A reflection can measure a surface, but first you need to know which surface it sampled.
A navigation antenna on a pier hears a satellite directly and also receives some signal reflected from below. The second path can be a nuisance for positioning. If you model the geometry and observe how the signal strength oscillates, that same reflection can carry information about the height of the surface.
GNSS interferometric reflectometry, or GNSS-IR, uses the interference between the direct and reflected signals. In a simple horizontal-surface model, the frequency of the oscillation against the sine of satellite elevation tells you the vertical distance from the antenna to the effective reflector. Turning that distance into a water level, snow depth or surface elevation requires more evidence.
We will derive the short equation, calculate a toy height and then test what happens when the reflecting patch moves, the antenna moves, or two surfaces compete. These are measurement checks you can carry to other kinds of remote sensing, where the instrument’s footprint is rarely a single point.
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Choose your reading route
Route 1 · Chapters 1–3: Understand why the signal oscillates
See the two paths before writing an equation
Connect the extra path, wavelength and measured frequency.
Route 2 · Chapters 4–5: Work out what was sampled
Work out where the reflecting patch lies
Compare elevation arcs and surface footprints.
Route 3 · Chapters 6–8: Turn distance into a level
Subtract from a declared antenna reference
Keep the antenna datum, timing and independent references straight.
Route 4 · Chapters 9–10: Recognise misleading peaks
Handle two plausible surfaces without picking the prettier peak
Test multiple surfaces and changes in instrument or processing.
Route 5 · Chapters 11–12: Transfer and report the result
Transfer the method from water to snow
Check a snow case and assemble a traceable observation record.
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Understand why the signal oscillates
See the two paths before writing an equation
Count cycles on the right horizontal axis
Separate the repeating component from the broad trend
Work out what was sampled
Work out where the reflecting patch lies
Choose arcs with enough information
Turn distance into a level
Subtract from a declared antenna reference
Keep the time support of a moving surface
Compare with a reference that can answer the question
Recognise misleading peaks
Handle two plausible surfaces without picking the prettier peak
Recognise changes that look like surface motion
Transfer and report the result
Transfer the method from water to snow
Write a result another observer can reproduce
CHAPTER 1 OF 12 · Understand why the signal oscillates
1. See the two paths before writing an equation
Imagine the same sound reaching your ear by a direct path and by a reflection from a wall. When the two waves arrive in step, their amplitudes can reinforce one another. When they arrive out of step, they can partly cancel. A GNSS antenna can receive a direct satellite signal and a weaker reflected contribution. As the satellite moves, the relative path length changes, and the measured strength can show a repeated pattern.
A receiver used for positioning usually seeks to limit the effect of reflections. Reflectometry asks a different question: does the changing reflection encode a useful property of a surface? Here we focus on estimating a height from signal strength, not on correcting navigation positions or assuming every kind of multipath comes from a flat surface.
Draw the antenna above a flat surface and imagine its mirror image below. In the far-field, specular, horizontal approximation, the extra path is about 2H sin e, where H is antenna height above the reflecting plane and e is the satellite elevation angle above the horizon. The two comes from the separation between the real antenna and its mirror image. The sine projects that separation along the incoming direction.
If H is 3 m, an elevation with sin e = 0.10 gives 0.60 m of extra path. At sin e = 0.20 it gives 1.20 m. The antenna and reflector did not move between these two calculations. The path changed because the viewing geometry changed. That is the source of information, not a reason to correct the oscillation away.
CHAPTER 2 OF 12 · Understand why the signal oscillates
2. Count cycles on the right horizontal axis
One cycle of interference corresponds to a relative path change of one wavelength λ in the ideal model. If x = sin e, the extra path grows by 2H for one unit increase in x. The oscillation frequency in cycles per unit x is therefore f = 2H/λ, or H = λf/2. This f is neither the radio carrier frequency in hertz nor the number of oscillations per second.
