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Slant range

Slant range, also known as slant distance, is the line-of-sight distance between two points that are not at the same elevation, forming the hypotenuse of a right triangle where the legs represent the horizontal separation and vertical height difference between the points.^[1]^[2] In radar systems, slant range refers to the direct line-of-sight path from the sensor to the target along the radar beam, and it differs from ground range—the horizontal projection onto the Earth's surface—which can lead to geometric distortions when imaging sloped terrain.^[2]^[3]^[4] This distinction is critical for accurate target positioning, as uncorrected slant range measurements can cause misalignment in displays overlaid with GPS data or maps, particularly at short ranges or steep incidence angles where the deviation is most pronounced.^[2] The concept is foundational in various technologies, including synthetic aperture radar (SAR) imaging, geospatial mapping, remote sensing, and navigation systems such as ultra-short baseline (USBL) for underwater positioning, where it accounts for elevation or depth differences.^[4]^[1]

Fundamentals

Definition

Slant range is defined as the straight-line distance along the line of sight between two points that are not at the same elevation, typically measured from an observer, such as a sensor or radar antenna, to a target.^[1] This measurement accounts for the direct path in three-dimensional space, distinguishing it from planar distances.^[2] A key distinction of slant range lies in its geometric role: unlike the horizontal ground range or vertical height, it represents the hypotenuse of a right triangle formed by elevation-separated points. For instance, in a basic two-dimensional scenario, it denotes the line-of-sight distance from a ground-based radar to an airborne target, which exceeds the ground range due to the target's altitude.^[5]

Geometric Interpretation

In the geometric interpretation of slant range, it represents the straight-line distance along the line of sight between an observer and a target, forming the hypotenuse of a right triangle in the observer-target plane. The two legs of this triangle consist of the horizontal ground range—the perpendicular projection of the slant range onto the horizontal plane—and the vertical height difference between the observer and the target. This configuration arises in scenarios such as radar or optical observations, where the line of sight deviates from the horizontal due to elevation disparities.^[6]^[2]^[4] Diagrams illustrating this geometry typically depict the observer at one vertex of the right triangle, with the right angle at the base where the ground range meets the vertical leg; the target lies at the opposite end of the hypotenuse. For instance, a side-view schematic shows the slant range line slanting upward or downward from the observer to the target, emphasizing how increasing height separation elongates the hypotenuse relative to the ground range. In cases where the observer and target share the same elevation, such as both at sea level, the vertical leg collapses to zero, causing the slant range to coincide exactly with the ground range along the horizontal. These visualizations highlight the obliquity introduced by non-zero elevations, aiding in understanding distortions in imaging or ranging systems.^[6]^[2] Related terms clarify this spatial relationship: ground range is defined as the horizontal distance from the observer's nadir point to the target's nadir projection on the surface, while the elevation angle is the acute angle between the horizontal plane at the observer and the line of sight to the target. A conceptual example occurs when observing two points at sea level, where the slant range simplifies to the direct Euclidean straight-line distance without any vertical offset; however, introducing elevation, such as from an airborne platform to a surface target, creates the characteristic slant, making the path longer than the ground range and introducing angular obliquity that affects resolution and projection.^[2]^[4]

Calculations

Basic Pythagorean Relation

In the context of slant range calculations under a flat-Earth approximation, the fundamental relationship arises from the geometry of a right triangle formed by the observer (such as a radar or sensor) and the target. The ground range d serves as one leg, representing the horizontal distance along the surface between the point directly below the observer and the target's projection on the ground; the height difference h forms the other leg, denoting the vertical separation between the observer and the target; and the slant range r is the hypotenuse, the straight-line distance through space.^[7]^[2] This configuration directly invokes the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. To derive the slant range formula, consider the right angle at the target's ground projection: the horizontal displacement d and vertical displacement h are perpendicular. Squaring both sides of the theorem gives r^2 = d^2 + h^2. Taking the positive square root (as distance is non-negative) yields the core formula:

