIG. 7  illustrates a variety of inputs that can be used to determine an optimum corrective action. As shown in  FIG. 7 , the inputs include formation hardness/unconfined compressive strength (UCS)  710 , formation structure  712 , inclination/azimuth  714 , current zone  716 , measured depth  718 , desired tool face  730 , vertical section  720 , bit factor  722 , mud motor torque  724 , reference trajectory  730 , vertical section  720 , bit factor  722 , torque  724  and angular velocity  726 . In  FIG. 7 , reference trajectory  730  of borehole  106  is determined to calculate a trajectory misfit in a step  732 . Step  732  may output the trajectory misfit to determine an optimal corrective action to minimize the misfit at step  734 , which may be performed using the other inputs described above. Then, at step  736 , the drilling rig is caused to perform the optimal corrective action. 
     It is noted that in some implementations, at least certain portions of steering control process  700  may be automated or performed without user intervention, such as using rig control systems  700  (see  FIG. 7 ). In other implementations, the optimal corrective action in step  736  may be provided or communicated (by display, SMS message, email, or otherwise) to one or more human operators, who may then take appropriate action. The human operators may be members of a rig crew, which may be located at or near drilling rig  210 , or may be located remotely from drilling rig  210 . 
     Referring to  FIG. 8 , one embodiment of a user interface  850  that may be generated by steering control system  168  for monitoring and operation by a human operator is illustrated. User interface  850  may provide many different types of information in an easily accessible format. For example, user interface  850  may be shown on a computer monitor, a television, a viewing screen (e.g., a display device) associated with steering control system  168 . 
     As shown in  FIG. 8 , user interface  850  provides visual indicators such as a hole depth indicator  852 , a bit depth indicator  854 , a GAMMA indicator  856 , an inclination indicator  858 , an azimuth indicator  860 , and a TVD indicator  862 . Other indicators may also be provided, including a ROP indicator  864 , a mechanical specific energy (MSE) indicator  866 , a differential pressure indicator  868 , a standpipe pressure indicator  870 , a flow rate indicator  872 , a rotary RPM (angular velocity) indicator  874 , a bit speed indicator  876 , and a WOB indicator  878 . 
     In  FIG. 8 , at least some of indicators  864 ,  866 ,  868 ,  870 ,  872 ,  874 ,  876 , and  878  may include a marker representing a target value. For example, markers may be set as certain given values, but it is noted that any desired target value may be used. Although not shown, in some embodiments, multiple markers may be present on a single indicator. The markers may vary in color or size. For example, ROP indicator  864  may include a marker  865  indicating that the target value is 50 feet/hour (or 15 m/h). MSE indicator  866  may include a marker  867  indicating that the target value is 37 ksi (or 255 MPa). Differential pressure indicator  868  may include a marker  869  indicating that the target value is 200 psi (or 1.38 kPa). ROP indicator  864  may include a marker  865  indicating that the target value is 50 feet/hour (or 15 m/h). Standpipe pressure indicator  870  may have no marker in the present example. Flow rate indicator  872  may include a marker  873  indicating that the target value is 500 gpm (or 31.5 L/s). Rotary RPM indicator  874  may include a marker  875  indicating that the target value is 0 RPM (e.g., due to sliding). Bit speed indicator  876  may include a marker  877  indicating that the target value is 150 RPM. WOB indicator  878  may include a marker  879  indicating that the target value is 10 klbs (or 4,500 kg). Each indicator may also include a colored band, or another marking, to indicate, for example, whether the respective gauge value is within a safe range (e.g., indicated by a green color), within a caution range (e.g., indicated by a yellow color), or within a danger range (e.g., indicated by a red color). 
     In  FIG. 8 , a log chart  880  may visually indicate depth versus one or more measurements (e.g., may represent log inputs relative to a progressing depth chart). For example, log chart  880  may have a Y-axis representing depth and an X-axis representing a measurement such as GAMMA count  881  (as shown), ROP  883  (e.g., empirical ROP and normalized ROP), or resistivity. An autopilot button  882  and an oscillate button  884  may be used to control activity. For example, autopilot button  882  may be used to engage or disengage autodriller  510 , while oscillate button  884  may be used to directly control oscillation of drill string  146  or to engage/disengage an external hardware device or controller. 
     In  FIG. 8 , a circular chart  886  may provide current and historical tool face orientation information (e.g., which way the bend is pointed). For purposes of illustration, circular chart  886  represents three hundred and sixty degrees. A series of circles within circular chart  886  may represent a timeline of tool face orientations, with the sizes of the circles indicating the temporal position of each circle. For example, larger circles may be more recent than smaller circles, so a largest circle  888  may be the newest reading and a smallest circle  889  may be the oldest reading. In other embodiments, circles  889 ,  888  may represent the energy or progress made via size, color, shape, a number within a circle, etc. For example, a size of a particular circle may represent an accumulation of orientation and progress for the period of time represented by the circle. In other embodiments, concentric circles representing time (e.g., with the outside of circular chart  886  being the most recent time and the center point being the oldest time) may be used to indicate the energy or progress (e.g., via color or patterning such as dashes or dots rather than a solid line). 
     In user interface  850 , circular chart  886  may also be color coded, with the color coding existing in a band  890  around circular chart  886  or positioned or represented in other ways. The color coding may use colors to indicate activity in a certain direction. For example, the color red may indicate the highest level of activity, while the color blue may indicate the lowest level of activity. Furthermore, the arc range in degrees of a color may indicate the amount of deviation. Accordingly, a relatively narrow (e.g., thirty degrees) arc of red with a relatively broad (e.g., three hundred degrees) arc of blue may indicate that most activity is occurring in a particular tool face orientation with little deviation. As shown in user interface  850 , the color blue may extend from approximately 22-337 degrees, the color green may extend from approximately 15-22 degrees and 337-345 degrees, the color yellow may extend a few degrees around the 13 and 345 degree marks, while the color red may extend from approximately 347-10 degrees. Transition colors or shades may be used with, for example, the color orange marking the transition between red and yellow or a light blue marking the transition between blue and green. This color coding may enable user interface  850  to provide an intuitive summary of how narrow the standard deviation is and how much of the energy intensity is being expended in the proper direction. Furthermore, the center of energy may be viewed relative to the target. For example, user interface  850  may clearly show that the target is at 90 degrees but the center of energy is at 45 degrees. 
     In user interface  850 , other indicators, such as a slide indicator  892 , may indicate how much time remains until a slide occurs or how much time remains for a current slide. For example, slide indicator  892  may represent a time, a percentage (e.g., as shown, a current slide may be 56% complete), a distance completed, or a distance remaining. Slide indicator  892  may graphically display information using, for example, a colored bar  893  that increases or decreases with slide progress. In some embodiments, slide indicator  892  may be built into circular chart  886  (e.g., around the outer edge with an increasing/decreasing band), while in other embodiments slide indicator  892  may be a separate indicator such as a meter, a bar, a gauge, or another indicator type. In various implementations, slide indicator  892  may be refreshed by autoslide  514 . 
     In user interface  850 , an error indicator  894  may indicate a magnitude and a direction of error. For example, error indicator  894  may indicate that an estimated drill bit position is a certain distance from the planned trajectory, with a location of error indicator  894  around the circular chart  886  representing the heading. For example,  FIG. 8  illustrates an error magnitude of 15 feet and an error direction of 15 degrees. Error indicator  894  may be any color but may be red for purposes of example. It is noted that error indicator  894  may present a zero if there is no error. Error indicator may represent that drill bit  148  is on the planned trajectory using other means, such as being a green color. Transition colors, such as yellow, may be used to indicate varying amounts of error. In some embodiments, error indicator  894  may not appear unless there is an error in magnitude or direction. A marker  896  may indicate an ideal slide direction. Although not shown, other indicators may be present, such as a bit life indicator to indicate an estimated lifetime for the current bit based on a value such as time or distance. 
     It is noted that user interface  850  may be arranged in many different ways. For example, colors may be used to indicate normal operation, warnings, and problems. In such cases, the numerical indicators may display numbers in one color (e.g., green) for normal operation, may use another color (e.g., yellow) for warnings, and may use yet another color (e.g., red) when a serious problem occurs. The indicators may also flash or otherwise indicate an alert. The gauge indicators may include colors (e.g., green, yellow, and red) to indicate operational conditions and may also indicate the target value (e.g., an ROP of 100 feet/hour). For example, ROP indicator  868  may have a green bar to indicate a normal level of operation (e.g., from 10-300 feet/hour), a yellow bar to indicate a warning level of operation (e.g., from 300-360 feet/hour), and a red bar to indicate a dangerous or otherwise out of parameter level of operation (e.g., from 360-390 feet/hour). ROP indicator  868  may also display a marker at 100 feet/hour to indicate the desired target ROP. 
     Furthermore, the use of numeric indicators, gauges, and similar visual display indicators may be varied based on factors such as the information to be conveyed and the personal preference of the viewer. Accordingly, user interface  850  may provide a customizable view of various drilling processes and information for a particular individual involved in the drilling process. For example, steering control system  168  may enable a user to customize the user interface  850  as desired, although certain features (e.g., standpipe pressure) may be locked to prevent a user from intentionally or accidentally removing important drilling information from user interface  850 . Other features and attributes of user interface  850  may be set by user preference. Accordingly, the level of customization and the information shown by the user interface  850  may be controlled based on who is viewing user interface  850  and their role in the drilling process. 
