The atmosphere can look different in different directions
Two satellites appear at the same elevation, but their signals arrive from opposite sides of the sky. A model depending only on elevation predicts the same atmospheric contribution for both. If the atmosphere is directionally uneven, that symmetry can fail. Part of the resulting pattern may then compete with the receiver’s horizontal coordinates.
This guide explains north and east tropospheric-gradient parameters through original, deliberately small observation models. We first hold position fixed, recover a directional pattern, and check it against an unused direction. Then we restore horizontal position and discover why a beautifully balanced sky can still leave two parameters inseparable.
Every residual, gradient, mapping coefficient and uncertainty in the worked cases is fictional. The examples are not weather observations, operational mapping functions or receiver settings. Use the existing zenith-delay guide for the elevation-only height problem and the water-vapour guide for meteorological conversion; this article develops the additional azimuth-dependent question.
Geodesy subject guide · Learning that transfers into life · How X Works hub
Choose your reading route
Route 1 · Chapters 1–2: Define the directional delay
Route 2 · Chapters 3–5: Recover and check a gradient
Route 3 · Chapters 6–8: Restore position and geometry
Route 4 · Chapters 9–10: Audit uncertainty and alternatives
Distinguish shared observation errors from directional contrasts
Route 5 · Chapters 11–12: Transfer and report
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Define the directional delay
Separate propagation delay from ground motion
Declare azimuth and the simplified parameter equation
Recover and check a gradient
Construct four observations from known teaching parameters
Predict an unused azimuth and test model adequacy
Change the azimuth convention without changing the observation
Restore position and geometry
Restore horizontal position and expose a hidden ambiguity
Use changed elevation to provide a different sensitivity ratio
Propagate noise through the cardinal-direction estimates
Audit uncertainty and alternatives
Distinguish shared observation errors from directional contrasts
Keep atmospheric gradients distinct from weather conclusions
CHAPTER 1 OF 12 · Define the directional delay
1. Separate propagation delay from ground motion
A radio signal can acquire an extra atmospheric path delay without the antenna moving. Conversely, a moving antenna changes the geometric path even if the propagation model is unchanged. Both contributions enter the observation comparison. A coordinate estimate is therefore an inference from a combined model, not a direct label attached to every unexplained range residual.
The International GNSS Service identifies zenith path delay and north/east troposphere gradients as related but distinct products. IERS atmospheric-propagation modelling includes azimuth-dependent gradient terms. These sources establish why direction belongs in the model; the arithmetic below is an original teaching construction rather than a reconstruction of a particular station or product.
A zenith-delay parameter describes a chosen vertical reference contribution. A gradient parameter describes a directional departure within a specified representation and mapping rule. It is not automatically a local humidity derivative in familiar weather-map units. Before comparing two reported gradients, retain their parameter definition, units, mapping function, reference direction and treatment of already-modelled delays.
A useful first question is: what changed between the observations being compared? Elevation, azimuth, time, signal, antenna environment and processing choices can all matter. Isolating one of these in a classroom example helps explain its signature. It does not show that the other contributions are absent from real measurements collected at different times.
CHAPTER 2 OF 12 · Define the directional delay
2. Declare azimuth and the simplified parameter equation
Measure azimuth A clockwise from north towards east: north is zero degrees, east ninety, south one hundred eighty and west two hundred seventy. At one stipulated elevation and epoch, write the simplified residual model r(A) = q+k[GN cosA+GE sinA]. Here q is the common residual at that elevation, and GN and GE are north/east gradient parameters.
For these exercises, q, GN, GE and r have millimetre units, while k is a known dimensionless sensitivity. Choosing k = 2 makes the arithmetic easy. It is not a claim that a real gradient mapping factor equals two at any universal elevation. A real analysis must use its actual propagation model and documented parameter definitions.
The sine and cosine terms encode a first directional harmonic. The gradient contribution changes sign in opposite directions: cos(A+180°) = −cosA and sin(A+180°) = −sinA. The common q does not change sign. That difference allows opposite-direction contrasts to distinguish a common component from a directional one when the other assumptions hold.
This representation is a model, not a promise that every atmospheric pattern has only one directional harmonic. Sharp spatial structure, timing differences or unmodelled site effects may leave a pattern it cannot describe. Keeping an independent direction for a check is more informative than assuming that an exact fit to the determining observations proves the atmosphere was completely represented.
