Fundamentals of Surveying

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This section covers the history, definitions and principles of surveying, its classification and applications, the concept of scale, linear and angular measurements, and the units used in surveying with their standardisation and conversion.

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Definition and History • Surveying is the art and science of determining the relative positions of points on, above or below the surface of the earth by measuring distances, directions (angles) and elevations, and of representing them on a plan, map or numerical/digital model.

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Levelling is the branch dealing with elevations.

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Geomatics is the modern, wider term covering the acquisition, processing, analysis, storage and presentation of spatially referenced data (surveying, photogrammetry, remote sensing, GNSS, GIS and cartography). • History: boundary re-establishment after Nile floods in Egypt (rope stretchers, ≈ 1400 BC);

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Greek and Roman instruments (groma, chorobates, dioptra) for roads and aqueducts; the plane table and chain in medieval Europe; the Great Trigonometrical Survey of India (from 1802) under Lambton and Everest — the source of the historic height of Mt Everest; invention of the telescope, vernier and theodolite; photogrammetry and aerial survey (20th century);

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EDM (1950s), satellite positioning/GPS (fully operational 1995), total stations, GNSS-RTK, LiDAR, UAVs and GIS. • In Nepal: the Survey Department (Department of Survey, under the Ministry of Land Management) carries out the national geodetic control, topographic mapping, cadastral survey and land-record work; the joint Nepal-China measurement announced the height of Sagarmatha (Everest) as 8 848.86 m in 2020.

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Principles of Surveying • 1.

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Work from the whole to the part.

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Establish a framework of high-accuracy control points first, then fix the details from that framework — this localises and prevents the accumulation of error (working from part to whole magnifies errors). • 2.

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Fix a point by at least two independent measurements (processes).

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A point is located by two distances, two angles, or one distance and one angle — a third, redundant measurement provides a check. • Supporting rules: always provide checks on field work and computation; maintain consistency of accuracy (angles and distances measured to matching precision); adopt the accuracy the purpose demands — no more, no less (economy of accuracy); record observations neatly, in the field, in ink, and never erase.

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Classification of Surveying Basis Classes Curvature of the earth Plane surveying — the earth is treated as flat, curvature neglected; suitable for small areas (customarily up to ≈ 250 km²; the difference between an arc and its chord of 18.5 km is only about 1 cm).

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Geodetic surveying — curvature and the earth's figure are taken into account; used for large areas and national control Purpose/nature Topographical, cadastral (property boundaries, land records), engineering (route, construction, setting out), hydrographic, mine, geological, archaeological, astronomical, city, military/defence, as-built Instrument used Chain/tape, compass, plane table, theodolite, tacheometric, levelling, total station, GNSS, photogrammetric, laser scanning/LiDAR, remote sensing Place of work Land (topographic, cadastral), marine/hydrographic, aerial (photogrammetric, UAV), underground (tunnel, mine) Method Triangulation, traversing, trilateration, photogrammetric, satellite • Applications: preparation of topographic and cadastral maps, land registration and taxation, route surveys for roads, canals, railways and transmission lines, setting out of buildings, bridges, tunnels and dams, deformation monitoring, volume computation for earthwork, hydrographic charting, resource and environmental mapping, disaster mapping, and the base data for GIS.

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Scale • Scale = map distance ÷ corresponding ground distance, usually written as a representative fraction (RF) such as 1:25 000 (dimensionless, so it holds for any unit) or as an engineer's scale (1 cm = 250 m).

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A large-scale map has a small denominator (cadastral 1:500–1:2 500) and shows much detail; a small-scale map has a large denominator (1:250 000 and smaller) and covers a large area. • Area scale = (linear scale)²: on a 1:1 000 map, 1 cm² represents 100 m².

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Shrinkage factor of an old paper map = shrunk length ÷ original length; corrected length = measured length ÷ shrinkage factor, and areas are corrected by its square. • A graphical (bar) scale drawn on the map remains valid even if the paper shrinks or the map is photo-reduced; diagonal and vernier scales allow finer reading.