     Referring to  FIG. 9 , one embodiment of a guidance control loop (GCL)  900  is shown in further detail GCL  900  may represent one example of a control loop or control algorithm executed under the control of steering control system  168 . GCL  900  may include various functional modules, including a build rate predictor  902 , a geo modified well planner  904 , a borehole estimator  906 , a slide estimator  908 , an error vector calculator  910 , a geological drift estimator  912 , a slide planner  914 , a convergence planner  916 , and a tactical solution planner  918 . In the following description of GCL  900 , the term “external input” refers to input received from outside GCL  900 , while “internal input” refers to input exchanged between functional modules of GCL  900 . 
     In  FIG. 9 , build rate predictor  902  receives external input representing BHA information and geological information, receives internal input from the borehole estimator  906 , and provides output to geo modified well planner  904 , slide estimator  908 , slide planner  914 , and convergence planner  916 . Build rate predictor  902  is configured to use the BHA information and geological information to predict drilling build rates of current and future sections of borehole  106 . For example, build rate predictor  902  may determine how aggressively a curve will be built for a given formation with BHA  149  and other equipment parameters. 
     In  FIG. 9 , build rate predictor  902  may use the orientation of BHA  149  to the formation to determine an angle of attack for formation transitions and build rates within a single layer of a formation. For example, if a strata layer of rock is below a strata layer of sand, a formation transition exists between the strata layer of sand and the strata layer of rock. Approaching the strata layer of rock at a 90 degree angle may provide a good tool face and a clean drill entry, while approaching the rock layer at a 45 degree angle may build a curve relatively quickly. An angle of approach that is near parallel may cause drill bit  148  to skip off the upper surface of the strata layer of rock. Accordingly, build rate predictor  902  may calculate BHA orientation to account for formation transitions. Within a single strata layer, build rate predictor  902  may use the BHA orientation to account for internal layer characteristics (e.g., grain) to determine build rates for different parts of a strata layer. The BHA information may include bit characteristics, mud motor bend setting, stabilization and mud motor bit to bend distance. The geological information may include formation data such as compressive strength, thicknesses, and depths for formations encountered in the specific drilling location. Such information may enable a calculation-based prediction of the build rates and ROP that may be compared to both results obtained while drilling borehole  106  and regional historical results (e.g., from the regional drilling DB  412 ) to improve the accuracy of predictions as drilling progresses. Build rate predictor  902  may also be used to plan convergence adjustments and confirm in advance of drilling that targets can be achieved with current parameters. 
     In  FIG. 9 , geo modified well planner  904  receives external input representing a well plan, internal input from build rate predictor  902  and geo drift estimator  912 , and provides output to slide planner  914  and error vector calculator  910 . Geo modified well planner  904  uses the input to determine whether there is a more optimal trajectory than that provided by the well plan, while staying within specified error limits. More specifically, geo modified well planner  904  takes geological information (e.g., drift) and calculates whether another trajectory solution to the target may be more efficient in terms of cost or reliability. The outputs of geo modified well planner  904  to slide planner  914  and error vector calculator  910  may be used to calculate an error vector based on the current vector to the newly calculated trajectory and to modify slide predictions. In some embodiments, geo modified well planner  904  (or another module) may provide functionality needed to track a formation trend. For example, in horizontal wells, a geologist may provide steering control system  168  with a target inclination as a set point for steering control system  168  to control. For example, the geologist may enter a target to steering control system  168  of 90.5-91.0 degrees of inclination for a section of borehole  106 . Geo modified well planner  904  may then treat the target as a vector target, while remaining within the error limits of the original well plan. In some embodiments, geo modified well planner  904  may be an optional module that is not used unless the well plan is to be modified. For example, if the well plan is marked in steering control system  168  as non-modifiable, geo modified well planner  904  may be bypassed altogether or geo modified well planner  904  may be configured to pass the well plan through without any changes. 
     In  FIG. 9 , borehole estimator  906  may receive external inputs representing BHA information, measured depth information, survey information (e.g., azimuth and inclination), and may provide outputs to build rate predictor  902 , error vector calculator  910 , and convergence planner  916 . Borehole estimator  906  may be configured to provide an estimate of the actual borehole and drill bit position and trajectory angle without delay, based on either straight line projections or projections that incorporate sliding. Borehole estimator  906  may be used to compensate for a sensor being physically located some distance behind drill bit  148  (e.g., 50 feet) in drill string  146 , which makes sensor readings lag the actual bit location by 50 feet. Borehole estimator  906  may also be used to compensate for sensor measurements that may not be continuous (e.g., a sensor measurement may occur every 100 feet). Borehole estimator  906  may provide the most accurate estimate from the surface to the last survey location based on the collection of survey measurements. Also, borehole estimator  906  may take the slide estimate from slide estimator  908  (described below) and extend the slide estimate from the last survey point to a current location of drill bit  148 . Using the combination of these two estimates, borehole estimator  906  may provide steering control system  168  with an estimate of the drill bit&#39;s location and trajectory angle from which guidance and steering solutions can be derived. An additional metric that can be derived from the borehole estimate is the effective build rate that is achieved throughout the drilling process. 
     In  FIG. 9 , slide estimator  908  receives external inputs representing measured depth and differential pressure information, receives internal input from build rate predictor  902 , and provides output to borehole estimator  906  and geo modified well planner  904 . Slide estimator  908  may be configured to sample tool face orientation, differential pressure, measured depth (MD) incremental movement, MSE, and other sensor feedback to quantify/estimate a deviation vector and progress while sliding. 
     Traditionally, deviation from the slide would be predicted by a human operator based on experience. The operator would, for example, use a long slide cycle to assess what likely was accomplished during the last slide. However, the results are generally not confirmed until the downhole survey sensor point passes the slide portion of the borehole, often resulting in a response lag defined by a distance of the sensor point from the drill bit tip (e.g., approximately 50 feet). Such a response lag may introduce inefficiencies in the slide cycles due to over/under correction of the actual trajectory relative to the planned trajectory. 
     In GCL  900 , using slide estimator  908 , each tool face update may be algorithmically merged with the average differential pressure of the period between the previous and current tool face readings, as well as the MD change during this period to predict the direction, angular deviation, and MD progress during the period. As an example, the periodic rate may be between 10 and 60 seconds per cycle depending on the tool face update rate of downhole tool  166 . With a more accurate estimation of the slide effectiveness, the sliding efficiency can be improved. The output of slide estimator  908  may accordingly be periodically provided to borehole estimator  906  for accumulation of well deviation information, as well to geo modified well planner  904 . Some or all of the output of the slide estimator  908  may be output to an operator, such as shown in the user interface  850  of  FIG. 8 . 
     In  FIG. 9 , error vector calculator  910  may receive internal input from geo modified well planner  904  and borehole estimator  906 . Error vector calculator  910  may be configured to compare the planned well trajectory to an actual borehole trajectory and drill bit position estimate. Error vector calculator  910  may provide the metrics used to determine the error (e.g., how far off) the current drill bit position and trajectory are from the well plan. For example, error vector calculator  910  may calculate the error between the current bit position and trajectory to the planned trajectory and the desired bit position. Error vector calculator  910  may also calculate a projected bit position/projected trajectory representing the future result of a current error. 
     In  FIG. 9 , geological drift estimator  912  receives external input representing geological information and provides outputs to geo modified well planner  904 , slide planner  914 , and tactical solution planner  918 . During drilling, drift may occur as the particular characteristics of the formation affect the drilling direction. More specifically, there may be a trajectory bias that is contributed by the formation as a function of ROP and BHA  149 . Geological drift estimator  912  is configured to provide a drift estimate as a vector that can then be used to calculate drift compensation parameters that can be used to offset the drift in a control solution. 