Take a wavelength of 0.190 m and a fitted oscillation frequency of 30 cycles per unit sine of elevation. The reflector height is 0.190 × 30/2 = 2.85 m. If x changes only by 0.15 over the selected arc, the model predicts about 30 × 0.15 = 4.5 cycles in that arc. The fitted frequency and the observed number of cycles are related, but they are not interchangeable.
Do not feed the raw elevation angle in degrees into the same formula. Sine is nonlinear, so equally spaced degrees do not generally give equally spaced values of x. Nor is the satellite’s rate of elevation change necessarily constant in time. Use the specified geometric coordinate and check whether the sampling method handles uneven spacing.
For a transfer check, hold the surface height at 2.85 m but change the wavelength to 0.244 m. The expected frequency is 2 × 2.85/0.244 ≈ 23.36 cycles per unit x. The two signals can indicate the same height with different oscillation frequencies. Check height after the conversion rather than expecting the peaks to line up on a raw frequency axis.
CHAPTER 3 OF 12 · Understand why the signal oscillates
3. Separate the repeating component from the broad trend
Recorded signal-to-noise or carrier-to-noise indicators can include a broad trend with elevation because antenna response and other factors change. A common analysis isolates an oscillatory component after suitable conversion and detrending. Do not assume the stored unit is linear amplitude. If values are reported in logarithmic units, identify the receiver convention and the quantity you need before converting.
The simple picture is a slow background plus a sinusoidal term in x. A spectral method can look for the frequency of that residual, including when x samples are uneven. A Lomb–Scargle analysis is one commonly used approach in GNSS-IR processing. A peak is a candidate frequency, not a verdict about the identity of a reflecting surface.
Consider a quiet teaching arc where changing the detrending window shifts the dominant peak from 30 to 31 cycles per unit x. For a 0.190 m wavelength the inferred height changes by 0.095 m. If your intended comparison needs a few centimetres, that processing sensitivity matters. A neat plot after one preferred detrending choice does not resolve the ambiguity.
Record the signal, wavelength, conversion, arc bounds, detrending choice and any rejection thresholds. Test a plausible range of settings before trusting a small change. A physical reflector should not disappear solely because you made a modest, defensible change to a preprocessing parameter, though complicated sites can offer several plausible interpretations.
CHAPTER 4 OF 12 · Work out what was sampled
4. Work out where the reflecting patch lies
In a simple plane geometry, the horizontal distance from the antenna’s vertical projection to the specular point is about H divided by tan e, in the satellite azimuth direction. Lower elevations move that point farther away. An actual reflection has a footprint rather than coming from a mathematical point, and the contribution depends on the surface, antenna and geometry.
At H = 3 m and e = 10°, the nominal horizontal distance is about 17 m. At e = 20° it is about 8.2 m. If the pier ends 12 m away along that azimuth, the two elevations may not be sampling the same kind of surface. A broad elevation arc could cross from water to decking even though the receiver never moved.
Change the azimuth by 90° and the same elevation and height can point to a different environment. One sector might be open water while another includes a seawall or roof. Plot the candidate footprint using a site sketch, and choose directions and elevations that can plausibly sample the intended reflector. A sector label such as “east” needs enough detail to be reproducible.
The specular point is a starting guide, not a hard footprint boundary. Surface roughness and the first Fresnel region matter. A small tide pool, snow patch or narrow drain can occupy only part of the reflecting area. If the target surface is fragmented, the inferred height can represent the dominant coherent contribution rather than the average height of all material within view.
CHAPTER 5 OF 12 · Work out what was sampled
5. Choose arcs with enough information
The sine-of-elevation span controls how much oscillation you can observe at a given height. An arc with x span 0.06 and f = 30 covers 1.8 cycles. An arc with x span 0.20 at the same height covers six cycles. More usable cycles generally give more leverage to distinguish frequencies, but only if the reflecting surface and signal remain appropriate over the longer arc.
A wider arc can run over the shoreline or span a rapidly changing tide. A narrower arc can isolate the intended patch but make the frequency estimate uncertain. Do not solve one problem by hiding the other. Compare separate arcs, check whether their inferred heights agree after accounting for time, and state where the data cannot separate alternatives.
