Question 1

Least-squares adjustment terminology: the equations obtained by multiplying each observation equation by the coefficient of its unknowns and summing are called what?

Accepted Answer

Introduction / Context: In surveying and geodesy, least-squares adjustment is used to estimate unknowns (e.g., coordinates) from redundant observations while minimizing the sum of squared residuals. Understanding the formation of the core equations is fundamental for reliable network adjustment. Given Data / Assumptions: A linearized observation model relating observations to unknown parameters. Matrix form: v = A x - l, with A as coefficient matrix. Weighted least-squares objective: minimize vᵀ W v. Concept / Approach: Setting the derivative of the objective with respect to the unknowns to zero yields the normal equations: (Aᵀ W A) x = Aᵀ W l. Conceptually, this is equivalent to multiplying each observation equation by its coefficients and summing to collect terms in the unknowns. Step-by-Step Solution: Start from observation equations relating measurements to unknowns.Form the least-squares objective and differentiate with respect to unknowns.Set derivatives to zero to obtain the normal equations.Recognize that the described algebraic process corresponds to constructing the normal equations. Verification / Alternative check: Textbook derivations show equivalence between matrix normal equations and coefficient-weighted summations. Why Other Options Are Wrong: Observation equations: original measurement relationships. Conditional equations: constraints independent of observations (e.g., geometric conditions). Residual equation/None: do not describe the forming step. Common Pitfalls: Confusing conditional constraints with observation models; ignoring weighting in forming normal equations. Final Answer: Normal equation

Question 2

Astronomical surveying with Polaris: for the determination of azimuth, Polaris is preferably observed when it is at which position?

Accepted Answer

Introduction / Context: Polaris (the Pole Star) lies close to the north celestial pole and is commonly used in astronomical surveying to establish true azimuth. Measurement geometry impacts sensitivity and observational error. Given Data / Assumptions: Clear sky observations near the horizon region where Polaris traces a small circle around the pole. Use of a theodolite or total station with astronomical capability. Concept / Approach: At elongation, Polaris attains its maximum angular displacement east or west of the meridian. The azimuth of Polaris varies most slowly near elongation, making azimuth determination less sensitive to timing errors and refraction, and therefore more accurate operationally. Step-by-Step Solution: Identify positions: culmination (upper/lower transit across the meridian) and elongation (max east/west azimuth).Recognize that at elongation, azimuth changes minimally with time compared to culmination.Therefore, observe at elongation for better azimuth accuracy. Verification / Alternative check: Survey manuals recommend elongation observations due to reduced rate of azimuth change and simpler corrections. Why Other Options Are Wrong: Culmination: higher rate of azimuth change; more timing sensitivity. Neither/either: less precise guidance than the established best practice. Lower transit only: unnecessarily restrictive and not optimal. Common Pitfalls: Neglecting refraction and instrument collimation; inadequate timekeeping; poor identification of elongation times. Final Answer: At elongation

Question 3

Stereophotogrammetry parallax measurement: the standard parallax equation is valid across the entire overlap only if the parallax is measured in which direction?

Accepted Answer

Introduction / Context: In aerial stereophotogrammetry, horizontal parallax (p) is used to derive elevations and relative positions from overlapping photographs. The direction of parallax measurement relative to the flight base line is crucial for the validity of standard equations. Given Data / Assumptions: Standard vertical photography with known base line (B) along the flight direction. Parallax is measured between conjugate image points in the overlap region. Concept / Approach: The classical parallax equation relates parallax to camera geometry when parallax is measured along the base line direction. Measuring parallax parallel to the base simplifies relationships and ensures consistency across the entire overlap area, where the geometry aligns with flight direction. Step-by-Step Solution: Identify the base line as the projection of flight path between exposure stations.Note that parallax fundamentally manifests along this direction in vertical photography.Therefore, measure parallax parallel to the base line for the standard parallax formulations to hold across the overlap. Verification / Alternative check: Photogrammetry texts emphasize measuring x-parallax along the base line; y-parallax (normal) is typically minimized by orientation. Why Other Options Are Wrong: Normal to base line: corresponds to y-parallax and is minimized; equations are not derived for this direction. Both/neither/45°: do not reflect the standard geometric derivation. Common Pitfalls: Failing to eliminate tilt/roll during relative orientation; mixing x- and y-parallax components. Final Answer: Parallel to the base line

Question 4

Celestial coordinate systems for astronomy and surveying: which coordinate pair is the most convenient standard for specifying positions of stars on the celestial sphere?