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Scale choice depends on the purpose, the smallest detail to be shown and the plotting accuracy (≈ 0.25 mm on paper).

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Linear and Angular Measurements • Linear: direct measurement by chain, tape or wire; indirect by tacheometry (stadia), EDM/total station (phase-difference or pulse timing) or satellite positioning.

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Horizontal distance is what is plotted, so slope distances are reduced. • Angular: horizontal angles (between the vertical planes through two lines) and vertical angles (above/below horizontal; the zenith angle is measured from the vertical).

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Instruments: compass, theodolite, total station, sextant. • Bearings: the whole-circle bearing (WCB) is measured clockwise from north, 0°–360°; the quadrantal (reduced) bearing is measured from north or south, 0°–90°, towards east or west (e.g., N 40° E).

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Fore bearing and back bearing differ by exactly 180° when there is no local attraction.

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Bearings may be true (from geographic north), magnetic (from magnetic north) or grid (from grid north); magnetic declination is the angle between true and magnetic north (east or west, varying with place and time), and azimuth is the clockwise angle from north (usually grid or true). • Relation: included angle = difference of bearings; and in a traverse the bearing of the next line = bearing of the previous line + included (clockwise) angle ± 180°.

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Units, Standardisation and Conversion Quantity Units and conversions Length SI metre;

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1 link (metric chain) = 0.2 m;

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Gunter's chain = 66 ft = 20.117 m (100 links); engineer's chain = 100 ft; metric chains of 20 m and 30 m;

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1 nautical mile = 1 852 m Angle Sexagesimal — 1 circle = 360°, 1° = 60′, 1′ = 60″; centesimal — 1 circle = 400 grads (gon), 1 grad = 100 c; radian — 1 circle = 2π rad, 1 rad = 57°17′44.8″ = 206 265″ Area 1 hectare = 10 000 m² = 2.471 acres;

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1 acre = 4 046.86 m²;

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1 km² = 100 ha Nepali land units Hill:

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1 ropani = 508.72 m² = 16 aana = 64 paisa = 256 daam.

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1 bigha = 6 772.63 m² = 20 kattha, 1 kattha = 20 dhur = 338.63 m²;

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1 bigha ≈ 13.31 ropani Volume m³;

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1 m³ = 35.315 ft³ • Standardisation: a tape or chain is compared at regular intervals with a standard length under stated conditions (usually a temperature of 20 °C and a specified pull, supported in a stated way); if its actual length differs from the nominal length, every measurement made with it must be corrected (see 1.2).

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Modern EDM instruments and total stations are calibrated on a baseline with known distances, and levelling staves and GNSS antennas are similarly checked.

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This section covers chain and tape surveying with the types of tapes and the corrections applied, plane-table surveying, compass surveying, the theodolite and its adjustments, and tacheometry.

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Chain and Tape Surveying • Chains: metric chains of 20 m (100 links) and 30 m (150 links), Gunter's (66 ft), engineer's (100 ft) and revenue chain (33 ft).

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Tapes: cloth/linen (rough work), metallic (cloth with wire), steel (accurate, α ≈ 11.5 × 10−6/°C) and invar (36% nickel, α ≈ 0.2–1.2 × 10−6/°C, for base lines — expensive and soft).

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Accessories: arrows, pegs, ranging rods, offset rod, plumb bob, cross staff and optical square (for setting out right angles), spring balance and thermometer. • Ranging: direct ranging when the ends are intervisible, reciprocal ranging when they are not (two surveyors direct each other).

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Chaining on sloping ground is done by stepping (short horizontal steps with a plumb bob) or by measuring along the slope and applying a slope correction. • Chain-survey field work: a framework of well-conditioned triangles (angles between about 30° and 120°) based on a long base line, with check lines (to verify plotting) and tie lines (to locate detail); detail is picked up by offsets — perpendicular (short, ≤ 15 m) or oblique — recorded in a single- or double-line field book, and plotted with a scale and set square. • Errors are cumulative (always of the same sign — wrong chain length, temperature, sag, slope not corrected) or compensating (equally likely + or − — fractional reading, plumbing).

