Earth surface · R_eff = R/(1-κ) geometric chord A Observer B Target h_min = +847 m VISIBLE · · · 2,000 integration points · · · d = 443 km · great-circle distance Computed from ERA5 L137: κ = 0.1574 κ_required = 0.1512 · VISIBLE ✓
Patent Pending · SA 1020263556 · PCT/SA2026/050034

The world's first
real-atmosphere
refraction engine

κ computed from ERA5 measured data — not assumed.

κ = −REarth × (dN/dh) × 10⁻⁶

Every surveyor, engineer, and scientist working with line-of-sight has been using κ = 0.13 — an empirical average from the 1960s. KAPPA replaces that assumption with the actual measured atmosphere from ERA5 reanalysis at 137 model levels, for any location, any date, any time.

137L
ERA5 model levels
2,000pt
Ray integration
10⁻¹⁵
Binary search precision
157+
PCT countries
ERA5 L137 · 0.25° Resolution
ITU-R P.453-14 Full Wet Term
Barrel & Sears EDM Correction
Rueger Reciprocal Levelling
80-Iteration Binary Search
ICAO Doc 7488 Baseline
Copernicus CDS Downstream Service
ISO-Structured PDF Reports
Patent Pending · SA 1020263556
PCT International · PCT/SA2026/050034
ERA5 L137 · 0.25° Resolution
ITU-R P.453-14 Full Wet Term
Barrel & Sears EDM Correction
Rueger Reciprocal Levelling
80-Iteration Binary Search
ICAO Doc 7488 Baseline
Copernicus CDS Downstream Service
ISO-Structured PDF Reports
Patent Pending · SA 1020263556
PCT International · PCT/SA2026/050034

A 60-year-old assumption
is costing you accuracy

The difference between an assumed κ and a measured κ is the difference between a survey that closes and one that does not.

Before KAPPA
κ = 0.13

An empirical average from 1960s European observations. Applied uniformly regardless of location, season, humidity, or elevation. Wrong by 15–40% in most real conditions.

With KAPPA
κ = measured

Derived from ERA5 reanalysis at 137 vertical levels using the full ITU-R P.453 wet-term formula. Location-specific, date-specific, time-specific. Correct.

Systematic Error at 10 km
±40 mm

Unmodelled refraction error in a first-order levelling network. Enough to fail closure tolerances and require expensive re-observation.

With KAPPA
< 2 mm

Real atmospheric correction applied per session. Network closures that meet first-order tolerances on the first attempt.

Eight steps from
coordinates to correction

Every computation is traceable to published scientific standards. No black boxes. No empirical shortcuts.

01
User Input
Observer and target coordinates, elevation, observation date and UTC time.
Claim 1
02
Pressure Level Selection
ICAO hypsometric conversion identifies the ERA5 pressure slice spanning the atmospheric path.
P = P₀(1 - Lh/T₀)^(g/RL)
03
ERA5 Data Retrieval
Temperature, specific humidity, and geopotential retrieved via Copernicus CDS API with server-side credentials.
e = qP / (0.622 + 0.378q)
Claim 1, 12
04
ITU-R P.453 Refractivity
Full wet-term refractivity computed at each pressure level. The wet term from ERA5 humidity contributes Δκ ≈ +0.015 to +0.030.
N = 77.6P/T + 3.73×10⁵·e/T²
Claim 1
05
OLS Gradient Regression
Least-squares linear regression of N against altitude derives the refractivity gradient and refraction coefficient.
κ = −R_Earth × dN/dh × 10⁻⁶
Claim 1
06
Spherical Ray Geometry
Observer and target placed on effective sphere R/(1−κ). Ray height evaluated at 2,000 integration points.
h(f) = |r_A + f(r_B−r_A)| − R_eff
Claims 2, 4, 5, 9
07
Required-κ Binary Search
80-iteration binary search determines the minimum κ required for geometric visibility, converging to ~10⁻¹⁵.
Claim 3
08
Results and ISO Report
Computed κ, required κ, verdict, EDM ppm correction, levelling c+r, angular refraction, reciprocal height difference — all in a professional PDF report.
Claims 4, 5, 6, 7, 8

One engine.
Six disciplines.

KAPPA serves every professional who needs to know what the atmosphere was actually doing at a specific place, on a specific date, at a specific time.