CHAPTER 3 OF 12 · Recover and check a gradient
3. Construct four observations from known teaching parameters
Choose fictional q = 18 millimetres, GN = 3 millimetres, GE = −2 millimetres and k = 2. The north observation is rN = 18+2×3 = 24. The south observation is rS = 18−2×3 = 12. Their different residuals reflect the directional term while sharing the same common contribution.
The east observation is rE = 18+2×(−2) = 14 millimetres. The west observation is rW = 18−2×(−2) = 22. A negative GE means the directional contribution is smaller towards east than towards west under this definition. It does not mean there is negative atmospheric water or that a total physical delay has become negative.
Recover the parameters from the observations rather than merely substituting the values used to generate them. The north–south difference is twelve millimetres, equal to 2kGN, so GN = 12/4 = 3. The east–west difference is minus eight, giving GE = −8/4 = −2. Either opposite pair has average eighteen, recovering q.
Finally reconstruct all four residuals from the recovered parameters. This reverse check tests sign and ordering, not just the final magnitudes. An implementation that silently interchanges north and east may still produce a plausible gradient length, but it will assign the pattern to the wrong direction and fail the directional reconstruction.
CHAPTER 4 OF 12 · Recover and check a gradient
4. Predict an unused azimuth and test model adequacy
Use a held-out azimuth A = 45 degrees at the same stipulated elevation and epoch. Since both sine and cosine equal 1/√2, the prediction is 18+2(3−2)/√2 = 18+√2, approximately 19.4142 millimetres. This value was not needed to recover the three parameters from the cardinal observations.
Suppose a fictional independent residual at that direction is 23 millimetres. Its observed-minus-predicted difference is approximately 3.5858. That discrepancy deserves an uncertainty comparison and investigation. It does not, by itself, establish whether the source is higher-order atmospheric structure, observation noise, an antenna-environment effect or a mismatch in the assumed effective epoch.
An exact cardinal fit can also conceal a missing directional term. Add H sin(2A) to the teaching model. It is zero at every cardinal direction, so none of the original four observations changes. At forty-five degrees it equals H. Choosing H = 3.5858 reproduces the held-out discrepancy without changing the recovered cardinal parameters.
This counterexample is deliberately algebraic. It proves that the selected observations do not test every possible directional pattern; it does not prove that a real discrepancy has that particular harmonic origin. A useful validation design chooses directions that discriminate between plausible models, rather than repeating observations only where the alternatives happen to agree.
CHAPTER 5 OF 12 · Recover and check a gradient
5. Change the azimuth convention without changing the observation
Another interface may measure azimuth anticlockwise from east. Call that angle B and define B = 90°−A. Then cosA = sinB and sinA = cosB. The same directional expression becomes GN sinB+GE cosB. The physical observation remains unchanged when the labels and trigonometric coefficients are translated together.
For the original north direction, A = 0 and B = 90 degrees. The translated expression gives GN×1+GE×0 = 3 millimetres before multiplication by k, just as before. For east, B = 0 and the expression gives GE. Checking two nonparallel directions exposes a swapped origin more reliably than comparing one gradient magnitude.
The gradient length is √(GN²+GE²), here √13, approximately 3.60555 millimetres. A consistent rotation of the horizontal coordinate description preserves that length while changing its components. Preserving the length alone is a weak convention test because a mistaken swap or sign reversal can preserve it too.
Record the reference direction rather than using the word azimuth as though it supplied an entire convention. The same habit applies to local coordinate axes, ocean-loading components and satellite orbit errors. Numbers become transferable when the directions attached to them travel with the values; a familiar unit is not a replacement for those directions.
CHAPTER 6 OF 12 · Restore position and geometry
6. Restore horizontal position and expose a hidden ambiguity
Now allow a north coordinate correction n and an east correction e. At a fixed elevation with horizontal direction-cosine magnitude cE, the first-order coordinate contribution is −cE[n cosA+e sinA]. Add that to the earlier model. The north-like coefficient becomes kGN−cE n, and the east-like coefficient becomes kGE−cE e.
Even complete azimuth coverage at that one elevation identifies only these combined coefficients when both position and gradients are free. Their sine and cosine signatures are exactly shared. The data cannot decide how much of the north-like pattern came from n and how much came from GN without another independent sensitivity or justified information.
Use fictional cE = 0.8 and k = 2. A combined north coefficient of 2.8 millimetres is consistent with n = 4, GN = 3 because −0.8×4+2×3 = 2.8. It is also consistent with n = 0, GN = 1.4. Both parameter pairs predict the same north-like pattern at every azimuth in this model.
