Atmospheric Refraction Coefficient — κ
Computes k from real ERA5 measured data · ITU-R P.453 · Exact spherical geometry
A
Observer
Instrument or eye position
Location name ?
Latitude (°)
Longitude (°)
Elevation (m ASL)
B
Target
Remote point being observed
Location name
Latitude (°)
Longitude (°)
Elevation (m ASL)
Observation
Date (UTC)
Time (UTC)
Great-circle distance
Options
Units
Atmosphere mode
Lo (hPa)
Mid (hPa)
Hi (hPa)
Computed κ
—
refraction coefficient
Required κ for visibility
—
minimum for LOS
dN/dh
—
N·km⁻¹
Avg N
—
N-units
Ray lift
—
m at target
Curvature
—
m over path
Ray clearance profile — 2,000 integration points
Geometric
Refracted
Refractivity N profile — atmospheric slice
Sensitivity analysis
| Perturbation | κ | Δκ |
|---|---|---|
| Run analysis to compute sensitivity | ||
Atmospheric layers — ERA5 profile
| hPa | Alt (m) | T (K) | RH (%) | e (hPa) | N | k (local) |
|---|---|---|---|---|---|---|
| No data | ||||||
Line-of-Sight Visibility Analysis
Exact spherical geometry · Visible portion · Implied κ from observation
Uses observer and target from the Refraction panel. Run Refraction first to populate κ.
Computed visibility
Horizon arrival elevation
—
m above sea level
Visible above horizon
—
metres of target visible
Mountain profile diagram
Back-calculate κ from observation
Enter how many metres of the target you observed above the horizon cut. KAPPA will back-calculate the implied atmospheric κ and compare with the ERA5-computed value.
Observed visible metres
Elevation angles
| Point | Geometric (°) | Apparent (°) | Δ (″) |
|---|---|---|---|
| Run analysis first | |||
Path statistics
| Quantity | Value | Unit |
|---|---|---|
| Run analysis first | ||
Geodetic Levelling Corrections
Curvature + refraction · Staff reading · Angular refraction · ERA5 real κ
Observation geometry
Sight distance (km)
κ source
ERA5 κ: — — run Refraction panel first to populate.
Staff reading correction
Observed staff reading (m)
Instrument height (m)
For first-order levelling, use the Reciprocal Levelling module which eliminates the refraction term entirely.
c + r correction
—
mm
Corrected staff reading
—
m
Correction breakdown
| Correction | Value | Description |
|---|---|---|
| Earth curvature | — | d²/(2R) × 1000 mm |
| Refraction | — | −k × d²/(2R) × 1000 mm |
| Combined c+r | — | d²(1−2k)/(2R) × 1000 mm |
| Angular refraction | — | 206265 × k × d/R arcsec |
Comparative: standard vs ERA5 κ
| κ value | c+r (mm) | Error vs ERA5 |
|---|---|---|
| Standard k=0.13 | — | — |
| ERA5 measured k | — | 0.0 mm |
| Difference | — | — |
c + r = d² × (1 − 2κ) / (2R)
d = sight distance (km) · κ = refraction coefficient · R = 6371 km
Total Station & EDM Atmospheric Correction
Barrel & Sears group refractivity · Wavelength-dependent ppm · ERA5 humidity included
Instrument settings
Instrument wavelength
Measured slope distance (m)
Atmospheric conditions
κ source
Using ERA5 atmospheric profile — run Refraction panel first.
Atmospheric ppm correction
—
parts per million
Corrected distance
—
metres
Distance correction
—
mm applied
Wavelength used
—
nm
Group refractivity breakdown
| Term | Value | Description |
|---|---|---|
| N_g (standard) | — | 287.6155 + 4.8866/λ² + 0.068/λ⁴ |
| N_dry (actual) | — | 77.6P/T |
| N_wet (actual) | — | 3.73×10⁵·e/T² |
| N_total (actual) | — | Full ITU-R P.453 |
Wavelength sensitivity
| Wavelength | N_g | ppm vs 589nm |
|---|---|---|
| Run analysis to compute | ||
N_g = 287.6155 + 4.8866/λ² + 0.068/λ⁴
Barrel & Sears formula · λ in μm · Δd/d = (N_actual − N_standard) × 10⁻⁶ ppm
Reciprocal Levelling — Rueger Formula
Refraction-independent height difference · Back-calculated κ · Cross-validation with ERA5
Simultaneous reciprocal observations eliminate the refraction term. The computed height difference is independent of κ and can be compared with the ERA5-derived result.