     In  FIG. 9 , slide planner  914  receives internal input from build rate predictor  902 , geo modified well planner  904 , error vector calculator  910 , and geological drift estimator  912 , and provides output to convergence planner  916  as well as an estimated time to the next slide. Slide planner  914  may be configured to evaluate a slide/drill ahead cost equation and plan for sliding activity, which may include factoring in BHA wear, expected build rates of current and expected formations, and the well plan trajectory. During drill ahead, slide planner  914  may attempt to forecast an estimated time of the next slide to aid with planning. For example, if additional lubricants (e.g., fluorinated beads) are indicated for the next slide, and pumping the lubricants into drill string  146  has a lead time of 30 minutes before the slide, the estimated time of the next slide may be calculated and then used to schedule when to start pumping the lubricants. Functionality for a loss circulation material (LCM) planner may be provided as part of slide planner  914  or elsewhere (e.g., as a stand-alone module or as part of another module described herein). The LCM planner functionality may be configured to determine whether additives should be pumped into the borehole based on indications such as flow-in versus flow-back measurements. For example, if drilling through a porous rock formation, fluid being pumped into the borehole may get lost in the rock formation. To address this issue, the LCM planner may control pumping LCM into the borehole to clog up the holes in the porous rock surrounding the borehole to establish a more closed-loop control system for the fluid. 
     In  FIG. 9 , slide planner  914  may also look at the current position relative to the next connection. A connection may happen every 90 to 100 feet (or some other distance or distance range based on the particulars of the drilling operation) and slide planner  914  may avoid planning a slide when close to a connection or when the slide would carry through the connection. For example, if the slide planner  914  is planning a 50 foot slide but only 20 feet remain until the next connection, slide planner  914  may calculate the slide starting after the next connection and make any changes to the slide parameters to accommodate waiting to slide until after the next connection. Such flexible implementation avoids inefficiencies that may be caused by starting the slide, stopping for the connection, and then having to reorient the tool face before finishing the slide. During slides, slide planner  914  may provide some feedback as to the progress of achieving the desired goal of the current slide. In some embodiments, slide planner  914  may account for reactive torque in the drill string. More specifically, when rotating is occurring, there is a reactional torque wind up in drill string  146 . When the rotating is stopped, drill string  146  unwinds, which changes tool face orientation and other parameters. When rotating is started again, drill string  146  starts to wind back up. Slide planner  914  may account for the reactional torque so that tool face references are maintained, rather than stopping rotation and then trying to adjust to an optimal tool face orientation. While not all downhole tools may provide tool face orientation when rotating, using one that does supply such information for GCL  900  may significantly reduce the transition time from rotating to sliding. 
     In  FIG. 9 , convergence planner  916  receives internal inputs from build rate predictor  902 , borehole estimator  906 , and slide planner  914 , and provides output to tactical solution planner  918 . Convergence planner  916  is configured to provide a convergence plan when the current drill bit position is not within a defined margin of error of the planned well trajectory. The convergence plan represents a path from the current drill bit position to an achievable and optimal convergence target point along the planned trajectory. The convergence plan may take account the amount of sliding/drilling ahead that has been planned to take place by slide planner  914 . Convergence planner  916  may also use BHA orientation information for angle of attack calculations when determining convergence plans as described above with respect to build rate predictor  902 . The solution provided by convergence planner  916  defines a new trajectory solution for the current position of drill bit  148 . The solution may be immediate without delay, or planned for implementation at a future time that is specified in advance. 
     In  FIG. 9 , tactical solution planner  918  receives internal inputs from geological drift estimator  912  and convergence planner  916 , and provides external outputs representing information such as tool face orientation, differential pressure, and mud flow rate. Tactical solution planner  918  is configured to take the trajectory solution provided by convergence planner  916  and translate the solution into control parameters that can be used to control drilling rig  210 . For example, tactical solution planner  918  may convert the solution into settings for control systems  522 ,  524 , and  526  to accomplish the actual drilling based on the solution. Tactical solution planner  918  may also perform performance optimization to optimizing the overall drilling operation as well as optimizing the drilling itself (e.g., how to drill faster). 
     Other functionality may be provided by GCL  900  in additional modules or added to an existing module. For example, there is a relationship between the rotational position of the drill pipe on the surface and the orientation of the downhole tool face. Accordingly, GCL  900  may receive information corresponding to the rotational position of the drill pipe on the surface. GCL  900  may use this surface positional information to calculate current and desired tool face orientations. These calculations may then be used to define control parameters for adjusting the top drive  140  to accomplish adjustments to the downhole tool face in order to steer the trajectory of borehole  106 . 
     For purposes of example, an object-oriented software approach may be utilized to provide a class-based structure that may be used with GCL  900  or other functionality provided by steering control system  168 . In GCL  900 , a drilling model class may be defined to capture and define the drilling state throughout the drilling process. The drilling model class may include information obtained without delay. The drilling model class may be based on the following components and sub-models: a drill bit model, a borehole model, a rig surface gear model, a mud pump model, a WOB/differential pressure model, a positional/rotary model, an MSE model, an active well plan, and control limits. The drilling model class may produce a control output solution and may be executed via a main processing loop that rotates through the various modules of GCL  900 . The drill bit model may represent the current position and state of drill bit  148 . The drill bit model may include a three dimensional (3D) position, a drill bit trajectory, BHA information, bit speed, and tool face (e.g., orientation information). The 3D position may be specified in north-south (NS), east-west (EW), and true vertical depth (TVD). The drill bit trajectory may be specified as an inclination angle and an azimuth angle. The BHA information may be a set of dimensions defining the active BHA. The borehole model may represent the current path and size of the active borehole. The borehole model may include hole depth information, an array of survey points collected along the borehole path, a gamma log, and borehole diameters. The hole depth information is for current drilling of borehole  106 . The borehole diameters may represent the diameters of borehole  106  as drilled over current drilling. The rig surface gear model may represent pipe length, block height, and other models, such as the mud pump model, WOB/differential pressure model, positional/rotary model, and MSE model. The mud pump model represents mud pump equipment and includes flow rate, standpipe pressure, and differential pressure. The WOB/differential pressure model represents draw works or other WOB/differential pressure controls and parameters, including WOB. The positional/rotary model represents top drive or other positional/rotary controls and parameters including rotary RPM and spindle position. The active well plan represents the target borehole path and may include an external well plan and a modified well plan. The control limits represent defined parameters that may be set as maximums and/or minimums. For example, control limits may be set for the rotary RPM in the top drive model to limit the maximum RPMs to the defined level. The control output solution may represent the control parameters for drilling rig  210 . 
     Each functional module of GCL  900  may have behavior encapsulated within a respective class definition. During a processing window, the individual functional modules may have an exclusive portion in time to execute and update the drilling model. For purposes of example, the processing order for the functional modules may be in the sequence of geo modified well planner  904 , build rate predictor  902 , slide estimator  908 , borehole estimator  906 , error vector calculator  910 , slide planner  914 , convergence planner  916 , geological drift estimator  912 , and tactical solution planner  918 . It is noted that other sequences may be used in different implementations. 
     In  FIG. 9 , GCL  900  may rely on a programmable timer module that provides a timing mechanism to provide timer event signals to drive the main processing loop. While steering control system  168  may rely on timer and date calls driven by the programming environment, timing may be obtained from other sources than system time. In situations where it may be advantageous to manipulate the clock (e.g., for evaluation and testing), a programmable timer module may be used to alter the system time. For example, the programmable timer module may enable a default time set to the system time and a time scale of 1.0, may enable the system time of steering control system  168  to be manually set, may enable the time scale relative to the system time to be modified, or may enable periodic event time requests scaled to a requested time scale. 
     Referring now to  FIG. 10 , a block diagram illustrating selected elements of an embodiment of a controller  1000  for performing surface steering according to the present disclosure. In various embodiments, controller  1000  may represent an implementation of steering control system  168 . In other embodiments, at least certain portions of controller  1000  may be used for control systems  510 ,  512 ,  514 ,  522 ,  524 , and  526  (see  FIG. 5 ). 
     In the embodiment depicted in  FIG. 10 , controller  1000  includes processor  1001  coupled via shared bus  1002  to storage media collectively identified as memory media  1010 . 
     Controller  1000 , as depicted in  FIG. 10 , further includes network adapter  1020  that interfaces controller  1000  to a network (not shown in  FIG. 10 ). In embodiments suitable for use with user interfaces, controller  1000 , as depicted in  FIG. 10 , may include peripheral adapter  1006 , which provides connectivity for the use of input device  1008  and output device  1009 . Input device  1008  may represent a device for user input, such as a keyboard or a mouse, or even a video camera. Output device  1009  may represent a device for providing signals or indications to a user, such as loudspeakers for generating audio signals. 