Accepted Answer

Introduction / Context: Accurate specification of star positions is essential for navigation, geodesy, and astronomy. Several coordinate systems exist, but their convenience differs for cataloging and long-term reference. Given Data / Assumptions: Positions are referenced on the celestial sphere. We seek a system stable with Earth’s rotation and suitable for star catalogs. Concept / Approach: The equatorial coordinate system uses declination (Dec) and right ascension (RA). Declination is analogous to latitude, measured north/south of the celestial equator. Right ascension is analogous to longitude but measured in time units eastward from the vernal equinox. This system is independent of observer location and time of night (aside from precession/nutation corrections) and is thus ideal for star catalogs and pointing. Step-by-Step Solution: Compare systems: Alt–Az depends on observer and time; Hour angle changes continuously with Earth’s rotation.RA–Dec remains fixed for a given epoch, enabling universal catalogs.Therefore, Declination and Right Ascension is the most convenient standard. Verification / Alternative check: Star atlases and telescope GoTo databases universally use RA/Dec for epoch references (e.g., J2000.0). Why Other Options Are Wrong: Alt–Az: topocentric and time-dependent; useful for local pointing, not for cataloging. Declination–Hour angle: hour angle varies with time. Declination–Altitude or Azimuth–Declination: mixed systems lacking standard global usage. Common Pitfalls: Forgetting to apply precession/nutation/proper motion when high precision is required. Final Answer: Declination and right ascension system

Question 5

Astronomical surveying: for a star to be circumpolar as seen by an observer at latitude φ (in either hemisphere), which condition must be satisfied?

Accepted Answer

Introduction / Context: In astronomical surveying and navigation, a star is called circumpolar if it never sets below the local horizon and remains perpetually above it throughout a sidereal day. Recognizing the condition for circumpolarity is essential for planning observations and understanding star trails and azimuth determinations. Given Data / Assumptions: Observer at latitude φ (north positive, south negative). Declination of the star is δ, and its angular distance from the nearest celestial pole is 90° − |δ|. Standard spherical astronomy geometry, neglecting refraction near the horizon for the conceptual condition. Concept / Approach: A star never sets if the lower culmination still lies above the horizon. Geometrically, the star’s circular path around the celestial pole remains entirely above the horizon when the angular radius of that circle (90° − |δ|) is less than the altitude of the pole above the horizon (|φ|). This yields the circumpolarity criterion: |δ| > 90° − |φ|, which is equivalent to the star’s angular distance from the pole being less than |φ|. Step-by-Step Solution: At the observer’s site, the pole altitude equals |φ|.A star’s polar distance = 90° − |δ|.For the star’s entire diurnal circle to be above the horizon: 90° − |δ| < |φ|.Rearrange: |δ| > 90° − |φ|. This means its distance from the pole is less than |φ|.Hence, the condition is: angular distance from pole < latitude. Verification / Alternative check: At the North Pole (φ = 90°), all stars with δ > 0° are circumpolar because polar distance is ≤ 90°. At the equator (φ = 0°), no star is circumpolar, matching the condition. Why Other Options Are Wrong: Declination = 0°: stars on the celestial equator rise and set. Declination = 90°: only the pole itself; overly restrictive. Hour angle = 180°: defines lower culmination moment, not circumpolarity. RA equals LST: specifies transit time, not circumpolar nature. Common Pitfalls: Confusing declination with altitude; overlooking sign conventions for southern hemisphere; ignoring that circumpolarity depends on both δ and φ. Final Answer: Its angular distance from the celestial pole must be less than the observer's latitude

Question 6

Triangulation practice: a subsidiary station chosen near a main triangulation station to bypass obstructions is commonly called a satellite/eccentric/false station. Which name is NOT used for such a subsidiary point?