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If the chain is too long, the measured distance is too small, and true length = measured length × (actual chain length ÷ nominal length); the same ratio squared applies to areas.

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Correction to a taped length Formula and sign Standardisation (absolute length) Ca = L × (c/ℓ), c = excess (+) or shortage (−) of the tape over its nominal length ℓ Slope Csl = −h²/2L (h = difference of level) or −L(1 − cos θ) — always negative Temperature Ct = α(Tm − T0)L — positive if the field temperature exceeds the standard temperature Pull (tension) Cp = (P − P0)L/(AE) — positive if the applied pull exceeds the standard pull Sag Cs = −W²L/(24P²) = −w²L³/(24P²) — always negative (a suspended tape measures too long); eliminated at the normal tension Reduction to mean sea level Cmsl = −L·H/R (H = mean height above MSL, R ≈ 6 370 km) — negative Plane-Table Surveying • Equipment: plane table with tripod, alidade (plain or telescopic), trough compass, spirit level, U-fork (plumbing fork) with plumb bob, drawing sheet, pins and water-proof cover. • Setting up has three steps: centring (plotted point over the ground station), levelling and orientation (making the plotted lines parallel to the ground lines) — by trough compass (rough) or by back sighting along a previously plotted line (accurate). • Methods: radiation (rays drawn from one station to detail points, distances measured); intersection (a point is fixed by rays from two stations — no distance measurement needed, used for inaccessible points); traversing (a plane-table traverse round the area); and resection (fixing the position of the table itself from plotted points) — the two-point and three-point problems, the latter solved by the mechanical (tracing-paper) method, the graphical (Bessel's) method or trial and error guided by Lehmann's rules; the great triangle of error is avoided when the station lies on the circle through the three known points (the 'danger circle'). • Merits: plotting is done in the field so nothing is omitted and errors are seen at once, no field book is needed, it is fast and cheap, and it suits small scales.

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Demerits: unsuitable in wet or windy weather, bulky equipment, no record of measurements for re-plotting at another scale, and lower accuracy.

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Compass Surveying Feature Prismatic compass Surveyor's compass Graduation 0°–360° whole-circle bearing, figures inverted 0°–90° in four quadrants (E and W interchanged) Graduated card attached to The magnetic needle (card rotates) The box (needle rotates over the card) Reading taken At the south end of the needle, through the prism, while sighting At the north end of the needle, by looking down after sighting Sighting and support Sight vane with fine wire; can be held in hand or on a tripod Plain sight vanes; usually needs a tripod • Magnetic declination changes with place and time (secular, annual, diurnal and irregular variations, the last caused by magnetic storms).

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True bearing = magnetic bearing ± declination (+ for east declination). • Local attraction — the deflection of the needle by nearby iron, steel, electric lines or magnetic rock — is detected when the difference between the fore and back bearings of a line is not 180°.

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Correction: start from a line whose FB and BB differ by exactly 180° (both stations free from attraction) and work round the traverse correcting the affected bearings, or use the included angles (which are unaffected if both stations are equally attracted) and recompute the bearings. • Compass traversing is quick but of low accuracy; it suits preliminary and small surveys, forest and geological work.

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Theodolite • Parts: levelling head and tribrach with foot screws, lower plate carrying the graduated horizontal circle, upper (vernier) plate with the verniers and plate bubble, lower and upper clamps with tangent screws, standards (A-frames) carrying the trunnion (horizontal) axis, telescope with vertical circle, altitude bubble and vertical-circle clamp, optical plummet or plumb bob, and the tripod.

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A transit theodolite is one whose telescope can be revolved (transited) through 180° in the vertical plane. • Least count = (value of the smallest main-scale division) ÷ (number of vernier divisions) — commonly 20″ or 1′ for vernier instruments, and 1″ for precise optical/electronic theodolites. • Temporary adjustments at every station: setting up and centring over the mark, levelling with the plate bubble and foot screws, and focusing — the eyepiece on the cross-hairs and the objective on the target, eliminating parallax. • Permanent adjustments:

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(i) the plate-bubble axis perpendicular to the vertical axis;

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(ii) the line of collimation perpendicular to the horizontal axis (collimation error, removed by changing face);

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(iii) the horizontal axis perpendicular to the vertical axis (trunnion/standards error, also removed by changing face);

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(iv) the altitude bubble and vertical-circle index in adjustment (vertical index error). • Measurement of horizontal angles: the method of repetition (the angle is measured several times by carrying the accumulated value round the circle — increases precision and eliminates graduation errors) and the method of reiteration (directions) (all directions are observed round the horizon from one station and closed on the first — suited to several angles at a station).