r = \sqrt{d^2 + h^2}

This equation quantifies the slant range for scenarios where the line of sight is unobstructed and the terrain is approximated as planar.^[7] From this relation, inverse formulas can be obtained by algebraic rearrangement, assuming r > \max(d, h) to ensure real-valued solutions. Solving for the ground range gives d = \sqrt{r^2 - h^2}, useful when the measured slant range and height are known to infer horizontal separation. Similarly, for height, h = \sqrt{r^2 - d^2}, applicable in elevation estimation from range data. These inverses maintain the same geometric foundation and are employed in basic range-to-position conversions.^[2]^[7] The derivation and formulas rely on key assumptions: a flat planar surface neglecting Earth's curvature, absence of atmospheric refraction or multipath effects, and direct line-of-sight propagation without obstacles. These simplifications hold for short-range applications or local approximations where the distances are small relative to the Earth's radius.^[7]^[2] For illustration, consider an observer at a height of 5 km above a target with a ground range of 10 km. Substituting into the formula yields r = \sqrt{10^2 + 5^2} = \sqrt{125} \approx 11.18 km, demonstrating how the slant range exceeds the ground range due to the vertical offset.^[7]

Accounting for Earth's Curvature

For long-distance measurements exceeding approximately 10-20 km, the flat-Earth approximation introduces significant errors in slant range calculations, necessitating models that incorporate Earth's sphericity to maintain accuracy.^[6] The basic Pythagorean relation provides a suitable starting point for short ranges but must be extended using spherical geometry for scenarios like satellite passes or over-the-horizon radar.^[8] The precise slant range c between an observer at height h_o above Earth's surface and a target at height h_t, separated by central angle \theta, is derived from the law of cosines applied to the triangle formed by Earth's center and the two points:

c = \sqrt{(R + h_o)^2 + (R + h_t)^2 - 2(R + h_o)(R + h_t)\cos\theta}

where [R](/page/R) is Earth's mean radius of approximately 6371 km.^[9] This formula computes the straight-line chord distance through space, accounting for the curvature by treating the positions as points on concentric spheres. Key factors include the inclusion of [R](/page/R), which scales the geometry; elevation angles, which relate to the line-of-sight orientation and influence \theta; and the subtended central angle \theta, representing the angular separation at Earth's center.^[6] To determine \theta from geographic coordinates, approximation methods such as the haversine formula—adapted from great-circle distance calculations—can be used to find the angular separation before substitution into the law of cosines.^[10] Alternatively, the law of cosines within spherical triangles provides a framework for more complex configurations involving multiple points or non-radial paths.^[11] These approaches prioritize the spherical model for ranges where curvature effects dominate, ensuring errors remain below 1% for distances up to thousands of kilometers. A representative example occurs in satellite ranging: for a low-Earth orbit satellite at 500 km altitude passing over a ground station (h_o = 0, h_t = 500 km), the minimum slant range at nadir (90° elevation, \theta \approx 0) is approximately 500 km, increasing to over 2000 km at low elevations (e.g., 10°), highlighting how \theta and elevation angles modulate the distance.^[12] Such variations are critical for link budget analyses in satellite systems. These models have limitations, notably ignoring atmospheric refraction, which bends propagation paths and extends effective ranges beyond geometric predictions, particularly at low angles.^[8] They are valid primarily for ranges greater than 10-20 km, where flat-Earth errors exceed typical measurement tolerances of 0.1-1 km.^[6]