Accepted Answer

Introduction / Context: In triangulation surveying, obstacles (buildings, trees, relief) may prevent direct observations from a primary station. A nearby auxiliary point is then selected to carry out the required sightings while mathematically relating it back to the main station. Given Data / Assumptions: Auxiliary points are near the primary station. Terminology in classical triangulation texts recognizes several synonyms. We must identify the term that is NOT standard for this purpose. Concept / Approach: Common synonyms for a nearby auxiliary point are: satellite station, eccentric station, or false station. These reflect the idea that the observed station is not the true main point but a nearby surrogate whose relationship to the main station is known. The term “pivot station” is not a recognized synonym in this context. Step-by-Step Solution: List known synonyms: satellite, eccentric, false station.Compare with “pivot station” which refers to rotating supports in instruments, not a subsidiary triangulation point.Therefore, “pivot station” is not used for the auxiliary point. Verification / Alternative check: Surveying manuals consistently employ the three accepted synonyms; “pivot” relates to instrument mechanics rather than station classification. Why Other Options Are Wrong: Satellite station: standard term. Eccentric station: standard term. False station: standard term in some texts. Offset station: also used in practical contexts. Common Pitfalls: Assuming “pivot” refers to station naming; it does not. Final Answer: Pivot station

Question 7

Aerial photography motion blur: if v is the ground speed of the aircraft (m/s), t is the camera shutter time (s), and the photograph scale is 1:S, what is the linear image displacement due to motion during exposure?

Accepted Answer

Introduction / Context: In vertical aerial photography, forward motion of the aircraft during the finite shutter opening time causes linear image displacement (motion blur) on the photograph. Controlling this blur is crucial for sharp imagery and accurate photogrammetric measurements. Given Data / Assumptions: Ground speed of aircraft = v (m/s). Shutter time (exposure) = t (s). Photo scale = 1:S (dimensionless; S = ground distance / photo distance). Nominal flying height is such that scale is constant across the small exposure time. Concept / Approach: During exposure, the aircraft advances a ground distance g = v * t. On the photo, this ground distance is reduced by the scale factor S. Hence the corresponding smear length on the film/sensor is d = g / S. Step-by-Step Solution: Compute ground travel during exposure: g = v * t.Convert to photo displacement using scale 1:S: d = g / S.Therefore: d = v * t / S.This is the standard first-order relation for motion blur in vertical photography. Verification / Alternative check: From lens geometry, f/H = 1/S for vertical photos; using image-space expression d = (v * t) * (f/H) also reduces to d = v * t / S, confirming the relationship. Why Other Options Are Wrong: d = v * t * S: inverts the scale; would imply larger blur for larger S, contrary to reality. d = v / (t * S): wrong units and dependence. d = S / (v * t): dimensionally incorrect. d = (v * t)^2 / S: introduces a non-physical square term. Common Pitfalls: Confusing scale 1:S with its reciprocal; ignoring along-track versus across-track components; neglecting image motion compensation (IMC) mechanisms in modern cameras. Final Answer: d = v * t / S

Question 8

Astronomical corrections: the standard correction for parallax (in arcseconds) expressed as a function of zenith distance α is which of the following?

Accepted Answer

Introduction / Context: Topocentric parallax arises because observations are made from the Earth’s surface rather than its center. For solar/lunar or stellar observations in classical navigation and surveying tables, small empirical corrections (in arcseconds) are applied to convert apparent positions to geocentric ones. Given Data / Assumptions: Correction magnitude given as a small constant multiplier (arcseconds). α denotes zenith distance (complement of altitude). Sign convention assumes subtracting the correction from the apparent altitude where appropriate. Concept / Approach: For small parallax corrections, a cosine dependence on zenith distance is common, since the projection of the observer’s displacement from Earth’s center onto the line of sight involves cos α. A negative sign indicates the observed (topocentric) altitude is greater than the geocentric altitude and needs reduction. Step-by-Step Solution: Recognize parallax term structure: proportional to cos α.Adopt the empirical coefficient 8.8 arcseconds for the stated context.Apply sign: correction = − 8.8 cos α arcsec. Verification / Alternative check: For α = 0° (object at zenith), cos α = 1, largest correction in magnitude; for α = 90° (horizon), cos α = 0, vanishing correction—consistent with geometry. Why Other Options Are Wrong: + 8.8 cos α: wrong sign. ± 0.8 sin α or + 8.8 sin α: wrong angular dependence and/or magnitude. Duplicate punctuation “- 8.8 cos α.” is effectively the same as option (a) but presented ambiguously; standard form is without trailing dot. Common Pitfalls: Mixing refraction and parallax corrections; confusing altitude (h) and zenith distance (α = 90° − h). Final Answer: - 8.8 cos α

Question 9

Longitude difference between two places: identify the correct rule(s) for computing the difference in longitudes depending on hemispheres.