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Observations are taken on both faces (face left and face right) and the mean taken, which cancels collimation, trunnion and vertical-index errors. • Other uses: prolonging a line, measuring vertical angles and magnetic bearings, and setting out angles and lines.

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Tacheometry • Tacheometry determines horizontal distances and elevation differences indirectly, from staff intercepts observed with a tacheometer (a theodolite with stadia hairs) — fast, and valuable in broken or steep ground where chaining is difficult; accuracy is about 1 in 500 to 1 in 1 000. • Stadia (fixed-hair) method: the staff intercept s between the upper and lower hairs gives, for a horizontal sight, D = K·s + C, where K = f/i ≈ 100 is the multiplying constant and C = f + d the additive constant, which is made zero by an anallactic lens in external-focusing instruments (and is negligible in modern internal-focusing ones). • For an inclined sight with a vertical staff and vertical angle θ:

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D = K·s·cos²θ + C·cos θ and the vertical component V = K·s·(sin 2θ)/2 + C·sin θ;

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RL of staff station = RL of instrument axis + V − central hair reading (with the sign of V following the angle).

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With the staff held normal to the line of sight, D = (Ks + C)cos θ ± h sin θ. • Other systems: the movable-hair (subtense) method (fixed staff intercept, variable stadia interval), the tangential method (two vertical angles to two targets, no stadia hairs) and the subtense bar (a horizontal bar of known length observed with a theodolite).

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Tacheometry is used for contouring, detail survey, rough traversing and checking chained distances; it has been largely replaced by total stations and GNSS.

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This section covers the establishment of horizontal control by triangulation, trilateration and traversing — their principles, methods, strength of figures, computations, error adjustment and applications.

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Horizontal Control • A horizontal control network is a framework of points whose plan positions (coordinates) are known to a stated accuracy; all later detail survey and setting out is referred to it, following the principle of working from the whole to the part.

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It is established by triangulation, trilateration, traversing or, today, by GNSS. • Networks are classified by order (first, second and third order / primary, secondary and tertiary) according to the accuracy and spacing of the stations.

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Triangulation • Principle: if one side (the base line) of a triangle and all its angles are measured, the other sides can be computed by the sine rule; a chain of connected triangles therefore fixes many points from one measured length — the classical method when distance measurement was difficult and angle measurement easy. • Classification (typical values): first (primary) order — triangle closure ≈ 1″, base accuracy ≈ 1 in 106, sides 30–150 km, for national geodetic control; second order — closure ≈ 3″, sides 8–65 km; third order — closure ≈ 6″–12″, sides 1.5–10 km, for local control. • Figures/layouts: chain of simple triangles (rapid and economical, but few checks — used for narrow strips), braced (geodetic) quadrilaterals — the strongest figure with the most checks and best-conditioned computation routes — and centred polygons (for area coverage).

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A well-conditioned triangle has angles near 60° and none smaller than about 30° or greater than 120°, because the computed length is least affected by angular error in that range. • Strength of figure (US Coast and Geodetic Survey):

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R = (D − C)/D × Σ(δA² + δAδB + δB²), where D is the number of directions observed, C the number of conditions, and δA, δB are tabulated factors depending on the distance angles; the smaller the value of R, the stronger the figure.