Applications in Technology

Radar Systems

In radar systems, slant range represents the direct line-of-sight propagation distance between the radar antenna and the target, determined from the round-trip time-of-flight of the transmitted electromagnetic pulse reflected back as an echo. This distance is calculated using the formula r = \frac{c \cdot t}{2}, where c is the speed of light (approximately $3 \times 10^8 m/s) and t is the measured time delay of the echo.^[13]^[14] The reliance on slant range enables precise target ranging but introduces geometric distortions when mapping to horizontal distances on the ground. The concept of slant range emerged during World War II radar developments, notably in the British Chain Home early warning system, which detected aircraft at ranges up to 80 miles by measuring echo delays to provide bearing, height, and distance for air defense.^[15] In modern applications, slant range is integral to air traffic control radars, where it supports aircraft tracking and separation by combining range data with azimuth and elevation angles, and to weather radars, which use it to locate precipitation echoes along beam paths for storm analysis.^[16]^[17] In synthetic aperture radar (SAR) imaging, slant range measurements cause distortions such as foreshortening, where terrain slopes facing the radar appear compressed because multiple ground points map to the same range resolution cell, and layover, where the tops of elevated features like mountains are imaged before their bases, leading to overlapping signals.^[18]^[13] To mitigate these, SAR data processing converts slant range to ground range maps using the relation d = r \sin \phi, where \phi is the local incidence angle between the radar beam and the surface normal, often requiring topographic corrections for accurate geolocation.^[19] For example, in airborne radar systems, a target at a 30° elevation angle yields a ground range approximately 86.6% of the measured slant range, making the horizontal distance appear shorter and potentially compressing features in the display.^[20] In aviation, slant range plays a critical role in distance-measuring equipment (DME), a radio navigation aid that determines the direct line-of-sight distance between an aircraft and a ground-based transponder station using ultrahigh frequency (UHF) signals in the 960–1215 MHz band.^[21] The aircraft's onboard interrogator sends pulses to the ground station, which replies after a fixed delay; the round-trip time, adjusted for the delay, yields the slant range in nautical miles, inherently incorporating the aircraft's altitude above the station and thus exceeding the true horizontal ground distance.^[22] This measurement forms the basis for positioning in systems like VOR/DME, where the VHF omnidirectional range (VOR) provides azimuthal bearing and DME supplies the radial distance, enabling pilots to fix their position at the intersection of a VOR radial and a DME range circle.^[23] The inclusion of the altitude component in DME slant range leads to overestimation of ground distance, with the error most pronounced when the aircraft is close to or overhead the station; for instance, at 10,000 feet above ground level directly over the station, the DME displays approximately 1.65 nautical miles instead of zero horizontal distance, as 6,076 feet equates to 1 nautical mile vertically.^[23] A common rule of thumb holds that slant range error becomes negligible when the horizontal distance from the station is at least 1 nautical mile per 1,000 feet of altitude, such as requiring a 5-nautical-mile separation at 5,000 feet to minimize discrepancies during en-route or terminal operations.^[24] In VOR/DME navigation, this error affects hyperbolic-like positioning in area navigation (RNAV) applications using multiple DME stations, where intersecting range circles define the aircraft's location, necessitating corrections to derive accurate ground distances for waypoint tracking and route adherence.^[25] Pilots routinely account for slant range in approach path calculations, particularly during DME arcs on non-precision instrument approaches, where the displayed distance guides turns and descent timing around the station; for example, maintaining a 10-nautical-mile arc at 5,000 feet requires adjusting the effective radius downward by the vertical component to ensure proper alignment with the runway threshold.^[26] The DME slant range error increases proportionally with aircraft altitude above the station, potentially displacing fixes by up to several nautical miles in high-altitude en-route scenarios without adjustment, prompting the use of altimeter data for manual compensation via the Pythagorean relation—ground distance equals the square root of (slant range squared minus altitude in nautical miles squared)—to compute true horizontal separation.^[27] Integration of DME with global positioning system (GPS) in flight management systems (FMS) enables hybrid slant-corrected navigation, where GPS-derived ground track distances supplement or validate DME slant ranges, and FMS algorithms apply altitude-based corrections to DME inputs for precise RNAV positioning during en-route and terminal phases.^[28] In such systems, DME serves as a backup to GPS, with slant range data updating inertial references, while GPS overlays on VOR/DME approaches allow substitution where the ground-projected distance aligns closely enough to disregard minor slant discrepancies.^[29] Federal Aviation Administration (FAA) and International Civil Aviation Organization (ICAO) standards address slant range errors in en-route navigation through accuracy requirements and procedural mitigations; ICAO Annex 10 specifies DME precision as no worse than 0.25 nautical miles plus 1.25% of the measured distance, while FAA guidelines in Advisory Circular 90-45A mandate altimeter compensation in three-dimensional RNAV systems above flight level 180 and recommend increased lateral separation buffers below that level to accommodate uncorrected errors up to 2.5 nautical miles cross-track en route.^[30] These standards ensure safe operations by requiring pilots to cross-check DME readouts against charts and apply corrections, particularly in VOR minimum operational network scenarios where DME remains essential for IFR navigation integrity.^[27]