Taking more observations around the same elevation circle can reduce noise in the measured combination, but it does not remove the exact ambiguity. This is a different limitation from missing eastern or western satellites. A balanced sky can be useful while still failing to distinguish parameters whose model columns are proportional under the chosen elevation restriction.
CHAPTER 7 OF 12 · Restore position and geometry
7. Use changed elevation to provide a different sensitivity ratio
Consider north–south half-differences d1 and d2 formed at two stipulated elevations. They remove the common terms under the same-epoch symmetric construction. Choose horizontal geometric coefficients c1 = 0.8 and c2 = 0.6, and fictional gradient sensitivities k1 = 2 and k2 = 0.5. The equations are d1 = −0.8n+2GN and d2 = −0.6n+0.5GN.
Generate d1 = 2.8 and d2 = −0.9 millimetres from n = 4 and GN = 3. The coefficient determinant is (−0.8)×0.5−2×(−0.6) = 0.8, which is nonzero. Unlike the one-elevation problem, these two contrasts distinguish the two parameters within the stated model.
Eliminating GN gives n = (0.5d1−2d2)/0.8. Substitution yields (1.4+1.8)/0.8 = 4 millimetres. The corresponding gradient solution is GN = (0.6d1−0.8d2)/0.8 = (1.68+0.72)/0.8 = 3. Reconstruct both contrasts before interpreting the result.
The selected k values are educational inputs, not a proposed operational mapping function. What matters structurally is that the ratio of gradient sensitivity to coordinate sensitivity changes between the two rows. If that ratio remained identical, changing the elevation label alone would not remove the ambiguity. Inspect the actual coefficients rather than treating different labels as independent information.
CHAPTER 8 OF 12 · Restore position and geometry
8. Propagate noise through the cardinal-direction estimates
First return to the fixed-position cardinal example and assume independent residual errors with equal standard uncertainty σ = 2 millimetres. Since GN = (rN−rS)/(2k), its variance is 2σ²/(4k²). For k = 2, that is 0.5 square millimetres, giving standard uncertainty approximately 0.707107 millimetres. GE has the same variance under these symmetric assumptions.
Estimate q using all four residuals: q = (rN+rS+rE+rW)/4. Its variance is 4σ²/16 = 1 square millimetre, so its standard uncertainty is one millimetre. The cardinal contrasts and this average are uncorrelated in this particular equal-noise independent construction. That property follows from symmetry, not from the names of the estimated parameters.
At the held-out forty-five-degree direction, the predicted residual is q+√2(GN+GE). Under the same simplified covariance account, its variance is 1+2×0.5+2×0.5 = 3 square millimetres. To compare it with an independent held-out observation of variance four, use the difference variance seven, not merely the uncertainty of one side.
These calculations describe the declared stochastic model. They do not include uncertainty in the sensitivity factor, unmodelled directional structure or shared atmospheric evolution. Reporting many decimal places in a formal uncertainty does not establish that those omissions are negligible. Keep the uncertainty of the fitted model separate from independent evidence about whether the model is adequate.
CHAPTER 9 OF 12 · Audit uncertainty and alternatives
9. Distinguish shared observation errors from directional contrasts
Add the same unknown residual offset a to all four cardinal observations. Their differences are unchanged, so the recovered GN and GE remain unchanged. Their average increases by a, so q absorbs the shared contribution. This algebra explains a cancellation in the synthetic design; it is not evidence that every clock or atmospheric error enters real data identically.
Now add a only to the north observation. GN changes by a/(2k), and q changes by a/4. With fictional a = 4 millimetres and k = 2, the fitted GN increases by one and q increases by one. A direction-specific observation problem can therefore contaminate both the directional and common estimates.
If errors at north and south have equal variance σ² and covariance C, the variance of their difference is 2σ²−2C. Its cancellation depends on the sign and size of the dependence. Applying an independence formula to reused or correlated observations can misstate the contrast uncertainty even when the subtraction itself is correct.
This is why repeated residual patterns need context. A nearby reflector, obstruction or processing artefact may recur with geometry. A model may fit part of that pattern using gradient parameters. The parameter estimate then describes the best fit under the chosen equation, not automatic proof that the entire pattern came from spatial variation in atmospheric refractivity.