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It is used to choose between alternative routes of computation and between alternative figures. • Field work: reconnaissance and selection of stations (intervisible, well-conditioned figures, stable ground, accessible); station marking with permanent pillars and reference marks; signals and towers where necessary; base-line measurement (invar tape or EDM) with all corrections, and base extension through a base net; measurement of angles by repetition or directions with a precise theodolite; satellite station and reduction to centre when a station cannot be occupied; intervisibility and height of stations — allowing for curvature and refraction, the required clearance is h = 0.0673 D² m with D in km. • Computation: triangle closure and its distribution, spherical excess for large triangles, side computation by the sine rule, azimuth transfer and coordinate computation, followed by least-squares adjustment of the network (1.6).

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Trilateration • Trilateration fixes the network by measuring all the sides (with EDM or GNSS baselines) and computing the angles from them; it became practical once EDM made distance measurement quicker and more accurate than angle measurement. • Advantages: fast, accurate over long distances, works in poor visibility (with EDM) and needs no elaborate angular observing programme.

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Limitations: fewer internal checks than triangulation unless redundant lines are measured, all lines must be intervisible for EDM, and scale errors propagate; height of the stations and meteorological corrections must be applied.

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In practice triangulateration (both angles and distances measured) is used, and today most control is established by GNSS, which needs no intervisibility.

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Traversing • A traverse is a series of connected lines whose lengths and directions are measured.

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Open traverse — starts at a known point and ends at an unknown one (no check, used for routes); closed traverse — either a loop returning to the starting point or a link (connecting) traverse running between two known points, both of which provide checks. • Angle measurement: by included angles (interior or exterior), by deflection angles (right or left of the prolonged line — for route surveys) or by bearings (compass or gyro). • Checks on angles: sum of interior angles = (2n − 4) × 90°; sum of exterior angles = (2n + 4) × 90°; algebraic sum of deflection angles = 360°; for a link traverse the computed final azimuth must agree with the known one. • Computation: reduce lengths to horizontal, compute the bearing of each line, then the latitude (ΔN) = l cos θ and departure (ΔE) = l sin θ; for a closed loop ΣΔN and ΣΔE should be zero.

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The closing error e = √[(ΣΔN)² + (ΣΔE)²], its direction tan α = ΣΔE/ΣΔN, and the relative precision = e ÷ perimeter, expressed as 1 in N (e.g., 1 in 5 000 for ordinary engineering work). • Adjustment: the angular error is distributed first (equally, or in proportion to the number of observations); then the linear misclosure is distributed by the Bowditch (compass) rule — correction to the latitude/departure of a line ∝ length of that line (used when angles and distances are of comparable accuracy) — or by the transit rule — correction ∝ the latitude/departure of the line (used when the angles are more accurate than the distances); also the axis method, and rigorous least-squares adjustment (1.6).

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Computations are tabulated in a Gale's traverse table, and coordinates are then obtained by successive addition; the area follows from the coordinates by the cross-multiplication rule A = ½|Σ(xi(yi+1 − yi−1))|. • Omitted measurements (a missing length or bearing) can be computed from the condition that the latitudes and departures must sum to zero — but then no check remains. • Applications: control for topographic and cadastral surveys, route and construction surveys, city surveys, boundary demarcation, and densification of GNSS control.

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This section covers levelling and its types, direct and indirect levelling (spirit, precise, reciprocal and trigonometric levelling), the errors and corrections involved, and the temporary and permanent adjustments of levelling instruments.

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Definitions • Level surface — a curved surface everywhere perpendicular to the direction of gravity (e.g., mean sea level); horizontal plane — tangent to it at a point; datum — the surface from which elevations are reckoned (usually mean sea level); reduced level (RL) or elevation — the height of a point above the datum; benchmark (BM) — a point of known RL:

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GTS benchmarks, permanent, arbitrary and temporary benchmarks. • Back sight (BS) — the first reading after setting up, on a point of known RL; fore sight (FS) — the last reading before moving, on a change point; intermediate sight (IS) — any other reading; change (turning) point — a point on which both an FS and a BS are taken; height of instrument (HI) = RL of the point + BS.

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Instruments and Types of Levelling • Instruments: dumpy level (telescope rigidly fixed), tilting level (fine levelling of the line of sight for each sight), automatic (self-levelling) level with a compensator, and the digital level with a bar-coded (invar) staff which reads and records automatically.