Satellite Communications

In satellite communications, slant range represents the line-of-sight distance between a satellite and a ground station, playing a pivotal role in signal propagation characteristics and overall system performance. This distance directly influences propagation delay, Doppler shift, and attenuation due to free-space path loss, making it essential for designing reliable links in geostationary (GEO), medium Earth orbit (MEO), and low Earth orbit (LEO) systems. Unlike nadir range, which is simply the difference in altitudes, slant range accounts for the angular position of the satellite relative to the observer, varying significantly with the elevation angle \epsilon observed from the ground station. Accurate computation of slant range is critical for optimizing antenna pointing, power allocation, and frequency planning to mitigate signal degradation over long orbital paths.^[12] The slant range r in a satellite-to-ground link can be calculated using geometric relations that incorporate Earth's curvature. For a circular orbit, the exact expression is

r = -(R + h_g) \sin \epsilon + \sqrt{(R + h_s)^2 - (R + h_g)^2 \cos^2 \epsilon},

where R is Earth's mean radius (approximately 6371 km), h_s is the satellite's orbital altitude, and h_g is the ground station's altitude above sea level.^[12] This formula derives from the law of cosines applied to the triangle formed by Earth's center, the ground station, and the satellite, with the angle at the ground station being $90^\circ + \epsilon. For LEO systems, where h_s \ll R, an approximation simplifies calculations under moderate to high elevation angles: r \approx \frac{h_s - h_g}{\sin \epsilon}, treating the path as nearly flat-Earth while still capturing the inverse relation to \sin \epsilon.^[12] Slant range reaches its minimum at nadir (\epsilon = 90^\circ), equaling h_s - h_g, and increases toward the horizon (\epsilon \to 0^\circ), where visibility limits typically cap usable ranges to avoid excessive atmospheric effects. In link budget analysis, slant range primarily determines free-space path loss (FSPL), a dominant factor in signal attenuation for satellite systems. The FSPL in decibels is given by

\text{FSPL} = 20 \log_{10} r + 20 \log_{10} f + 32.44,

with r in kilometers and f in megahertz; this equation quantifies the geometric spreading of the signal wavefront over the propagation distance.^[31] Variations in slant range thus directly impact received signal strength, requiring higher transmit power or gain at maximum ranges to maintain bit error rates. In LEO constellations like Starlink, where satellites move rapidly relative to ground users, slant range fluctuations necessitate frequent handovers between satellites—typically triggered when \epsilon drops below 25–40°, increasing path loss by 10–20 dB compared to overhead passes and prompting switches to maintain low-latency connectivity.^[32] For GEO satellites at 36,000 km altitude, the slant range at the equator (nadir) is approximately 36,000 km, rising to about 42,000 km for ground stations at 30° latitude due to the off-nadir geometry.^[33] As of 2025, slant range considerations have become integral to 5G non-terrestrial networks (NTN), where dynamic beamforming adjusts antenna patterns to compensate for range-induced path loss and timing advances in LEO and MEO deployments. In these systems, real-time slant range estimates enable adaptive precoding and resource allocation, supporting seamless integration of satellite segments with terrestrial 5G infrastructure under 3GPP Release 17 and beyond.^[34]