Accepted Answer

Introduction / Context: Accurate computation of longitude differences is fundamental in navigation, astronomy, and geodetic computations. Depending on whether locations lie in the same or opposite hemispheres (east or west), the arithmetic changes to obtain the smallest longitudinal separation. Given Data / Assumptions: Longitudes measured east or west from Greenwich (0° to 180°E/W). Interest is in the minimal angular separation along parallels. All values expressed in degrees for simplicity. Concept / Approach: When both places are in the same hemisphere (both E or both W), the absolute difference gives the longitude separation. When they are in opposite hemispheres, the separation is the sum of magnitudes. If the computed separation exceeds 180°, its supplement to 360° represents the shorter path across the globe. Step-by-Step Solution: Case 1: Same hemisphere → Δλ = |λ1 − λ2|.Case 2: Opposite hemispheres → Δλ = |λ1| + |λ2|.Case 3: If Δλ > 180° → Use 360° − Δλ for the smaller arc. Verification / Alternative check: Test with examples (e.g., 170°E and 170°W): naive sum is 340°, but true smallest separation is 360° − 340° = 20°. Why Other Options Are Wrong: Each individual rule is correct, so “All the above” is the most complete choice. Common Pitfalls: Forgetting to reduce to the smaller arc; mixing E/W signs; not converting minutes/seconds consistently. Final Answer: All the above

Question 10

Relief (height) displacement on a vertical aerial photograph: identify how it varies with position, terrain elevation, and flying height.

Accepted Answer

Introduction / Context: Relief displacement is the apparent radial shift of image points on vertical aerial photographs caused by object height above the reference datum. Understanding its dependencies is crucial for accurate planimetry and for deriving elevations photogrammetrically. Given Data / Assumptions: Vertical photograph with principal point well defined. Object point at elevation h above datum, flying height above datum H, focal length f. Radial distance r from the principal point to image point. Concept / Approach: The first-order relation for relief displacement d on a vertical photo is d ≈ (r * h) / H. Thus, displacement grows with radial distance r and object height h, and diminishes with larger flying height H. The direction of displacement is radially outward for elevated points. Step-by-Step Solution: Start with d ≈ (r * h) / H.As r increases (farther from principal point), d increases linearly.As h increases (higher terrain/objects), d increases linearly.As H increases (higher flight), d decreases inversely. Verification / Alternative check: At the principal point (r = 0), d = 0, consistent with zero displacement there. Doubling flying height halves displacement, confirming inverse dependence. Why Other Options Are Wrong: Each individual statement (a), (b), and (c) is correct; therefore, “All the above” is the correct aggregate choice. Common Pitfalls: Confusing relief displacement with lens distortion; ignoring sign convention (outward for elevated points, inward for depressions). Final Answer: All the above

Question 11

Timekeeping: how many sidereal days occur in one tropical year (the interval between successive vernal equinox passages)?

Accepted Answer

Introduction / Context: A sidereal day is measured relative to the distant stars, while a solar (mean) day is measured relative to the Sun. A tropical year is defined by the cycle of seasons (vernal equinox to vernal equinox). Because Earth orbits the Sun, there are slightly more sidereal days than solar days in one tropical year. Given Data / Assumptions: Tropical year length ≈ 365.2422 mean solar days. Sidereal day is about 23 h 56 m 4 s (≈ 0.99727 solar days). We want the count of sidereal days within one tropical year. Concept / Approach: As Earth completes one revolution around the Sun, the planet must rotate approximately one extra turn relative to the stars to realign the Sun to the local meridian, leading to ~1 additional sidereal day over the count of solar days in the year. Step-by-Step Solution: Number of mean solar days per tropical year ≈ 365.2422.Number of sidereal days ≈ number of solar days + 1 (approximately).Hence ≈ 366.2422 sidereal days in one tropical year. Verification / Alternative check: Compute with day lengths: (365.2422 solar days) / (0.9972696 solar days per sidereal day) ≈ 366.2422 sidereal days. Why Other Options Are Wrong: 365, 365.2224, 365.2422: these are solar-day counts or close variants, not sidereal-day counts. 366.2224: close but not as accurate as 366.2422. Common Pitfalls: Confusing sidereal day length with the number of sidereal days per year; neglecting the Earth’s orbital motion. Final Answer: 366.2422

Question 12

Aerial photo geometry: identify the correct statements about photographic tilt, principal line direction, and the principal plane on a tilted photograph.