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Levelling staves: telescopic/folding E-type staff, target staff and invar staff for precise work; tripod and change plate complete the outfit. • Direct (spirit) levelling — heights compared with a horizontal line of sight: simple levelling, differential (compound/fly) levelling between distant points, profile (longitudinal section) and cross-section levelling for routes, check levelling, precise levelling and reciprocal levelling. • Indirect levelling — trigonometric levelling (heights from vertical angles and distances, V = D tan θ, with curvature and refraction correction for long sights), barometric levelling (from air pressure — rough) and hypsometric levelling (from the boiling point of water). • Reciprocal levelling is used when the instrument cannot be set midway — across a river or valley: observations are taken from both banks, and the mean of the two apparent differences of level gives the true difference, eliminating collimation error, curvature and refraction together. • Precise levelling: invar double-scale staff with micrometer, short and balanced sights (≤ 50 m), line of sight kept well above the ground, double runs in opposite directions, with a permissible misclosure of about ±4√K mm (K in km) for first-order work and ±12√K mm for ordinary work.

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Booking and Reduction • Height of instrument (collimation) method:

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RL of any point = HI − (IS or FS).

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Fast, but intermediate sights are not fully checked.

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Arithmetic check: ΣBS − ΣFS = last RL − first RL. • Rise and fall method: the difference between consecutive staff readings gives a rise (previous reading larger) or a fall;

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RL of a point = previous RL ± rise/fall.

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Slower but checks every reading.

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Arithmetic check: ΣBS − ΣFS = ΣRise − ΣFall = last RL − first RL. • Closing error is distributed in proportion to the distance (or to the number of set-ups) round the circuit.

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Errors and Corrections Source Effect and remedy Collimation error Line of sight not horizontal when the bubble is centred — eliminated by balancing back-sight and fore-sight distances; checked by the two-peg test Curvature of the earth Makes readings too large; correction = −d²/2R = −0.0785 d² m (d in km) Refraction Bends the ray downwards, making readings too small; correction ≈ + (1/7) of the curvature correction = +0.0112 d² m Combined curvature and refraction −0.0673 d² metres with d in kilometres (e.g., −6.73 mm at 300 m, −67 mm at 1 km) — also eliminated by equal sight lengths Instrumental Defective staff graduation, unequal bubble sensitivity, loose tripod, index error — calibration and careful use Personal Parallax (focus the eyepiece and objective carefully), staff not held vertical (use a staff bubble or wave the staff and take the minimum reading), wrong booking, reading the wrong hair Natural Wind, heat shimmer near the ground, settlement of the tripod or change point — short sights, firm change plates, shaded instrument Adjustments of the Level • Temporary adjustments (at every set-up): setting up the tripod, levelling with the foot screws (bubble parallel to two foot screws, then perpendicular), and focusing the eyepiece on the cross-hairs and the objective on the staff to remove parallax. • Permanent adjustments of a dumpy level:

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(i) the bubble-tube axis perpendicular to the vertical axis (so the bubble stays centred when the telescope is turned) — adjusted by the reversal method;

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(ii) the horizontal cross-hair perpendicular to the vertical axis;

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(iii) the line of collimation parallel to the bubble-tube axis — the main adjustment, tested by the two-peg test: readings are taken on two pegs from a midpoint (which gives the true difference of level regardless of collimation error) and then from a point outside the line; the difference between the observed and expected readings gives the collimation error, which is removed with the diaphragm screws.

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This section covers the planning, reconnaissance, monumentation, control survey and detailing of a topographical survey, together with contouring, mapping and drafting.