     Controller  1000  is shown in  FIG. 10  including display adapter  1004  and further includes a display device  1005 . Display adapter  1004  may interface shared bus  1002 , or another bus, with an output port for one or more display devices, such as display device  1005 . Display device  1005  may be implemented as a liquid crystal display screen, a computer monitor, a television or the like. Display device  1005  may comply with a display standard for the corresponding type of display. Standards for computer monitors include analog standards such as video graphics array (VGA), extended graphics array (XGA), etc., or digital standards such as digital visual interface (DVI), definition multimedia interface (HDMI), among others. A television display may comply with standards such as NTSC (National Television System Committee), PAL (Phase Alternating Line), or another suitable standard. Display device  1005  may include an output device  1009 , such as one or more integrated speakers to play audio content, or may include an input device  1008 , such as a microphone or video camera. 
     In  FIG. 10 , memory media  1010  encompasses persistent and volatile media, fixed and removable media, and magnetic and semiconductor media. Memory media  1010  is operable to store instructions, data, or both. Memory media  1010  as shown includes sets or sequences of instructions  1024 - 2 , namely, an operating system  1012  and surface steering control  1014 . Operating system  1012  may be a UNIX or UNIX-like operating system, a Windows® family operating system, or another suitable operating system. Instructions  1024  may also reside, completely or at least partially, within processor  1001  during execution thereof. It is further noted that processor  1001  may be configured to receive instructions  1024 - 1  from instructions  1024 - 2  via shared bus  1002 . In some embodiments, memory media  1010  is configured to store and provide executable instructions for executing GCL  900 , as mentioned previously, among other methods and operations disclosed herein. 
     In other embodiments, stratigraphic information may also be used to steer a wellbore into one or multiple geological target formations when one or multiple wells have already been drilled in the vicinity. In particular, stratigraphic misfit heat maps can be computed to determine and/or display the misfit between the characteristic functions of the subject and offset wells. 
     The methods disclosed herein involving the use of stratigraphic misfit heat maps can be executed in fully automated, semi-automated and/or manual modes. Systems for performing such methods and/or steering a well may be separate computer systems or may be combined with some or all of the computer systems described above. For example, the methods described herein involving the generation and use of stratigraphic misfit heat maps may be implemented automatically, and may form a portion of steering control system  168 , rig control system  520 , or may be a part of another computer system, or may be a separate and distinct computer system. It is to be noted that in some implementations, at least certain portions of the stratigraphic misfit heat map methods may be performed without user intervention, and in some implementations, an optimal corrective action, interpretation, or analysis may be used to update or modify a well plan automatically, and/or may be provided or communicated (such as by display, SMS message, email, or otherwise) to one or more human operators, who may then take appropriate action. The human operators to whom the information is provided may be members of a rig crew, which may be located at or near drilling rig  210 , and/or may be located remotely from drilling rig  210 . 
     The subject wellbore can be steered into one or multiple geological stratigraphic targets. The directional drilling process typically follows a spatial well plan, in which the position of the desired wellbore trajectory is given in spatial coordinates. However, the desired geological target may not be at exactly the depth assumed when creating the well plan, due to unknown lateral variations and other uncertainties in geological stratigraphy. Therefore, it is common practice to update the well plan based on new stratigraphic information from the wellbore, as it is being drilled. This stratigraphic information is gained on one hand from Measurement While Drilling (MWD) and Logging While Drilling (LWD) sensor data, but could also include drilling dynamics data giving information, for example, on the hardness of the rock, and/or on properties like WOB, ROP, MSE, and the like. To identify the stratigraphic depth of the subject wellbore, the measurements from the subject well can be compared with corresponding measurements from existing well logs and local geologic information in the vicinity. 
     In the simplest implementation of the disclosed method, the measurements on the subject wellbore at increasing measured depth (MD) along the wellbore are compared with a single offset well log, organized by Stratigraphic Vertical Depth (SVD). Taking the statistical properties of the measurements into account, a misfit can be computed for every pair of subject well MD and offset log SVD, representing each possible SVD that the gamma measurement could correspond to for each MD along the wellbore. This misfit can be displayed as a stratigraphic misfit heat map, for example with MD on the X axis, SVD on the Y axis and the misfit color coded at the (MD, SVD) position on the display. Geologically plausible interpretations are then visible as paths of minimal misfit through the heat map. Any such path SVD(MD) specifies the stratigraphic depth along the wellbore trajectory, which can be used to steer the wellbore toward a desired stratigraphic target. 
     Better use of the available information can be made by an extended implementation, in which the measurements of the subject wellbore are simultaneously compared with multiple offset well logs. A pre-drill geomodel can be constructed to provide a mapping between the stratigraphic depths of all the offset well logs. The wellbore position at MD can then be compared simultaneously with multiple offset well logs at corresponding stratigraphic depths to compute a combined misfit. 
     The heat map may also be used to identify stratigraphic anomalies, such as structural faults, stringers and breccia. An automated algorithm may be used to identify SVD(MD) paths of minimal misfit through the heat map. These paths can further be guided through stratigraphic control points set by an operator. The identified paths may directly be used in applying corrections to the well plan to steer the subject wellbore into a geological target or keep it in a stratigraphic target zone. 
     To make optimal use of the geological information of multiple offset wells in the vicinity of the subject well, it is useful to first create a mapping which relates the stratigraphic depths between the offset wells. 
     In a simple form, this mapping can specify the depths of corresponding geologic marker horizons on each offset well and then interpolate these marker horizons into a 2D surface between the offset wells. Any stratigraphic depth on any offset wellbore between two marker horizons can then be related to the corresponding depth on another offset well by linear interpolation. For example, if the depth of interest on one offset well is at one third of the depth interval between marker horizons A and B, we can assign the corresponding stratigraphic depth on the other offset well as one third of the depth interval between the known marker horizons A and B. 
     A more accurate mapping extends the concept of marker horizons by identifying one of the offset wells as the stratigraphic master well, which defines the stratigraphic depth axis, SVD. For each offset well we can then define a function relating the stratigraphic depth of the master well to the stratigraphic depth of the offset well. Interpolation of stratigraphic depth in the area between the offset wells then takes the form of a 2D interpolation between functions, as a generalization of the 2D interpolation between markers. 
     The result of either the simple marker horizon approach or the more accurate continuous approach is a 3D geomodel, that relates the stratigraphic depth SVD(x0, y0) at one location (x0, y0) to the corresponding stratigraphic depth SVD(x, y) at any other location (x, y) within the coverage area of the 3D geomodel. This mapping enables the measurements on the subject well to be compared simultaneously with the corresponding measurements on all offset wells. 
     If the geomodel were perfectly accurate and the spatial position of the subject wellbore position were perfectly known, then the geomodel would inform the driller of the exact stratigraphic depth of any position along the subject wellbore. However, due to inaccuracies in the geomodel and the surveyed location of the subject wellbore, the actual stratigraphic depth of the subject wellbore typically has to be verified and corrected during drilling. To geosteer the well, one can find the corrected stratigraphic depth of the subject well at which the subject measurements optimally agree with the measurements at the corresponding stratigraphic depths on the offset wells. 
     Finding the most likely stratigraphic depth of the subject wellbore can be done with a statistical model. We can characterize the measurement uncertainties and spatial variation of a specific parameter (such as gamma ray intensity) by an empirical covariance model. The model specifies the lateral covariance and the vertical covariance between sensor measurements for the same, as well as for different well logs. 
     The lateral covariance across well logs can empirically be estimated by using a large sample of vertical wells in a representative area and fitting a model specifying the covariance as a function of spatial separation. Correspondingly, the lateral covariance along a single well log can empirically be estimated from horizontal wells in a representative area, again fitting a model specifying the covariance as a function of spatial separation. The vertical covariances along a vertical wellbore and across horizontal wellbores can be empirically estimated in the same way. 
     The empirical covariance model enables assembling a covariance matrix for a specific combination of measurements on subject and offset wells. This matrix specifies the covariances for all relevant pairs of differences between subject and offset well measurements. 
     The covariance model can be used to specify a misfit m(MD) between a measurement on the subject well and the measurements at the corresponding stratigraphic depth on the offset wells, using a statistical norm, such as m(MD, SVD)=d T  cov −1  d, where d is the vector of differences between subject well measurement at MD and offset well measurements at the depths mapped to SVD, and cov −1  is the inverse of the covariance matrix given by the statistical model. 
     The stratigraphic heat map displays the misfit between the measurements of the subject wellbore and the offset wells. Different choices for the X and Y coordinates may be made when displaying the misfit as a heat map. One choice is to display MD on the X axis, while displaying SVD on the Y axis. 
       FIG. 11A  shows a conventional vertical section display  1102  of a wellbore, in which the X axis denotes the Vertical Section  1106  (distance from the wellhead in the direction of the virtual section azimuth) while the Y axis represents True Vertical Depth  1104 . The True Vertical Depth of the specific wellbore  1108  is measured and compared to the true vertical depth of the geology  1110  represented by the stratigraphic layers. Stratigraphic layers may be shown as dashed horizontal lines, as illustrated in  FIG. 11A . 