CHAPTER 10 OF 12 · Audit uncertainty and alternatives
10. Keep atmospheric gradients distinct from weather conclusions
The north/east parameters in a GNSS delay model are not a direct map of cloud location, rainfall or surface humidity. Their relation to meteorological quantities requires the retrieval definitions and appropriate atmospheric information. The existing water-vapour guide explains another conversion interface; do not use a fitted directional coefficient as a substitute for that physical interpretation.
A gradient changing between two processing solutions may reflect changed observations, constraints, mapping, coordinates or parameter intervals. Check those interfaces before describing it as an atmospheric event. A numerical difference is useful evidence, but its cause is not supplied by the parameter label. Preserve the underlying rows or residual summaries needed to investigate.
Also distinguish simultaneous geometry from temporal coverage. North and south observations collected at different times need not share the same q or gradient. If q changes while the pair is assembled, their difference can contain time evolution. Our clean opposite-direction cancellation assumed compatible effective epochs; relaxing that assumption changes what the contrast observes.
For a practical comparison, first align time and conventions, then examine azimuth/elevation coverage, residual structure and sensitivity to credible processing alternatives. Add independent meteorological or geodetic evidence where the claim requires it. The right next step is the one that discriminates competing explanations, not the one that merely makes the fitted gradient look smoother.
CHAPTER 11 OF 12 · Transfer and report
11. Test whether a constraint adds independent knowledge
A coordinate prior can help resolve the one-elevation position–gradient ambiguity. Its usefulness depends on whether the reference point, frame, epoch and uncertainty are compatible. A north coordinate copied from a different antenna endpoint or epoch can force the gradient to absorb a discrepancy that is not atmospheric.
For d = −0.8n+2GN, solve GN = 0.5d+0.4n. If d has standard uncertainty one millimetre and an independent n prior has standard uncertainty two, the gradient variance is 0.25×1²+0.16×2² = 0.89 square millimetres. Its standard uncertainty is approximately 0.943398 millimetres.
Treating that coordinate prior as exact would report only 0.5 millimetres. The omitted uncertainty did not disappear because the prior came from a separate file. If the prior shares observations with d, a covariance term is also needed. Independence is a property of the evidence account, not of separate filenames or different software outputs.
A useful transfer question asks what happens if the coordinate prior is biased high by five millimetres. Under this equation, GN shifts high by two millimetres. A stable gradient can therefore coexist with a wrong external constraint. Compare the source of the constraint and its effect on predictions rather than equating numerical stability with independent physical validation.
CHAPTER 12 OF 12 · Transfer and report
12. Report the model, coverage and inference separately
A clear report specifies the azimuth origin and sense, elevation mapping, gradient units and definitions, coordinate reference, parameter intervals and constraints. It distinguishes the estimated directional coefficients from physical atmospheric attribution. Retain enough observation geometry and covariance information for another reader to reproduce the parameter-separation claim.
For the fixed-position example, q = 18, GN = 3 and GE = −2 millimetres reproduce the cardinal residuals. For the restored-position example at one elevation, those observations identify only combined directional signatures. With the changed sensitivity ratios in chapter seven, the synthetic north coordinate and gradient become separately identifiable. Each statement keeps its assumptions.
The existing height–atmosphere guide demonstrates an elevation trade-off with vertical position and zenith delay. This guide adds a different directional mechanism. Neither small model replaces a full GNSS adjustment. They teach how to read a design matrix as an account of what the observations can distinguish, rather than a black box promising every fitted parameter is independently measured.
The transferable lesson is to ask which signatures remain different after all unknowns are restored. Opposite directions help identify a pattern; changed elevation can alter the position–atmosphere ratio; independent evidence can support its interpretation. A useful estimate combines those steps without letting one attractive fitted number stand in for all three.
Independent worked case and evidence checks
Build a fresh fixed-position case with q = 10 millimetres, GN = −1, GE = 4 and k = 3. Predict north, south, east and west residuals before reading the answers: they are seven, thirteen, twenty-two and minus two millimetres. Negative residuals are allowed because these are adjustments relative to an already-applied model, not total physical path lengths.
Recover GN from (7−13)/6 = −1 and GE from [22−(−2)]/6 = 4. The four-way average is ten. An unused forty-five-degree direction is predicted as 10+3(−1+4)/√2, approximately 16.36396 millimetres. Reconstructing a different case tests whether the method transfers beyond the original signs and coefficients.