Surveying and Remote Sensing

In photogrammetry, slant range refers to the oblique line-of-sight distance from the aerial camera's perspective center to ground features, which introduces scale distortions in oblique photographs where the camera axis is tilted relative to the nadir. These distortions cause objects farther from the nadir point to appear smaller and stretched radially, varying the photo scale as approximately s \approx \frac{c}{Z}, where c is the camera's focal length and Z is the slant range to the feature. Corrections for these effects rely on the collinearity equations, which enforce that the image point, object point, and perspective center lie on a straight line, relating measured image coordinates to ground coordinates through exterior orientation parameters (position and angles) and interior orientation parameters (focal length and principal point).^[35] In remote sensing applications such as LiDAR and multispectral imaging, slant range influences spatial resolution and necessitates geometric rectification to align data with map projections. For LiDAR systems, the laser footprint size expands with increasing slant range due to beam divergence, degrading horizontal and vertical resolution over longer distances and uneven terrain, as the effective spot diameter grows proportionally to the range. Multispectral sensors experience similar issues, where off-nadir viewing angles elongate features along the slant path, requiring rectification to mitigate positional errors in derived products like vegetation indices or land cover maps.^[36]^[37] Surveying applications leverage slant range for topographic modeling, particularly in drone-based surveys where it is calculated as r = \sqrt{d^2 + h^2}, with d as the horizontal distance and h as the flight altitude, to generate accurate three-dimensional point clouds from overlapping images or laser returns. This relation ensures precise georeferencing of points during structure-from-motion processing or direct ranging, enabling high-fidelity digital elevation models for terrain analysis.^[35] A representative example occurs in slant-range synthetic aperture radar (SAR) imaging for terrain mapping, where the range resolution \Delta r = \frac{c}{2B} (with c as the speed of light and B as the signal bandwidth) results in foreshortening and layover effects that compress elevated terrain features along the line of sight. Orthorectification addresses these by projecting slant-range data onto a ground plane using a digital elevation model to remove distortions and produce map-consistent outputs suitable for surveying.^[19]^[38]

Measurement Techniques

Direct Measurement Methods

Direct measurement of slant range relies on sensors that capture the line-of-sight distance between an observer and a target, often using electromagnetic waves or light pulses. Time-of-flight methods are among the most common, where a signal is transmitted to the target, reflects back, and the round-trip time is used to compute the distance.^[39] In radar systems, time-of-flight measurement involves transmitting short radio frequency pulses toward the target and detecting the echoed signal. The slant range r is calculated as r = \frac{c t}{2}, where c is the speed of light and t is the measured round-trip time, accounting for the signal's propagation to the target and back. This approach provides the direct line-of-sight distance in applications such as air traffic control and remote sensing.^[39] LiDAR employs a similar time-of-flight principle but uses laser pulses in the optical spectrum for higher resolution. The system emits pulses and measures the return time to determine the slant range, with the distance derived from r = \frac{c t}{2}, adjusted for instrument-specific timing calibrations like clock counts and analog converter factors. For instance, space mission LiDARs achieve sub-meter accuracy over kilometers by interpolating between pulse timings.^[40] Phase-based methods, such as interferometry, measure slant range by detecting phase differences in signals along the propagation path. In Interferometric Synthetic Aperture Radar (InSAR), the phase shift \Delta \phi = \frac{4\pi \Delta r}{\lambda} between two radar images reveals path length differences \Delta r, where \lambda is the wavelength; this enables precise slant range estimation for terrain mapping. GPS interferometry complements this by providing point-wise phase data to refine slant path measurements in satellite observations.^[41] Optical techniques utilize line-of-sight instruments for direct slant range acquisition in surveying. Laser rangefinders emit modulated laser beams and measure the phase shift or time-of-flight of the reflection to compute distance, offering millimeter accuracy up to several kilometers even in adverse conditions. Theodolites, often integrated into total stations with electronic distance measurement (EDM), combine angular readings with laser-based ranging to determine slant distances along inclined lines of sight.^[42]^[43] Modern drone-mounted systems enhance slant range measurement by integrating altimeters or LiDAR with GPS and inertial measurement units (IMUs). These compute real-time slant ranges by fusing position data from GPS, orientation from IMU, and direct ranging from onboard sensors, enabling applications in precision agriculture and topographic surveys. For example, LiDAR on drones measures slant range to ground features, georeferenced via GPS/IMU for accurate 3D mapping.^[44] A practical example is Distance Measuring Equipment (DME) used in aviation, which operates on time-of-flight principles in the 960–1215 MHz band. Aircraft interrogators send pulse pairs to ground stations, which reply after a fixed delay; the aircraft decodes the slant range from the total propagation time, supporting navigation with updates typically at rates of 30 to 150 pulse pairs per second once locked on.^[21]