The nominal width of a spectral peak should not be confused with the precision of a height estimate. A denser grid of trial frequencies gives a more finely drawn plot; it does not create additional observations. Measurement noise, arc geometry, multiple reflections and changing surface height still limit what you can infer.
Treat rising and setting arcs as separate opportunities with different time and spatial support. If both produce similar results, document whether they actually illuminate the same surface under comparable conditions. If they disagree, look for a changing level, different footprint, signal issue or site asymmetry before simply averaging them.
CHAPTER 6 OF 12 · Turn distance into a level
6. Subtract from a declared antenna reference
The reflector height H is a separation. To estimate a surface elevation z on a given vertical reference, use z = zA − H in the horizontal-plane model, where zA is the antenna elevation on that reference. Specify what antenna reference or effective observation model is used. A distance alone does not establish a height above a chart or land datum.
Suppose an antenna elevation is 5.20 m on a local vertical reference and the reflector height is 2.85 m. The inferred surface elevation is 2.35 m. Later H falls to 2.55 m while zA is unchanged. The inferred surface elevation rises to 2.65 m, a change of +0.30 m. The shorter reflector height means the surface moved toward the antenna.
Now suppose the antenna was lowered by 0.10 m between observations. Using the unchanged antenna elevation would attribute the full +0.30 m to the surface. Using zA = 5.10 m for the later observation gives 2.55 m, a rise of +0.20 m relative to the earlier 2.35 m. The relative measurement combined both movements.
This is why station maintenance records and independent surveys matter. If a mounting, antenna, radome or receiver changes, it may affect the geometry or how the signal is observed. Determine which part of the comparison changed before applying a datum correction. A break can be a real surface change, an instrument step or both.
CHAPTER 7 OF 12 · Turn distance into a level
7. Keep the time support of a moving surface
A satellite arc occupies an interval. A fitted single height compresses information gathered across that interval. If the reflector height changes substantially during it, the phase is no longer described by one constant H multiplying sin e. The fitted peak can reflect the evolving geometry and level together.
A tide illustrates the issue. In a fictional 40-minute arc the independent gauge rises from 1.10 to 1.42 m. Comparing the GNSS-IR result with the gauge value only at the beginning or only at the end could misrepresent the method. Compare compatible time support or use an appropriate time-varying model if the data and purpose support it.
With 2H(t) sin e(t) as a simple path model, its time derivative includes both a change in elevation and a change in height. An analysis that attributes every phase change to viewing geometry alone is vulnerable when the surface moves rapidly. This does not imply that every short arc needs a dynamic fit. It tells you what assumption to inspect.
A high-frequency level product should say which measurements support each time and where intervals were filled or estimated. Do not treat a smooth interpolated curve as a set of directly observed values. Where the original observations do not resolve the desired time variation, report that limit instead of reading the curve as evidence.
CHAPTER 8 OF 12 · Turn distance into a level
8. Compare with a reference that can answer the question
A nearby tide gauge or independently surveyed surface can help check the result. First confirm vertical datum, time alignment and location. A difference may be real if the gauge is in a sheltered basin and the reflected footprint is outside it, or if they represent different time intervals. An independent reference can still be the wrong reference for the question.
Use the same sign convention for all differences, for example GNSS-IR surface elevation minus gauge elevation. In a fictional matched series, three differences are +0.04, +0.05 and +0.06 m. They suggest a positive offset for those observations. They do not distinguish between an antenna-height error, a datum mismatch and a persistent site effect without another check.
Separate method calibration from validation. If the gauge was used to choose a datum offset or tune arc filters, checking those same observations mostly tells you how well the fitted relationship worked there. Hold out another period, tide range or viewing sector. Record where the reference itself is uncertain or disturbed.
A useful test can target a specific uncertainty. To check sign, compare a known rise and ask whether H fell. To check a station step, look at an independent antenna coordinate and simultaneous gauge. To check spatial representativeness, compare water sectors or nearby instruments. A single overall score cannot substitute for all three.