Accepted Answer

Introduction / Context: Even small departures from strict verticality in aerial photography introduce tilt, which affects scale, relief displacement patterns, and orientation procedures. Understanding the geometric elements—tilt angle, principal line, principal plane—is essential for accurate photogrammetric corrections. Given Data / Assumptions: Tilted photograph with perspective centre (camera station), principal point, and plumb point identifiable. Perspective axis is the line through the lens perpendicular to the image plane (at exposure). Concept / Approach: Tilt: angle between the negative plane and a horizontal plane that contains the perspective axis. Principal line: intersection of the photograph plane with the principal plane; it gives the direction of maximum tilt. Principal plane: truly vertical plane through the perspective centre that contains the principal point and the plumb point; it defines the geometry of tilt. Step-by-Step Solution: Identify definitions: (a) tilt definition matches standard photogrammetric usage.(b) Principal line indicates maximum tilt direction on the photo.(c) Principal plane includes perspective centre, principal point, and plumb point and is vertical.All three statements are correct. Verification / Alternative check: Photogrammetry texts depict these relationships consistently; tilt corrections rely on these definitions. Why Other Options Are Wrong: Any subset (e.g., only (a) and (c)) omits a correct statement; hence “All the above” is required. Common Pitfalls: Confusing the principal point with the nadir point; misidentifying the plumb point location. Final Answer: All the above

Question 13

Relation between air base and photographic base: for successive vertical photographs with flying height H and focal length f, how are the ground air base B and the image photographic base b related?

Accepted Answer

Introduction / Context: In aerial photogrammetry, the air base B is the ground distance between successive exposure stations, while the photographic base b is the distance between corresponding principal points on the overlapping photographs. Their relation under vertical photography links image geometry to ground geometry. Given Data / Assumptions: Vertical photographs with negligible tilt. Flying height above datum = H; camera focal length = f. Scale approximately uniform: scale ≈ f / H at the principal point. Concept / Approach: Linear distances on the photo are related to ground distances by the scale factor f / H. Thus, a ground distance equals the corresponding photo distance multiplied by H / f. Applying this to the inter-exposure distance yields B = (H / f) * b. Step-by-Step Solution: Scale at principal point: photo length / ground length = f / H.Rearrange: ground length = (H / f) * photo length.Let photo length = b → ground length (air base) B = (H / f) * b. Verification / Alternative check: Dimensional reasonableness: larger H increases B for a fixed b, matching intuition; larger f reduces scale and thus reduces computed B for fixed b. Why Other Options Are Wrong: B = (f / H) * b or b = (H / f) * B invert the scale mapping. B = (H * f) / b is dimensionally inconsistent. b = (f / H) * B restates the inverse but fails to answer the form asked. Common Pitfalls: Confusing scale f/H with its reciprocal H/f; ignoring minor variations in scale away from the principal point. Final Answer: B = (H / f) * b

Question 14

Measurement science: which statements correctly describe systematic errors in observations and how they affect results?

Accepted Answer

Introduction / Context: Errors in measurement are broadly classified as systematic, random, and blunders. Systematic errors are particularly important because they bias results in a predictable way and must be identified and corrected in surveying, astronomy, geodesy, and engineering metrology. Given Data / Assumptions: Well-calibrated instruments may still have residual systematic effects. Environmental influences (temperature, refraction) may introduce predictable biases. Observer biases (e.g., reaction time) can be modeled. Concept / Approach: Systematic errors have determinate causes and generally obey known functional forms (e.g., scale error, zero error). Because they are predictable, we can apply corrections to observations to remove their effect. These errors tend to bias results consistently high or low, hence are often termed cumulative. Step-by-Step Solution: Identify determinism: systematic errors follow laws (temperature coefficient, index error, etc.).Apply corrections: once quantified, correct observations to mitigate bias.Bias direction: results skew consistently in one direction.Terminology: cumulative errors is a common synonym in classical texts. Verification / Alternative check: Instrument calibration certificates provide correction tables precisely to handle systematic components, confirming their determinate nature. Why Other Options Are Wrong: Here, none are wrong; all characterize systematic errors accurately, so the aggregate choice is correct. Common Pitfalls: Confusing random scatter (handled statistically) with systematic bias; neglecting periodic errors (a subset with sine/cosine forms) that are still systematic. Final Answer: All the above