       FIG. 11B  illustrates a display  1112  that has been transformed (such as from the same data as displayed in  FIG. 11A ) to instead show the Measured Depth  1116  along the subject wellbore on the X axis and the Stratigraphic Vertical Depth (SVD)  1114  on the Y axis. The SVD of the specific wellbore  1118  is measured and compared to the SVD of geological formations  1110 .  FIG. 11B  shows the case where only a single offset well log is used for reference. This corresponds to assuming a 1D geomodel with horizontal horizons, as shown by the horizontal dashed lines. The advantage of showing MD on the X axis is that every point on the X axis is then uniquely associated with only one point along the subject wellbore. This helps display misfit heatmaps because every point in the heat map is then associated with only one measured depth. 
       FIG. 12  illustrates a simple form of a stratigraphic misfit heat map  1202 , for a single offset wellbore and assumed 1D geomodel with flat stratigraphic layers (horizontal dashed lines)  1212 . With Measured Depth  1116  plotted on the X axis and stratigraphic vertical depth  1114  on the Y axis, every (x,y) coordinate corresponds to a pair of depths, one on the subject well and one on the offset well. The misfit between those two points is the normalized misfit  1206 , which can then be color coded, for example with a dark color for low misfit and a bright color for high misfit, resulting in a stratigraphic misfit heatmap. In this heat map  1202 , the dark points  1210  and the light points  1208  represent the normalized misfit  1206  values. The measured gamma  1204  is displayed for the user below the heat map  1202 . It is to be noted that the heat map  1202  may include a variety of colors and/or brightness levels, such as bright red for a high misfit value and dark blue for a low misfit value, with colors of the spectrum (e.g., green, yellow, orange, etc.) representing intermediate misfit values. 
       FIG. 13  shows the MD/SVD display  1302  for a hypothetical 3D geomodel. The geomodel is defined by marker horizons  1306  that connect the markers of multiple offset well logs, which may be gamma ray intensity logs  1308 . The 3D geomodel maps the SVD  1118  for any MD  1116  point along the wellbore to the corresponding SVD points on each of the offset wells. This enables, for example, comparison of the gamma ray intensity  1304  measured on the subject well  1310  with all of the corresponding values on the offset wells simultaneously. The combined misfit of the subject well against the offset wells is subsequently plotted as a stratigraphic misfit heatmap (not shown here). 
       FIG. 14  shows a heat map  1402  plotted in MD/SVD for single offset well (1D geomodel). The stratigraphic horizons appear as horizontal stripes. The wellbore trajectory  1404  is plotted, along with the interpretation thereof using a conventional geosteering technique  1406 , and the new automated interpretation  1408 . A heat map display like those shown and described herein enables the visual or automated identification of interpretation paths (MD, SVD(MD)). Such a path with the lowest misfit between subject and offset well measurements specifies a continuous interpretation of the stratigraphic depth along the wellbore. In addition, the surveyed wellbore trajectory can be displayed by plotting its TVD as SVD  1118 . The heat map then shows the misfit in stratigraphy between the subject well and offset wellbore. If the pre-drill geomodel and the surveyed TVD of the wellbore are both accurate, the TVD of the wellbore is identical to the SVD  1118 . Consequently, the wellbore will coincide with a path of minimal misfit in the heat map. Due to errors in the geomodel and the surveyed wellbore position, it is however more likely that the path of minimal misfit is displaced from the wellbore trajectory. For example, if the surveyed wellbore lies below the path of minimum misfit, it means that the actual stratigraphic depth is shallower, requiring a downward steering correction to achieve the intended stratigraphic depth. 
     To directly illustrate the deviation between actual and supposed stratigraphic depth, another possible choice is to display MD on the X axis, while displaying the Relative Stratigraphic Vertical Depth (RSVD=SVD−TVD) on the Y axis.  FIG. 15  illustrates a stratigraphic misfit heat map  1502  for the same well as in  FIG. 14 , but displayed using an MD/RSVD plot. In  FIG. 15 , the wellbore TVD has been subtracted from the SVD, so that the Y axis shows the difference between the correction of the SVD relative to the TVD of the wellbore, or the RSVD  1504 . A previous manual conventional interpretation using the stratigraphic misfit heat map methods disclosed herein. 
     The heat map shows the misfit in stratigraphy between the subject and offset wellbores as a function of a vertical displacement between the actual and supposed stratigraphic depth. For the ideal situation of an accurate geomodel and accurately surveyed wellbore trajectory, the path of minimum misfit would follow the Y=0 line. In this case, a visual or automated identification of interpretation paths (MD, RSVD(MD)) with the lowest misfit then specifies a continuous correction of the stratigraphic depth along the wellbore, relative to the stratigraphic depth given by the geomodel. 
     An optimal interpretation can be identified as a path SVD(MD) with minimal cumulative misfit for all the points along the path. Instead of simply adding the misfits along the path, it may be advantageous to compute the misfit of the path as a whole, further taking the correlation between subsequent measurements into account. This can be achieved by computing the pathwise misfit M(path)=D T  COV −1  D, where D is the vector of all differences between subject and offset well measurements along the interpretation path and COV −1  is the inverse of the covariance matrix for all pairs of pointwise misfits along the interpretation path. Depending on the correlation between subsequent measurements, the path with the lowest pathwise misfit may not coincide with the lowest integrated pointwise misfit over the MD range. The concept of pathwise misfit can be extended into a cost function, where the cost takes further undesirable attributes into account, such as strong formation curvature or large fault offsets. For example, an interpretation that assumes multiple large faults may have low misfit, but a higher cost. These considerations may be quantitatively incorporated into a Bayesian prior that penalizes paths that are deemed unlikely from knowledge gained about the underlying formation prior to drilling. 
     An algorithm can be employed by a computer system to automatically determine paths SVD(MD) of minimal cost. Any such path constitutes a possible interpretation of the vertical stratigraphic offset between the subject wellbore and geomodel. The path can be parameterized using any suitable basis functions, such as harmonic functions or splines. Finding a path with lowest cost then translates into finding the optimal parameters of the representation of the path. The prior covariances of the coefficients can be computed empirically by generating a large number of random realizations governed by the Bayesian prior model and estimating the resulting parameter covariances. Possible algorithms for finding the parameters for the optimal SVD(MD) paths include maximum a posteriori estimation, least squares and optimization methods from graph theory. Due to the stochastic nature of the geosteering problem, one or multiple optimal paths can be displayed on the heat map display to enable quality control by the user. A heat map display  1602  with corresponding vertical and horizontal correlation plots is shown in  FIG. 16 .  FIG. 16  shows a fully automated interpretation  1604  of heat map against a previous interpretation using a conventional manual geosteering technique  1608 . 
     Stratigraphic information along a wellbore can directly be inferred from sensor data, such gamma ray, neutron density, electrical resistivity and acoustic velocity. On the other hand, stratigraphic information may also be inferred from drilling dynamics data, such as changes in the rate of penetration, torque, weight on bit, mechanical specific energy, vibrations, differential mud pressure, and combinations of the foregoing. Further ancillary information may be available from monitoring the composition of the drilling fluid returning from down hole. The following discusses the use of drilling dynamics data in more detail. 
     For each offset wellbore, common data channels can be identified. For example, one offset wellbore may only have gamma log data, while another may additionally have drilling dynamics data. A characteristic function is then defined specific to the given set of data channels. For example, this may be a linear combination of the channels, which optimally weights the contribution of each channel. Using the same formula, a characteristic function along the wellbore can then be computed for the subject wellbore as well as for the offset wellbore. The characteristic function F subj (MD) for the subject wellbore can be parameterized by measured depth (MD), while the characteristic function F offset (TVD) for the offset wellbore is parameterized by the true vertical depth (TVD) along the offset wellbore. Since the available channels may differ between different offset wellbores, one pair of characteristic functions F subj (MD), F offset (TVD) can be computed for each offset wellbore. 
     The characteristic function of an offset well can be projected to any location on the subject well using a 3D stratigraphic model. Instead of interpreting against a separate characteristic function for every offset wellbore, one may instead combine the characteristic functions of multiple offset wellbores into a single characteristic function (also called typelog) at any location along the subject wellbore. 
     It may also be possible to use data channels that are not common to both the subject well and the offset well. For example, the offset wellbore may have no drilling dynamics data but may instead include data from an acoustic logging tool from which a user or computer system can infer the hardness of the rock, especially when such information is taken together with information regarding MD and/or TVD, and information regarding geological formation(s) expected or already encountered by the well, such as may be included in the well plan. The drilling dynamics data channels from the subject well can then provide a measure of rock hardness that can then be related to the rock hardness estimated from the acoustic data channels of the offset well. 
     A heat map may be used to display the difference between the characteristic functions of the subject wellbore and an offset wellbore as a measure of the stratigraphic misfit. As a measure of the difference one may use any of the norms commonly used in data analysis, such as the relative difference computed as: 
       Misfit(MD,TVD)=| F   subj (MD)− F   offset (TVD)|/( F   subj (MD)+ F   offset (TVD))
 