CHAPTER 9 OF 12 · Recognise misleading peaks
9. Handle two plausible surfaces without picking the prettier peak
Suppose a pier deck and a water surface both provide coherent reflections. In a toy spectrum at λ = 0.190 m, one peak at 18 cycles per unit x corresponds to 1.71 m, and another at 30 corresponds to 2.85 m. The stronger peak need not be from the water. Antenna response, surface roughness and the illuminated patch can alter their relative amplitudes.
If the water rises while the deck is fixed, look for a peak that shifts in the expected direction, to a smaller reflector height. If a candidate remains stable through a documented rise, it may be tied to a fixed feature, but check timing and sector before labelling it. The shape of the spectrum is one piece of evidence, not the whole inference.
Split the observations by azimuth and elevation where sensible. If the 1.71 m peak occurs mainly in the deck-facing sector and the 2.85 m peak occurs across open water, that pattern supports the competing-surface explanation. Changing your angle cuts after seeing the result can tune the fit. Test the chosen cuts on other arcs.
What if both candidate peaks are plausible and independent evidence is missing? Keep the observation ambiguous. A method can report no reliable surface height for an arc rather than producing one number at all costs. Include the ambiguous arcs in performance reporting so the apparent reliability is not inflated by selective omission.
CHAPTER 10 OF 12 · Recognise misleading peaks
10. Recognise changes that look like surface motion
A storm can alter both the water level and its roughness. It can change the coherence of the reflection and the set of arcs that survive quality checks. If the storm causes more missing values around the highest water, a plot of accepted estimates may underrepresent the event. Availability is part of the measurement, not a nuisance to delete.
The receiving setup matters too. Firmware and receiver changes can alter the reported strength indicator or tracking behaviour. An antenna replacement can change gain patterns and the effective receiving reference. Compare overlapping periods, station logs and independent observations before treating a step as a geophysical signal.
A sloping or uneven surface violates the flat-plane geometry used for the simple conversion. Different azimuths may then produce systematic height differences without any time change. Do not assume the horizontal model provides a literal vertical distance to every point on a slope. A more appropriate model or a more limited claim may be needed.
Keep rejected data with reason codes: no usable arc, conflicting peaks, mixed footprint, poor agreement or known equipment change. A clean time series whose difficult conditions disappeared from the record is not a complete picture of performance. Report the conditions under which the method could not identify a reflector.
CHAPTER 11 OF 12 · Transfer and report the result
11. Transfer the method from water to snow
Imagine a stable antenna above a broad open field. In a snow-free period the reflector height is 3.25 m. In a winter arc with comparable support the estimated reflector height is 2.92 m. The apparent rise is 0.33 m. Under a stable antenna and a comparable, surface-dominated reflection, it can be interpreted as an estimate of snow depth relative to the snow-free reference.
Now change the scene. The winter footprint partly includes a ploughed lane and partly includes deep snow. An oscillation may be dominated by one of those patches. The 0.33 m need not equal the mean depth across the field. Add an independent snow transect or compare sectors known to share similar surface cover.
Snow depth is not snow water equivalent. If a separately measured snow density were 250 kg per cubic metre and a uniform depth of 0.33 m applied, the illustrative water equivalent would be 0.33 × 250/1000 = 0.0825 m, or 82.5 mm. Reflector height alone does not supply that density or establish uniformity.
The changed context exposes the transfer limit: the geometric idea carries over, but the interpretation depends on what reflected the signal and how much area it represents. Do not convert one kind of inference into a second physical variable without the extra observations that variable needs.
CHAPTER 12 OF 12 · Transfer and report the result
12. Write a result another observer can reproduce
Retain a short evidence ledger: station and equipment, signal and wavelength, time and azimuth–elevation bounds, processing and spectral candidates, selected footprint, reference elevation, comparison instrument and uncertainty. If the reported height is a change rather than an absolute elevation, state the baseline interval and stability assumption.