Now pretend the west observation is unavailable. North and south still determine q and GN in this restricted model, and east then determines GE. The solution remains unique, but its uncertainty and redundancy differ. There is no independent west-side check. The same number of estimated parameters does not imply the same strength of evidence.
With independent equal residual uncertainty σ, the north–south average has variance σ²/2. Since GE = (rE−q)/3, its variance is (σ²+σ²/2)/9 = σ²/6. Using a known q would have given σ²/9 instead. Estimating a common term from observations imports its uncertainty into the subsequent directional estimate.
Next restore a north position correction using cE = 0.6. The north-like coefficient is 3GN−0.6n. The pair GN = −1,n = 0 gives minus three millimetres; GN = 0,n = 5 gives the same coefficient. No number of exact azimuth observations at that one sensitivity ratio distinguishes those pairs.
Add an independent coordinate constraint rather than silently selecting one pair. Solve GN = (d+0.6n)/3. A three-millimetre change in n shifts GN by 0.6 millimetres. State the prior uncertainty and any shared evidence before propagating it. The physical interpretation remains conditional even when a numerical routine returns a unique result.
As a model-discrimination exercise, add a fictional term H sin(2A) with H = 5 millimetres. It vanishes in all four cardinal directions, so the recovered cardinal parameters do not change. At forty-five degrees the predicted residual increases by five. At one hundred thirty-five degrees it decreases by five. Those two directions distinguish this alternative from a simple common offset.
A common offset of five would increase every direction, including north and south. A north-gradient change would follow cosA, not sin(2A). Writing these competing signatures makes a proposed extra observation purposeful. The best additional direction is one where the candidate explanations predict different outcomes under compatible timing.
Finally consider a temporal failure: the north residual is observed while q = 10, but the south residual is observed later after q rises to sixteen, with the gradient unchanged. Treating them as a simultaneous pair makes the difference six millimetres too small. With k = 3, GN is biased by minus one millimetre. The error came from pairing incompatible events, not from a trigonometric calculation.
Keep the raw effective epochs in the teaching record. A second reader should be able to reproduce the valid pair and then deliberately break the timing assumption. This shows why coverage is more than points drawn on a sky plot: the points must also belong to the model interval whose parameters are being estimated.
Worked comparison
| Evidence pattern | What it can distinguish | What remains unresolved |
|---|---|---|
| Opposite azimuths at one elevation | Common versus directional residuals under compatible timing | Horizontal position versus gradient if both have proportional signatures |
| Another azimuth | A new test of the directional model | Every higher directional harmonic |
| Changed elevation sensitivity ratio | Position and gradient in the stated two-row model | Unmodelled errors or physical attribution |
| Coordinate prior | An extra constraint on a weak combination | Its own reference, uncertainty and possible bias |
| Small formal residuals | Agreement with the fitted equation | Independent accuracy or complete atmospheric realism |
Questions and practical checks
Does a north gradient mean the ground moved north?
No. A directional delay parameter and a coordinate correction have different definitions, even though their observation signatures can overlap.
Is full azimuth coverage sufficient?
Not always. At one elevation, free horizontal coordinates and gradients can enter proportional model columns.
Are negative gradient values unphysical?
No. They describe signed directional departures under a chosen convention, not negative total atmospheric water.
Does an exact cardinal fit validate every direction?
No. The independent sin(2A) counterexample agrees at all cardinal directions but differs elsewhere.
Can a coordinate prior be treated as exact?
Only when genuinely justified. Its uncertainty and any dependence belong in the gradient uncertainty account.
Continue with the connected subjects
Establish the delay and observation definitions
Separate zenith delay from height and clock parameters
Read GNSS wet-delay and water-vapour conversion with pressure-bias checks
Investigate competing geometric patterns
Test persistent multipath patterns instead of assigning every residual to atmosphere
Compare clock-reference choices and parameter observability
Sources and scope
International GNSS Service: zenith delay and north/east gradient products
IERS atmospheric propagation chapter: radio-delay mapping and directional gradients
IERS Conventions Centre: chapter version and update context
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 31 October 2026.
Compare the connected observation pathways
Keep signal-path effects separate from physical station displacement. These companion guides develop the related models with their own worked examples, units and reference checks.
Work through solid-Earth tidal deformation and coordinate conventions
Reconstruct ocean-tide loading from harmonic phases and component axes
Trace atmospheric-pressure loading through spatial anomalies and reference ledgers