     Here, MD is taken along the subject wellbore, whereas TVD is taken along the offset wellbore. 
     Different choices for the X and Y coordinates may be made when displaying the misfit function as a heat map. One possible choice is to display MD on the X axis, while displaying the difference in TVD between the subject wellbore and the offset wellbore (δTVD) on the Y axis. The heat map then shows the misfit in stratigraphy between the subject and offset wellbores as a function of a vertical displacement. The heat map enables the visual or automated identification of paths (MD, δTVD) with the lowest misfit. Such a path with the lowest misfit then identifies the vertical stratigraphic displacement between the two locations. If the offset wellbores had been combined into a single characteristic function using a 3D stratigraphic model, then the path of minimal misfit shows the vertical displacement of the true stratigraphic depth from the stratigraphic model depth at the location of the subject wellbore. 
     An algorithm can be employed on a computer system to automatically determine and/or follow the peaks or valleys in the heat map. Any such valley constitutes a possible interpretation of the vertical stratigraphic offset between the subject wellbore and the offset wells or the between the subject wellbore and the stratigraphic model. Possible algorithms to identify valleys, for example, include maximum likelihood estimation, least squares and optimization methods from graph theory. The problem can be considered similar to tracking streams in topographical maps or finding the quickest route to a destination and can be solved using well known methods. After identifying one or multiple paths, the solutions can be displayed on the heat map to enable quality control by the user. A simple example showing the result of using least squares estimation for linear segments is shown in  FIG. 17 . The display  1702  shown in  FIG. 17  can be automatically generated and displayed by a computer system using data obtained from a wellbore being drilled. The deviation between the mapped TVD and the subject TVD  1704  is plotted against the measured depth  1116 . The fully-automated interpretation  1706  is plotted over the heat map shown on the display  1702 . The normalized misfit  1206  is measured on the heat map, wherein the light area  1708  represents a high normalized misfit  1206  and the dark area  1710  represents a low normalized misfit  1206 . An improved display may be achieved by using splines to account for curvature of the well trajectory and the geological formation of interest. It is to be noted that the heat map  1202  may include a variety of colors and/or brightness levels, such as bright red for a high misfit value and dark blue for a low misfit value, with colors of the spectrum (e.g., green, yellow, orange, etc.) representing intermediate misfit values. 
     The fully automated algorithm may not always identify the correct path, due to stratigraphic anomalies and noise in the data. Human expert intervention may therefore be required to guide the automated interpretation. One possibility is to set user-defined way points through which the path must pass. The automated algorithm may then be constrained to only consider paths passing through these way points. This may be called a semi-automated interpretation. An example of such a semi-automated interpretation using way points in shown in  FIG. 18 . The display  1802  shown in  FIG. 18  displays a fully-automated interpretation  1706  and a semi-automated interpretation  1806 . The semi-automated interpretation  1806  features way points  1808 . The normalized misfit  1206  is measured on the heat map, wherein the light area  1810  represents a high normalized misfit  1206  and the dark area  1812  represents a low normalized misfit  1206 . The heat map also shows a stratigraphic anomaly zone  1804 . It is to be noted that the heat map  1202  may include a variety of colors and/or brightness levels, such as bright red for a high misfit value and dark blue for a low misfit value, with colors of the spectrum (e.g., green, yellow, orange, etc.) representing intermediate misfit values. 
     It is also possible to use heat maps for an entirely manual interpretation without any automation. An example is shown in  FIG. 19 . The display  1902  shown in  FIG. 19  displays a fully-automated interpretation  1706  and a manual interpretation  1904 . The display  1902  also includes plotted marker  1   1906  and plotted marker  2   1908 . The normalized misfit  1206  is measured on the heat map, wherein the light area  1910  represents a high normalized misfit  1206  and the dark area  1912  represents a low normalized misfit  1206 . It is to be noted that the heat map  1202  may include a variety of colors and/or brightness levels, such as bright red for a high misfit value and dark blue for a low misfit value, with colors of the spectrum (e.g., green, yellow, orange, etc.) representing intermediate misfit values. The manual interpretation of a single or combined heat map could include the following steps: 
     (1) Display the heat map with the Y axis centered on the most likely δTVD, as inferred for example from a 3D stratigraphic model linking the subject wellbore stratigraphy to the offset wellbore stratigraphy. 
     (2) Add lines for geological formation or marker horizons ( 1906  and  1908 ) and label these. 
     (3) Determine paths through valleys of low heat, corresponding to good stratigraphic agreement between the subject wellbore and the offset data. These valleys are the possible interpretations. 
     (4) Employ a user interface (UI) to define one or multiple paths along the selected valleys. 
     (5) Export the paths as stratigraphic interpretations, such as may be used to adjust a drill plan and/or adjust drilling operations to drill to a target formation. 
     Multiple heat maps may be displayed simultaneously along the subject wellbore using 3D visualization. A suitable user interface, for example using a game controller (e.g., an Xbox controller), may be used to set way points or define manual interpretations in a 3D display. An illustration of this approach is shown in  FIG. 20 . The user interface  2002  features a heat map for typelog  1   2006  and a heat map for typelog  2   2008 . The heat map for typelog  1   2006  and a heat map for typelog  2   2008  cross one another at the subject wellbore  2004 . The typelog  1   2010  and typelog  2   2012  are combined to create a combined typelog  2014 . Note that in the lateral section of the subject well the display of δTVD as a vertical offset is intuitive. However, in the vertical section, MD progresses downward. While δTVD is displayed horizontally, it also refers to the vertical direction. The user interface should be designed to intuitively communicate this to the user. 
     By analyzing the heat map image to identify zones of low misfit, interpretations can be constrained to the most promising regions. This can significantly speed up the automated interpretation by a computer system by constraining the region in which it is to search for solutions. The image analysis may also identify connected channels which are indicative of potential interpretations. 
     The result of the heat map interpretation is a mapping from the MD of the subject wellbore to the stratigraphy defined by offset wellbores or a 3D stratigraphic model. This information allows the user (or automated steering system) to determine the stratigraphic position of the wellbore and to make real-time corrections to the wellbore while it is being drilled to optimally steer the wellbore to the geological target and keep the wellbore in the stratigraphic target zone. 
     The generation, display, and use of the heat map interpretation, including uses such as updating the well plan and/or altering or adjusting one or more drilling parameters to drill to the target zone and/or stay in the target zone, including automatically or in a semi-automatic fashion, may be done with a programmed computer system which may be connected to one or more of the drilling rig control systems, such as described above, including steering control system  168  or CGL  900 . 
     Drilling dynamics parameters carry information about rock properties. However, this geological information must be separated from noise and unrelated drilling events. Described here are methods and systems to optimally extract the geological information from the various drilling dynamics parameters in such a way that the geological specificity is maximized and can be used. 
     Drilling dynamics data from previously drilled wells can be used to infer an optimal combination of re-scaled parameters to enhance the common geology signal. The data can be processed in the following way: 
     1. Identify existing wellbores which have one or multiple adjacent wellbores. All wells ideally will have the same drilling dynamics parameters to be used when steering a new wellbore.
 