Question 15

Longitude–time conversion in astro-surveying: the difference in longitude between two places, when expressed in time units, equals the difference in which time scales at those places?

Accepted Answer

Introduction / Context: Longitude is fundamentally linked to time because Earth completes one rotation (360 degrees) in 24 hours. This principle underpins astronomical navigation and geodetic surveying, where longitude differences are routinely converted into time differences. Given Data / Assumptions: Earth's rotation rate is assumed uniform for practical purposes. Time can be referenced to different celestial markers: mean sun, apparent (true) sun, or distant stars (sidereal). Concept / Approach: The angular–temporal relationship is 360 degrees ↔ 24 hours, or 15 degrees ↔ 1 hour. Therefore, 1 degree of longitude corresponds to 4 minutes of time. This conversion is independent of the chosen time reference (sidereal, apparent solar, or mean solar) because each is ultimately tied to Earth's rotation relative to a celestial reference. Step-by-Step Solution: Relate angle to time: 1 degree = 4 minutes.Apply to any longitude difference Δλ to obtain Δt = Δλ * (4 minutes/degree).Recognize that the same Δt manifests in sidereal, apparent solar, and mean solar time comparisons between the two places. Verification / Alternative check: Standard astro-navigation texts define local time = Greenwich time ± (longitude in time). This formulation is valid across the three time scales with the appropriate celestial reference. Why Other Options Are Wrong: No single option is exclusive: the relationship holds for all listed time systems. Common Pitfalls: Using longitude in degrees without converting to time (4 minutes per degree). Confusing east/west longitude sign conventions when adding/subtracting time. Final Answer: all the above

Question 16

Hour angle transformation: the hour angle of a celestial body at the Greenwich meridian equals the hour angle at any local meridian plus the longitude correction for which references?

Accepted Answer

Introduction / Context: Hour angle (HA) is the angular distance of a celestial object west of a reference meridian. Converting between local hour angle (LHA) and Greenwich hour angle (GHA) is central to astronomical observations and navigation. Given Data / Assumptions: LHA is measured at the observer's meridian; GHA is measured at Greenwich. Longitude is used algebraically (E positive or W positive per convention). Concept / Approach: The universal relation is GHA = LHA + longitude (with sign convention). This applies to bodies referenced to the mean sun (mean solar time), true sun (apparent solar time), the vernal equinox (sidereal time), and fixed stars. Step-by-Step Solution: Start with LHA(body) at local meridian.Add longitude (east +, west −, or as defined) to obtain GHA(body).Recognize identical form for mean sun, true sun, vernal equinox, and any star. Verification / Alternative check: Navigation formulae: LHA = GHA − longitude; rearranged gives GHA = LHA + longitude for every celestial reference. Why Other Options Are Wrong: No single reference is unique; the relationship holds for all listed references. Common Pitfalls: Incorrect longitude sign (east vs west) leading to hour-angle errors. Final Answer: all the above

Question 17

Horizon definitions in celestial surveying: the circle formed by a plane tangent to Earth at the observer’s location intersecting the celestial sphere is known as which horizon?

Accepted Answer

Introduction / Context: Multiple horizon definitions exist in astronomy and geodesy. Distinguishing them is essential for reducing observations and applying corrections. Given Data / Assumptions: A plane tangent to Earth at the observer's point is considered. The celestial sphere is the imaginary sphere on which celestial bodies are projected. Concept / Approach: The sensible horizon is the small circle on the celestial sphere formed by the intersection of the observer's tangent (horizontal) plane with that sphere. The visible horizon is the apparent line where sky meets terrain/sea; the true (rational) horizon is an ideal plane through Earth's center and the observer's zenith–nadir line; the term celestial horizon is often used synonymously with the rational horizon, not the local tangent definition. Step-by-Step Solution: Construct the tangent plane at observer's location.Intersect that plane with the celestial sphere.Identify the resulting circle as the sensible horizon. Verification / Alternative check: Standard astronomical surveying texts define sensible horizon precisely in this manner. Why Other Options Are Wrong: Visible horizon: depends on obstructions and refraction. Celestial/true horizon: refers to the rational horizon, not the local tangent circle. Common Pitfalls: Interchanging visible and sensible horizons in reductions. Final Answer: sensible horizon