Error Sources and Corrections

Atmospheric refraction represents a primary source of error in slant range measurements, as it bends the propagation path of signals due to variations in atmospheric density, resulting in an overestimation of the true geometric distance. This effect is particularly pronounced in radio and microwave frequencies used in radar and satellite systems, where the refractive index gradient causes the signal path to curve downward, effectively lengthening the measured slant range. For paths over approximately 100 km, uncorrected refraction can introduce errors on the order of several meters, though simplified models indicate range errors below 1 meter for dry atmospheric conditions up to 120 km ground range.^[45]^[46] Multipath reflections further degrade accuracy by creating multiple signal paths, such as ground bounces in low-elevation radar scenarios, which superimpose delayed echoes onto the direct signal and cause range ambiguities or ghost targets. In ground-based radar systems with broad elevation beam widths, multipath interference can lead to significant tracking errors, especially for upward-looking geometries over reflective surfaces.^[47] Elevation misestimation exacerbates these issues by introducing uncertainties in the vertical component of the target position, often due to imprecise sensor calibration or terrain variations, which propagate into overall slant range distortions.^[48] In radar applications, slant-to-ground range conversion errors arise prominently from unknown target heights, as assuming a zero-height target yields a slant range that overestimates the horizontal ground distance by the height-induced offset, leading to positional inaccuracies in tracking. Beam spreading, or broadening, compounds this by increasing the effective beam footprint with range, diluting signal energy and introducing resolution errors that affect precise slant range determination, particularly beyond 60 km where curvature effects amplify the distortion. To mitigate these, refraction models based on ITU-R standards, such as Recommendation P.834, provide methods to compute ray bending and path delays using vertical refractivity profiles, enabling corrections for tropospheric effects in propagation predictions.^[49]^[50]^[51] Kalman filtering techniques address dynamic tracking errors by recursively estimating target states, including slant range and elevation, from noisy measurements, effectively smoothing multipath and misestimation effects in real-time radar scenarios. Differential GPS enhances height accuracy for slant range computations, achieving vertical precisions of 0.07 to 0.16 feet in surveying applications by differencing signals from reference stations to cancel common errors. The impact of height errors on ground range can be approximated by the formula

\Delta d \approx r \sin \phi \cdot \frac{\Delta h}{r} = \sin \phi \cdot \Delta h,

where r is the slant range, \phi the elevation angle, and \Delta h the height error, highlighting how small vertical uncertainties amplify horizontal displacements at low angles. In satellite ranging, ionospheric delay corrections via dual-frequency signals substantially reduce slant path errors; by differencing phase measurements at L1 and L2 frequencies, the first-order ionospheric refraction is eliminated, cutting overall errors by over 90% from typical single-frequency values of tens of meters to residual higher-order effects of centimeters.^[52]^[53]^[54]^[55]