2. When extracting the drilling dynamics parameters from the wellbore database, also extract the Gamma. Then adjust the measured depth of the Gamma ray measurements for the distance between the Gamma sensor and the bit, so that all parameters are referenced to the measured depth of the bit.
 
3. Optionally compute derivative parameters such as the Mechanical Specific Energy (MSE).
 
4. Using the Gamma ray values for each well, identify marker horizons indicating a specific stratigraphic depth. Use the marker horizons to “flatten” the true vertical depth (TVD) between the wells. This can be achieved by choosing one well as the primary well and adjusting the TVDs of the other wells in such a way that their markers line up with the primary well.
 
5. Analyze the statistical distribution of each drilling parameter, for example by plotting histograms of the data from prior wells. Histograms show skewness in the distribution and they can be used to identify multiple populations, arising for example from different drilling modes. It may also be important to identify any non-stationarity of the parameters, such as trends of the mean and/or the variance with increasing MD.
 
6. Prior to incorporating the drilling dynamics data, it is advantageous to transform such data into stationary normal distributed parameters with zero mean and unit variance. This typically involves (a) breaking up multiple populations, such as periods of rotary drilling versus slide drilling, (b) removal of trends, (c) eliminating skewness in the distribution and (d) removing outliers. Essentially, a function generally should be defined for each drilling parameter that approximately maps it to a stationary random series following a Gaussian normal distribution with zero mean and unit variance.
 