Question 18

Atmospheric refraction corrections: which statements correctly describe how refraction corrections vary with the altitude of a celestial body?

Accepted Answer

Introduction / Context: Atmospheric refraction bends incoming light, shifting the apparent altitude of celestial bodies. Accurate astronomical observations require altitude-dependent refraction corrections. Given Data / Assumptions: Standard atmosphere; observations near horizon are most affected. Altitude measured from the sensible/visible horizon upward. Concept / Approach: Refraction increases sharply as altitude approaches the horizon and diminishes to negligible at the zenith. Practical tables show approximately 33 arcminutes near the horizon and nearly zero directly overhead; in general, correction follows functions proportional to cotangent of altitude. Step-by-Step Solution: At zenith: path through atmosphere is minimal → correction ≈ 0.At horizon: longest path → correction ≈ 33′.Between these, use altitude-based tables or formulae to interpolate corrections. Verification / Alternative check: Refraction tables in nautical almanacs/survey manuals confirm the quoted limiting values and altitude dependence. Why Other Options Are Wrong: Each individual statement is correct; thus the combined choice is appropriate. Common Pitfalls: Neglecting refraction at low altitudes, leading to large altitude errors. Final Answer: all the above

Question 19

Invar tape material in baseline measurement: the nickel content of the nickel–iron alloy (Invar) used for precision tapes is typically closest to which percentage?

Accepted Answer

Introduction / Context: Invar tapes are used for high-precision baseline measurements because their thermal expansion is extremely low, minimizing temperature-induced length errors. Given Data / Assumptions: Invar is a nickel–iron alloy engineered for near-zero thermal expansion. Typical composition is narrowly controlled to achieve this property. Concept / Approach: Near-zero thermal expansion occurs when nickel content is about 36%, with iron comprising the balance. This composition keeps the coefficient of thermal expansion around 0.7 × 10^-6 per °C, far below that of steel. Step-by-Step Solution: Recall definition of Invar and its target thermal property.Match the composition that delivers this behavior → ~36% Ni. Verification / Alternative check: Materials references for metrology equipment list Invar 36 (UNS K93600) as the standard alloy for tapes and rods. Why Other Options Are Wrong: 24%, 40%, or 60% nickel do not yield the characteristic near-zero expansion needed for surveying tapes. Common Pitfalls: Confusing Invar with stainless or Monel alloys that have higher expansion. Final Answer: 36%

Question 20

Aerial photogrammetry scale: for a truly vertical photograph with camera focal length f taken from flying height H above mean sea level (MSL), at a ground point of reduced level h, what is the local photo scale at that point?

Accepted Answer

Introduction / Context: Photo scale varies with terrain elevation. Converting image distances to ground distances requires the correct scale at each ground point, especially over undulating terrain. Given Data / Assumptions: Vertical photograph (tilt ≈ 0). Camera focal length = f. Flying height above MSL = H. Ground point reduced level = h above MSL. Concept / Approach: Local scale S equals focal length divided by flying height above ground at that point. Flying height above ground is H − h (since both are referenced to MSL for a vertical photo). Thus: S = f / (H − h) Step-by-Step Solution: Compute height above ground: H_g = H − h.Apply vertical photo scale formula: S = f / H_g.Therefore S = f / (H − h). Verification / Alternative check: Derivation from similar triangles between camera center, image point, and ground point yields the same relation. Why Other Options Are Wrong: f/H ignores ground elevation (valid only if h ≈ 0). (H − h)/f is the inverse (not the scale). f/(H + h) uses the wrong sign; ground elevation reduces flying height above ground. Common Pitfalls: Using a single average scale over significant relief, causing planimetric and height errors. Final Answer: f / (H - h)

Advanced Surveying Questions