Question 5 of 5: Hydrographic Terms Defined

Question 5: Hydrographic Terms Defined (25 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Lead-line (leadline) method. The oldest direct depth-measurement technique: a marked line weighted with a lead sinker is lowered from the vessel until the lead touches bottom, and the depth is read where the line meets the water surface. The line is graduated (traditionally in marks and "deeps") and the lead's hollow base ("arming" with tallow) can sample the bottom material. It is slow, gives only spot depths, and is limited to shallow, slow-current water, but it remains a valid independent check on echo-sounder depths and for confirming a least depth over a small obstruction where an acoustic result is doubtful.

(b) Induced heave. True heave is the vertical translation of the vessel. Induced heave is the additional apparent vertical motion of the transducer caused by the vessel's roll and pitch acting through the lever arm between the motion-reference unit (MRU) and the transducer: when the boat rolls or pitches, a transducer offset horizontally from the rotation centre swings up and down even if the hull's centre does not translate. For a lever-arm offset $\ell$ and a roll (or pitch) angle $\theta$, the induced vertical throw is $\Delta z=\ell\sin\theta$.

Given. A transducer lever-arm offset $\ell=3.0\ \text{m}$ with a roll $\theta=5^\circ$ (this part), and a $-3$ dB beam width $BW=2^\circ$ at depth $d=40\ \text{m}$ (part e). Find. the induced heave, and later the half-power level in decibels and the bottom footprint of the beam.

1. Induced-heave magnitude. With a transducer offset $\ell=3.0\ \text{m}$ from the roll axis and a roll $\theta=5^\circ$: $$\Delta z=\ell\sin\theta = 3.0\sin 5^\circ = \boxed{0.26\ \text{m}}$$ This is why the MRU must know the lever-arm vector accurately: induced heave is computed from the attitude and the offset and removed, rather than being filtered like real heave.

(c) Shoal-biased surfaces. A gridded/binned depth surface (e.g. a CUBE or weighted-grid surface) built by, within each cell, keeping the shoalest (shallowest) depth rather than the mean. Because charts exist for safety of navigation, the surface is deliberately biased toward the shallow side so that a real shoal or obstruction inside a cell is never averaged away into a deeper, dangerous value. It is the digital equivalent of the cartographic rule "chart the least depth."

(d) Secchi disk. A weighted white (or black-and-white quartered) disk, typically 20–30 cm in diameter, lowered on a graduated line until it just disappears from view; the depth at which it vanishes is the Secchi depth $Z_S$, a simple field measure of water transparency (turbidity). It matters in hydrography because optical systems are clarity-limited: bathymetric-LiDAR penetration is roughly $2$–$3\,Z_S$ (Q1), so the Secchi disk is a quick predictor of whether an optical survey will reach the bottom.

(e) −3 dB points. The −3 dB (half-power) points define the beam width of a transducer: they are the angular directions where the radiated/received intensity has fallen to half its on-axis maximum. On the decibel scale a power ratio of one-half is

1. Half-power in decibels. $$10\log_{10}(0.5)=\boxed{-3.01\ \text{dB}}$$ so the −3 dB angular span is the beam width $BW$.
2. Bottom footprint from the beam width. The −3 dB beam illuminates a patch of seabed of diameter $$F = 2\,d\,\tan\!\left(\tfrac{BW}{2}\right)$$ For $BW=2^\circ$ at water depth $d=40\ \text{m}$: $$F = 2(40)\tan(1^\circ)=\boxed{1.40\ \text{m}}$$ so a "2° beam" at 40 m resolves the bottom to a ~1.4 m footprint — the fundamental horizontal resolution limit of that channel.