Geodetic Surveying
Replace assumed κ = 0.13 with the real measured value for your specific session. Close first-order networks on the first attempt. Apply accurate curvature-and-refraction corrections at every sight distance.
c+r = d²(1−2κ)/(2R) — with real κ
Total Station / EDM
Compute wavelength-dependent atmospheric ppm corrections including the humidity wet term that all instruments ignore. At 10 km, the wet term alone contributes 30–80 mm of scale error. KAPPA eliminates it.
Ng = 287.6155 + 4.8866/λ² + 0.068/λ⁴
Radio Link Engineering
Characterise the refractivity gradient dN/dh for any microwave link path. Determine ducting risk, compute fade margin requirements, and build seasonal statistical models for link reliability.
dN/dh from real ERA5 — not 1970s maps
Atmospheric Science
Predict extreme long-distance optical observations. Back-calculate implied κ from documented sightings. Build seasonal visibility calendars using ERA5 historical data. Publish scientifically defensible results.
κ_required via 80-iteration binary search
Forensic Engineering
Determine whether a specific object was geometrically visible from a specific location at a specific historical date and time. ERA5 archive covers 1940 to present. ISO-structured PDF reports suitable for court submission.
Historical ERA5 archive from 1940
Long-Range Photography
Stop guessing when to drive to the mountain. KAPPA computes κ for every morning of the coming week and tells you exactly which dates exceed the required threshold for your target pair. The photograph of a lifetime — planned scientifically.
When κ_computed ≥ κ_required — GO

Built on the world's
best atmospheric dataset

ERA5 is ECMWF's fifth-generation global atmospheric reanalysis. It is the only freely available dataset with sufficient vertical resolution to compute a path-specific refraction coefficient from real measured data. KAPPA is the first operational tool to connect ERA5 directly to geodetic practice.

Contains modified Copernicus Climate Change Service information 2024. Neither the European Commission nor ECMWF is responsible for any use that may be made of this information.

// data source
ERA5 L137
ECMWF · Copernicus Climate Data Store
137
Model levels
0.25°
Spatial resolution
1h
Temporal resolution
1940—
Archive depth

Field validation
in progress

KAPPA is currently being validated against real-world professional survey observations and documented long-range photographic records. Results will be published here and in a peer-reviewed FIG Commission 5 technical paper.

// Geodetic Survey Validation
Phase I

First-order levelling network comparison. KAPPA-computed κ versus standard assumed κ = 0.13 over measured traverses. Network closure analysis.

In preparation
// Long-Range Optical Validation
Phase II

ERA5-predicted κ compared against documented extreme long-distance photographic observations. Blind prediction of visibility windows versus photographic record.

Collaborations open
// FIG Technical Paper
Phase III

Full scientific validation paper submitted to FIG Working Week 2026, Commission 5 — Positioning and Measurement. Peer-reviewed publication of methodology and field results.

Submission pending
// Are you a surveyor or long-range photographer?

We are looking for professional surveyors and long-range observation specialists to validate KAPPA against real field data. Early collaborators receive complimentary Professional access and co-authorship on the FIG paper.

Contact us to collaborate →
Saudi Patent
SA 1020263556
Filing Date
28 April 2026
PCT International
PCT/SA2026/050034
PCT Filing Date
01 May 2026
Countries Covered
157 PCT States
Search Authority
EPO (European)
Claims Filed
18 Claims

Access the real atmosphere.
At every level.

Consumer
Free
ICAO standard atmosphere
  • ICAO dry baseline κ
  • Line-of-sight visibility
  • Required-κ computation
  • Basic PDF report
  • — ERA5 real data
  • — EDM correction
  • — Levelling modules
Get started
Explorer
$7/month
or $59/year
  • ERA5 real atmospheric data
  • Full sensitivity analysis
  • N profile charts
  • Ray clearance diagrams
  • Professional PDF reports
  • — EDM correction
  • — Levelling modules
Start free trial
Professional
$490/year
per user · billed annually
  • Everything in Explorer
  • EDM wavelength correction
  • Geodetic levelling c+r
  • Reciprocal levelling
  • Simultaneity validation
  • ISO-structured PDF reports
  • Priority support
Start free trial
Enterprise / API
$2,900/year
10 seats · API access
  • Everything in Professional
  • 10 team seats
  • REST API access
  • Bulk analysis
  • OEM integration options
  • SLA support
  • Custom onboarding
Contact us

Stop assuming.
Start measuring.

Every surveyor, engineer, and scientist working with line-of-sight deserves the correct κ — not a 60-year-old guess. KAPPA gives you the real atmosphere, for any location, any date, any time.

▶ Launch KAPPA Pro — Free Read the technical paper →