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Slant range measurements combined with sensor position and attitude data can be used to derive three-dimensional point clouds—collections of georeferenced.
[39]
[PDF] mitigating the impact of the laser footprint size on airborne lidar data ...
The laser footprint size plays a crucial role in both the vertical and horizontal accuracy of the data collected over sloped or non-uniform terrain, as well as ...
[40]
[PDF] A Principles of Synthetic Aperture Radar
where L is the length of radar antenna, ρ is the slant range, and λ is the wavelength of the radar. If the scatterer on the ground remains stationary as the ...
[41]
Geometric Distortion in Imagery - Natural Resources Canada
Jan 8, 2025 · All remote sensing images are subject to some form of geometric distortions, depending on the manner in which the data are acquired.Missing: slant range photogrammetry
[42]
Range or distance measurement - Radartutorial.eu
The actual range of a target from the radar is known as slant range. Slant range is the line of sight distance between the radar and the object illuminated.
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Ground calibration tests of the laser altimeter (LIDAR) for MMX mission
May 15, 2025 · The MMX LIDAR is an active instrument for measuring the slant range to a target; it emits light pulses and detects their return, with the slant ...
[44]
[PDF] InSAR Principles: Guidelines for SAR Interferometry Processing and ...
Feb 19, 2007 · the slant range direction is one of the key parameters of SAR interferometry. The SAR interferogram is generated by cross-multiplying, pixel ...
[45]
Laser rangefinder for precise measurement and monitoring - Jenoptik
With Jenoptik's laser rangefinder, you can measure distances and positions in the range of a few centimetres up to 60 kilometres - contact-free, fast and with ...Missing: slant theodolite
[46]
Total Station and Laser Scanner Solutions for Precision Surveying
May 6, 2024 · Total stations measure angular and sloping distances using theodolite and EDM. Laser scanners use LIDAR to capture spatial data for 3D models.About Total Stations · Total Stations Types · About 3d Laser Surveys
[47]
Principles of Georeferencing and UAVs - AEVEX Geodetics
Georeferencing converts 1D/2D data to 3D position. Direct georeferencing is real-time, using GPS and IMU. Indirect georeferencing is post-mission.
[48]
[PDF] Radar Range Measurements in the Atmosphere - OSTI.GOV
This range error is a function of atmospheric constituents, such as water vapor, as well as the geometry of the radar data collection, notably altitude and ...
[49]
[PDF] Correction of atmospheric refraction errors in radio height finding
slant range, Ro is extremely small compared to the , height error; the slant range and radio range are assumed to be identical to the geometric range, R.
[50]
Multipath Interferences in Ground-Based Radar Data: A Case Study
Dec 5, 2017 · Multipath interference can occur in ground-based radar data acquired with systems with a large antenna beam width in elevation in an upward looking geometry.
[51]
https://journals.ametsoc.org/view/journals/atot/27/4/2009jtecha1359_1.xml
[52]
[PDF] Effects of tropospheric refraction on radiowave propagation - ITU
Recommendation ITU-R P.834 provides methods for the calculation of large-scale refractive effects in the atmosphere, including ray bending, ducting layers, the ...
[53]
Slant range vs. ground range - ResearchGate
Download scientific diagram | Slant range vs. ground range from publication: A general approach for altitude estimation and mitigation of slant range errors ...
[54]
Impacts of Beam Broadening and Earth Curvature on Storm-Scale ...
It appears that, within a range of 60 km, both the impacts of beam broadening and earth curvature can be neglected. As the distance increases beyond 60 km, more ...
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[PDF] Analysis of Kalman Filter Mechanizations for an Idealized ... - DTIC
Jan 3, 1972 · The tracking radar measures slant range and elevation of the vehicle and yields an output which is corrputed by noise. The noisy output is ...
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[PDF] HEIGHTING WITH GPS: POSSIBILITIES AND LIMITATIONS
These applications cover the range of achievable GPS heighting accuracy. INTRODUCTION. Global Positioning System (GPS) surveying has been used extensively and ...
[57]
Plane error caused by elevation error. R, h, and θ are the slant ...
R, h, and θ are the slant range, elevation, and incidence angle of a target, respectively. ∆h is the height error, and ∆x is the geolocation error caused by ∆h.<|control11|><|separator|>
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Chapter 4: GNSS error sources - NovAtel
For receivers that can only track a single GNSS frequency, ionospheric models are used to reduce ionospheric delay errors. Due to the varying nature of ...