Observations
A
From A, looking toward B
Zenith angle z_A (°′″ or decimal)
B
From B, looking toward A (simultaneous)
Zenith angle z_B (°′″ or decimal)
Slope distance S (m)
Instrument height A (m)
Reflector height B (m)
Observation timing (simultaneity check)
Per Rueger (1996) p.113 — refraction cancels only when both observations are truly simultaneous. Enter the time of each observation to verify.
Time of obs A (UTC)
Time of obs B (UTC)
Height difference A→B
—
metres (refraction-free)
Back-calculated κ
—
from reciprocal angles
ERA5 computed κ
—
from measured data
Single-direction error
—
mm vs reciprocal
Computation details
| Quantity | Value | Unit |
|---|---|---|
| z_A (decimal) | — | degrees |
| z_B (decimal) | — | degrees |
| cos(z_A) − cos(z_B) | — | dimensionless |
| Curvature term | — | m |
| Δh (A→B) | — | m |
| Corrected Δh (+ inst. heights) | — | m |
Δh = (S/2)(cos z_A − cos z_B) + (S²/2R)·sin²((z_A+z_B)/2)
Rueger formula · S = slope distance (m) · z in radians · R = 6,371,000 m
Refractivity N Profile
Full atmospheric slice · ERA5 measured profile · Gradient analysis
Profile populated automatically after running Refraction analysis.
Refractivity N vs Altitude
Temperature T vs Pressure
Full layer data
| hPa | Alt (m) | T (K) | T (°C) | RH (%) | e (hPa) | e_sat (hPa) | N | k (local) |
|---|---|---|---|---|---|---|---|---|
| Run Refraction analysis to populate profile | ||||||||
ERA5 Data Settings
Copernicus CDS connection · API key management · Data source status
Copernicus CDS API Key
Your free key from cds.climate.copernicus.eu fetches real measured ERA5 data (temperature, specific humidity, pressure at 137 levels) for your exact observation. Without it, KAPPA uses ICAO standard atmosphere — pressure and temperature only.
API Key
No key — ICAO standard atmosphere will be used.
How to get your free ERA5 key
1
Register at Copernicus CDS
Go to cds.climate.copernicus.eu → Register. Enter name, email, password. Verify email.
2
Accept ERA5 licence
Datasets → ERA5 hourly data on pressure levels → Licences tab → Accept Copernicus licence.
3
Copy your API key
Username → Profile → API Key. Copy the UUID string and paste above.
Alternative: Python script
For offline use, run the included
era5_fetch.py script. Paste the output JSON here:Connection status
ERA5 API key
Not entered
Data source
ICAO standard atm
Last retrieval
—
Levels retrieved
—
About ERA5 L137
ERA5 is ECMWF's fifth-generation global reanalysis — the highest-quality freely available atmospheric dataset. L137 refers to the 137 model levels, providing vertical resolution of 10-300m. KAPPA uses variables:
temperatureT(K) — dry refractivity term 77.6P/Tspecific_humidityq(kg/kg) → e(hPa) — wet term 3.73×10⁵e/T²geopotentialZ(m²/s²) → alt(m) = Z/g₀ — regression axisSaved Location Profiles
Store frequently used observer-target pairs for repeat observations
Save observer-target pairs from the Refraction panel for quick recall.
No saved profiles yet. Run a calculation and click "Save current".
Professional Reports
ISO-structured PDF · Full methodology · ERA5 data provenance · Citable output
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The KAPPA Pro report is a fully-cited professional document suitable for survey deliverables, legal proceedings, research publications, and engineering designs. It includes all computed metrics, raw ERA5 profile data, methodology derivations, and data provenance.
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Report contents
A
Executive summary — computed κ, required κ, verdict, data source
B
Computed κ analysis — full derivation, dN/dh, N profile
C
Required κ and visibility determination
D
Ray path clearance diagram — 2,000 point profile
E
Sensitivity analysis ±ΔT, ±ΔRH perturbations
F
Elevation angles — geometric and apparent
G
Surveyor corrections (EDM, levelling, reciprocal) if computed
H
Full ERA5 atmospheric profile table with all variables
I
Methodology, formulas, standards, data provenance
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