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Purpose and Stages • A topographic survey records the natural and artificial features (rivers, vegetation, roads, buildings, boundaries, utilities) and the relief (height information, usually by contours or a DEM) of an area, and presents them on a map or in a digital database. • Stages:

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(1) planning — purpose and users of the map, extent, scale and contour interval, accuracy standards, coordinate system and datum, method (ground survey, GNSS, photogrammetry, LiDAR, UAV), equipment, manpower, cost and time;

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(2) reconnaissance — study of existing maps and control, a field inspection of the area, preparation of an index sketch showing proposed stations and routes, checking intervisibility and access;

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(3) monumentation — establishing permanent station marks (concrete pillars, iron pins, marked bolts), numbered and described on a station description card with a reference sketch and ties;

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(4) control survey — horizontal control by traverse, triangulation or GNSS, and vertical control by levelling connected to national benchmarks;

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(7) mapping, drafting and field check. • Detailing methods: radiation with a total station (the standard method — coordinates and codes recorded electronically), offsets from chain lines, plane tabling, GNSS-RTK, and photogrammetric/LiDAR plotting for large areas.

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Feature coding and field sketches are essential so that the points can be joined correctly in the office.

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Contours and Contouring • A contour is a line joining points of equal elevation; the contour interval is the constant vertical distance between successive contours, and the horizontal equivalent is the horizontal distance between them (which varies with slope). • Choice of contour interval depends on the scale of the map (smaller scale → larger interval), the nature of the ground (flat ground needs a small interval:

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0.2–1 m; rolling 1–2 m; hilly and mountainous 5–25 m), the purpose and the time and funds available. • Characteristics of contours: contours close together indicate a steep slope and widely spaced contours a gentle slope; equally spaced contours mean a uniform slope; a contour is a closed curve either within the map or beyond it; two contours cannot cross except at an overhanging cliff, and they coincide only at a vertical cliff; contours cross a watershed (ridge) and a valley line at right angles, forming a V pointing upstream (uphill) in a valley and a U or rounded shape pointing downhill on a ridge; closed contours with higher values inside show a hill and with lower values inside a depression; a saddle or col appears where two hills meet. • Methods of contouring: direct method — the contour points themselves are located in the field by a level or total station (accurate but slow, suited to small areas and important projects); indirect methods — levels are taken at points of a grid of squares, along cross-sections of a route, by radial lines from a station, or by tacheometry/total station spot heights, and the contours are then interpolated between the spot heights (by estimation, arithmetic calculation or graphical construction).

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Today DEMs from photogrammetry, LiDAR and UAV photogrammetry generate contours automatically. • Uses of contour maps: determining the nature of the ground and drainage pattern, fixing the route and gradient of roads, canals and pipelines, computing earthwork volumes and reservoir capacity, delineating catchment areas, checking intervisibility between points, site selection and preparing sections.

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Mapping and Drafting • Plotting is done from coordinates (by computer, in CAD/GIS) or by protractor and scale in manual work; detail is drawn with conventional symbols and line weights, with contours as smooth continuous curves (every fifth contour thickened and labelled). • A complete map sheet carries a title, scale (RF and bar scale), north arrow, grid or graticule with coordinates, legend of symbols, contour interval, projection and datum, sheet number and index, date of survey and the surveying authority; the Survey Department of Nepal publishes topographic maps at 1:25 000 and 1:50 000, and cadastral maps at large scales. • Quality is ensured by field checks of plotted detail, independent check measurements, edge matching between sheets, and stated horizontal and vertical accuracy standards (for example, that 90% of well-defined points are within a stated tolerance and that contours are correct to half the contour interval).

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Digital output is stored with metadata, layers and attributes for use in GIS.

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This section covers the theory of measurements and errors, sources and types of error, accuracy and precision, least-squares adjustment by observation and condition equations with linearisation and its application to intersection, resection, traverse, triangulation and trilateration, the propagation of errors, and variance, covariance, correlation and regression.

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Theory of Measurements and Errors • No measurement is exact: the true value is unknown, and the most probable value (MPV) is estimated from redundant observations.

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Observations may be direct, indirect (computed from others) or conditioned.

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Error = observed value − true value; residual (v) = observed value − most probable value; discrepancy = difference between two measurements of the same quantity. • Sources: instrumental (imperfect construction or adjustment — collimation, index error, tape length), natural (temperature, refraction, wind, magnetic field, curvature) and personal (imperfect sight, judgement, setting). • Types: mistakes (blunders) — gross errors from carelessness, removed by checks and repetition; systematic (cumulative) errors — follow a physical law, the same sign, can be computed and corrected (tape standardisation, curvature, collimation); random (accidental, compensating) errors — small, equally likely to be + or −, they obey the laws of probability and are treated by least squares. • Laws of accidental error (normal distribution): small errors occur more frequently than large ones, positive and negative errors of the same size are equally likely, and very large errors do not occur. • Accuracy = closeness to the true value (freedom from systematic error and blunders); precision = closeness of repeated observations to each other (small random scatter).