Then explain one important limit in plain language. “The estimate applies to the open-water sector during these arcs. It does not represent the sheltered basin or the period without a usable reflection.” That is more useful than attaching a generic warning to a table of highly precise numbers.
Test a changed scenario before delivery. Would a rising reflector shorten the distance? Would a new signal change frequency while giving the same height? Could a different azimuth reach land? Would an antenna step look like a surface rise? If you can answer those questions from the report, the interpretation is less likely to be lost downstream.
The central habit is simple: name the measured separation, identify the reflecting support, and establish the reference before reporting a level. A repeatable oscillation gives a powerful clue. A well-tested geometry and independent evidence turn the clue into a defensible measurement.
Independent worked case and evidence checks
Use an independent fictional arc with wavelength 0.244 m and spectral frequency 22.0 cycles per unit sine of elevation. The reflector height is 0.244 × 22.0/2 = 2.684 m. With a compatible antenna elevation of 4.900 m, the inferred surface is 2.216 m on the local reference.
A second arc at 0.190 m gives 28.3 cycles per unit sine of elevation. Its inferred height is 2.6885 m and its surface elevation 2.2115 m. The separation between the two surface estimates is 4.5 mm. That agreement is useful only after confirming whether the arcs observed comparable times and footprints, and after accounting for shared errors.
Now change the second arc so it observes a sector with an obstruction and a dominant peak at 19.0 cycles per unit x. It gives a height of 1.805 m and an apparent elevation of 3.095 m. Do not conclude that the water rose by almost 0.9 m simply because the arithmetic worked. Check the footprint, independent level and alternate peaks. The appropriate result may be a rejected arc.
Finally, let both original arcs be correct in relative height but suppose the antenna elevation is actually 4.850 m. Both inferred surface elevations fall by 0.050 m. Cross-signal agreement is unchanged. This is a useful example of an error shared by two apparently independent calculations.
Worked comparison
| Observed clue | Check next | Do not conclude yet |
|---|---|---|
| Same peak frequency in different signals | Convert with each wavelength | Both surfaces have the same height |
| Peak stable through a tide | Check time and footprint | The water was motionless |
| Two methods agree on relative change | Check the shared datum | Absolute elevation is correct |
| No useful peak during a storm | Inspect sampling and rejection | No high water occurred |
Questions and practical checks
Is reflector height the water level?
No. It is the separation between the antenna reference and an effective reflecting surface under the chosen model. Use a compatible antenna elevation to obtain a level.
Why plot against the sine of elevation?
The simple horizontal-reflector path difference is linear in that quantity, so a stationary plane produces an approximately constant oscillation frequency in it.
Can a single antenna estimate several surfaces?
Different arcs or peaks can contain information about different surfaces. The surfaces must be distinguished with geometry, site evidence and appropriate checks before assigning heights.
Does a taller spectral peak mean a more accurate height?
Not necessarily. It can reflect a stronger unwanted surface or an artefact. Check the footprint, plausible height range, stability and independent comparison.
What if the ground is sloped?
The horizontal-plane formula may no longer represent the desired height. Consider a site-appropriate geometry or report the result as an effective estimate with its limit.
Continue with the connected subjects
Check the antenna and reflection
GNSS multipath repeatability: when reflections persist with satellite geometry
GNSS antenna reference points: understand the physical measurement reference
Satellite antenna offsets: distinguish endpoint geometry from the observed range
Check a changed vertical reference
Coordinate epochs: compare positions at compatible reference times
Geoid gradients: distinguish a height difference from shared reference uncertainty
Monument thermal expansion: test whether the antenna support moved
Sources and scope
Roesler and Larson: GNSS interferometric reflectometry processing tools and geometry
Original field study: GNSS-IR surface elevation, soil moisture and snow depth
Original Greenland study: GPS reflections and snow accumulation
The numerical cases in this guide are original fictional teaching examples, not measurements from a real site. This is an educational explanation, not a field safety instruction, engineering approval or an official certification. For a real project, use qualified practitioners, current product documentation and site-specific evidence. Sources were checked on 4 November 2026.