7. The same clean-up and transformation data may also be advisable for Gamma and any other logging data.
 
     Once the drilling parameters have been transformed into an equivalent number of parameters with zero mean and unit variance, one can infer the optimal linear combination of the re-scaled parameters using a statistical model as follows. 
     Let us assume that the drilling dynamics data vectors X(TVD) and Y(TVD) of two wells are the sum of a common geology signal vector G(TVD) and uncorrelated noise U(TVD) and V(TVD): 
         X (TVD)= G (TVD)+ U (TVD) 
         Y (TVD)= G (TVD)+ V (TVD) 
     The covariance matrix cov(G) can then be computed as: 
       &lt; X   i   ,Y   j   &gt;=&lt;G   i   ,G   j   &gt;+&lt;G   i   ,V   j   &gt;+&lt;U   i   ,G   j   &gt;+&lt;U   i   ,V   j   &gt;=&lt;G   i   ,G   j &gt;=&lt;cov( G )&gt; 
     Where &lt; &gt; denotes statistical expectation. Thus, the covariance matrix of the common geology signal vector G(TVD) can be estimated from the covariance between the drilling parameters of adjacent wells. To make the estimated covariance matrix symmetric for a pair of wells: 
       &lt;cov( G ) ij &gt;=½(&lt; X   i   ,Y   j   &gt;+&lt;Y   i   ,X   j &gt;)
 
     In case there are more than two adjacent wells available, the covariance matrix of the common geology signal can be estimated from the average over all possible combinations of pairs of wells. 
     The stratigraphic misfit heat map displays the misfit between the vector a of parameter values of the subject well against the corresponding vector b of parameter values of the offset well. These parameters can include for example rescaled gamma measurements as well as rescaled drilling dynamics parameters. Given the covariance matrix cov(G) of the common geology vector, we can then define the stratigraphic misfit between a and b as: 
       ∥ a−b ∥=(( a−b ) T  cov( G )( a−b )) 1/2  
 
     This allows a wide range of parameters to be included into the misfit heat map to make use of all relevant information to optimally display an accurate and helpful heat map and to steer the wellbore into the desired target formation. 
     In summary, various reference data from reference wells may be used to generate stratigraphic misfit heat maps that enable analysis of actual log data from a well being drilled. Generation of the heat maps may include an analysis of drilling dynamics parameters to identify multiple populations, trends, drift, skewness, and to eliminate outliers. The data logs from the reference data may be transformed into stationary normal distributed random variables. Certain statistical properties of a common geology vector may be identified by cross-correlation of data logs from one or more reference wells. A covariance matrix (cov(G)) of the common geology vector may be estimated. The covariance matrix may be used to define a stratigraphic misfit between data logs of the well being drilled and the one or more reference wells. The data logs combined with the common geology vector may be displayed and analyzed using 3D stratigraphic misfit heat maps. From the stratigraphic misfit heat map, a most likely stratigraphic trajectory of the well being drilled may be obtained, such as in one example from a valley of the minimum misfit in the heat map. 
     At any measured depth (MD) on the subject well, stratigraphic heat maps can be used to infer a likelihood of the wellbore being at a particular stratigraphic depth. Furthermore, the probability-weighted stratigraphic depth, the most likely depth, and the uncertainty of depth can be inferred by performing the following operations.
         Select a starting depth MD 1  on the subject wellbore. MD 1  could be, for example, the wellhead, a uniquely identified crossing of a stratigraphic marker or crossing of a stratigraphic fault.   Identify possible interpretations of sequences of stratigraphic depth along the wellbore, corresponding to valleys in the stratigraphic heat map.   Order the interpretations by their stratigraphic vertical depth SVDi(MDL) at the target measured depth MDL. Different interpretations can end up at approximately the same SVD at the target MDL. Treat all interpretations with similar SVD as a subset. Index these subsets with different depths SVDj(MDL) by the index j=1 . . . M, where M≤N is the number of such subsets. The average SVDj(MDL) of each subset j is given by SVDj=Av(SVDi(MDL) where the average is over all interpretations i in subset j.   The likelihood P(SVDj) of the subject wellbore being at the particular stratigraphic depth SVDj at the target MDL can then be inferred from the cumulative misfits as P(SVDj)=Sumi(f(Mi)|i belongs to subset j)/Sumi(f(Mi)), where f(Mi) is a suitable function of the misfit, which increases as the misfit decreases, such as its reciprocal value. The first sum is over the interpretations in the subset, while the second sum is over all interpretations.   The probability-weighted average SVDp is then given by       

       SVD p =Sum j (SVD jP (SVD j ))         Note that this depth could inadvertently fall between two valleys in the heat map onto a peak with maximum misfit.   The stratigraphic depth with the highest likelihood is given by       
       SVDML=(SVD j  where  P (SVD j ) is largest)         Uncertainty of depth can then be computed as       
       sigma(SVD)=(Sum j ( P (SVD j )(SVD j −SVD p )2)) 1/2  
 
     These discrete values at the stratigraphic depths SVDj of the interpretation subsets j can be extrapolated into a continuous probability density p(SVD) by the following further steps:
         Define a modified path (MDk, SVDi,dSVD(MDk)) for interpretation i belonging to subset j as depicted in  FIG. 17 , which also explains the computation of the likelihood of depth:
           Starting at MD L , decrease the index k until reaching the last inflection point MD Inf . The last inflection point being defined as the last point (highest k) where the second derivative   
               

         D ″( k )=SVD i (MD k−1 )−2SVD i (MD k )+SVD i (MD k+1 )
             changes sign from D″(k−1) to D″(k), which is equivalent to the product being negative:           

         D ″( k− 1) D ″( k )&lt;0.
             For the given modification dSVD, define a modified path as:           

       For MD k &lt;MD Inf :SVD i,dSVD (MD k )=SVD i (MD k ) 
       For MD k ≥MD Inf :SVD i,dSVD (MD k )=SVD i (MD k )+(MD k −MD inf )/(MD L −MD inf ) d SVD
             Compute the misfit M i,dSVD  of the modified path           

         M   i,dSVD =sum( M (MD k ,SVD i,dSVD (MD k ))         Specify the range SVDmin to SVDmax in which the probability density p(SVD) is defined:
           Define SVD min =SVD 0 =SVD 1 −SVD C , where SVD C  is a chosen constant offset with a magnitude similar to a typical SVD offset between two adjacent subsets of interpretations.   Define SVD max =SVD M+1 =SVD M +SVD C      
           Select the range of modified paths for each interpretation i belonging to subset j as the SVD interval as SVDj−1 to SVDj+1.   Define the total value of the function f( ) of the misfit M i,dSVD  as:       
         f   total =Sum i (integral( f ( M   i,dSVD ) d SVD),         where the sum is over all interpretations i. The integral is over the interval from SVD j−1  to SVD j+1 , where j is index of the subset that the interpretation i belongs to. This integral can be substituted by a discrete sum over small enough increments of dSVD.   Define the probability density p(SVD) as:       
         p (SVD)=sum i ( f ( M   i,dSVD ), if SVD j−1 &lt;SVD&lt;SVD j+1 )/ f   total            The probability-weighted mean stratigraphic depth is then given by       
       SVD p =integral(SVD  p (SVD))         Note that this depth could fall between two valleys in the heat map onto a peak with maximum misfit.   The stratigraphic depth with the highest likelihood is given by       
       SVD ML =(SVD where  p (SVD) is largest)         The uncertainty of stratigraphic depth can then be computed as       
       sigma(SVD)=(integral((SVD−SVD p ) 2   p (SVD))) 1/2  
 
     The above disclosed subject matter is to be considered illustrative, and not restrictive, and the appended claims are intended to cover all such modifications, enhancements, and other embodiments which fall within the true spirit and scope of the present disclosure. Thus, to the maximum extent allowed by law, the scope of the present disclosure is to be determined by the broadest permissible interpretation of the following claims and their equivalents, and shall not be restricted or limited by the foregoing detailed description.