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A precise survey may still be inaccurate.

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Statistical Measures and Weights • For n equally weighted observations of one quantity: most probable value = arithmetic mean x̄; residuals vi = xi − x̄; standard deviation σ = √(Σv²/(n − 1)); standard error (deviation) of the mean σm = σ/√n; probable error = 0.6745 σ (50% probability).

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Probability limits: ±1σ ≈ 68.3%, ±2σ ≈ 95.4%, ±3σ ≈ 99.7%. • Weights: w ∝ 1/σ² — a more precise observation gets a greater weight; in levelling w ∝ 1/(length of route) or 1/(number of set-ups); the weighted mean = Σwx/Σw, and σ0 = √(Σwv²/(n − 1)). • Propagation of errors: for y = f(x1, x2, …, xn) with independent observations, σy² = Σ(∂f/∂xi)² σi².

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Special cases: the error of a sum or difference of two quantities = √(σ1² + σ2²); the error of the sum of n equal measurements = σ√n; the error of a mean = σ/√n; for a product the relative errors combine in quadrature. • Variance, covariance and correlation: variance σ² measures dispersion; covariance σxy = Σ(x − x̄)(y − ȳ)/(n − 1) measures how two quantities vary together; the correlation coefficient r = σxy/(σxσy) lies between −1 and +1 (0 = uncorrelated).

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In adjustment, the dispersion of all the unknowns is held in a variance-covariance matrix, whose diagonal gives the variances and whose off-diagonal terms give the covariances; error ellipses are drawn from it. • Regression fits a relationship to observed pairs by least squares — for a straight line y = a + bx, b = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)² and a = ȳ − b x̄; used for calibration, deformation trends and transformations.

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Principle and Methods of Least Squares • Principle: the most probable values of the observed quantities are those that make the sum of the weighted squares of the residuals a minimum — Σpv² = minimum (for normally distributed random errors this also gives the maximum-likelihood estimate). • Redundancy (degrees of freedom) = n − u, where n = number of observations and u = number of unknown parameters (or the number of independent conditions in the condition method).

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Adjustment requires n > u. • Method of observation (parametric) equations: each observation is written as a function of the unknown parameters — v = A x̂ − l, where A is the design matrix, x̂ the corrections to the approximate parameter values and l the observed-minus-computed vector.

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Minimising vTPv gives the normal equations ATPA x̂ = ATPl, hence x̂ = (ATPA)−1ATPl, with the cofactor matrix Qx = (ATPA)−1 and the a-posteriori variance factor σ0² = vTPv/(n − u); the variance-covariance matrix of the parameters is σ0²Qx.

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This is the method used in almost all modern adjustment software. • Method of condition equations (correlates): the observations must satisfy geometric conditions (angles of a triangle summing to 180°, closure of a level loop or a traverse).

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The conditions are written as B v = w (w = misclosure), and minimising vTPv subject to them gives the correlates k = (BP−1BT)−1w and the residuals v = P−1BTk.

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It is convenient when the conditions are few and the parameters many. • Linearisation: most survey equations (distances, angles, azimuths) are non-linear in the coordinates, so they are expanded by a Taylor series about approximate values, keeping only the first-order terms; the adjustment is then iterated with the improved values until the corrections become negligible. • Applications: intersection (a point fixed by directions from two or more known stations), resection (the station fixed by directions to known points), traverse adjustment (rigorous alternative to Bowditch), triangulation (angle and station-adjustment conditions), trilateration and combined networks, level networks, GNSS baseline networks, and coordinate transformations.

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Least squares also provides statistical testing — the χ² test on σ0², data snooping for blunders, and reliability and error-ellipse analysis.