KAPPA

Atmospheric Optics — Line of Sight

Refraction Coefficient
Calculator

ERA5 L137

Snell's Law Integration

Haversine Distance

● Free to use

⚖ Patent Pending · SA 1020263556 · PCT/SA2026/050034

ICAO Standard Atmosphere

Refraction Analysis

ICAO Standard Atmosphere

Free Plan

Computed κ

—

atmospheric refraction coefficient

Required κ

—

minimum for line of sight

Refraction Curve — Ray height along path ICAO baseline

Where was the observer, and what was the target?

Decimal degrees — positive = North/East, negative = South/West.

Observer (Point A)

Where you were standing

Location name

Latitude °N

Positive = North, negative = South. E.g. 40.530 for northern Turkey.

Longitude °E

Positive = East, negative = West. E.g. 38.200 for eastern Turkey.

Elevation (metres)

Height above sea level in metres. Determines which atmospheric slice the light ray passes through.

Target (Point B)

What you were looking at

Location name

Latitude °N

Longitude °E

Elevation (metres)

When was the observation?

Use UTC. ERA5 data is available every hour.

Date

Format: YYYY-MM-DD. ERA5 covers 1940 to ~5 days ago.

Time (UTC)

HH:MM in UTC. The nearest available ERA5 hour will be used.

Great-circle distance (km)

Auto-calculated using the Haversine formula from your coordinates. Override if you have a more precise measurement.

Units

Metric (metres, km) or Imperial (feet, miles). Applies to all inputs and outputs.

Observation mode

LOS: standard line-of-sight visibility. Surveying: precise levelling corrections. EDM: electronic distance measurement with wavelength correction.

Atmospheric slice

The calculation uses only the air between your two elevations — the exact layer the light ray travels through.

Lower bound (hPa)

Pressure at the observer's elevation. Higher altitude = lower hPa.

Mid-path (hPa)

Upper bound (hPa)

Automatically derived from your elevations. Only this pressure slice is used in the calculation.

ERA5 API Key

Paste your free Copernicus CDS key to fetch real measured data. Without it the tool uses the ICAO standard atmosphere. Guide below — 2 minutes to set up.

Or load ERA5 data from Python script

Run the included era5_fetch.py script, then paste the contents of era5_output.json below. This gives you full real ERA5 data without needing a server.

How to get your free ERA5 API key

Free service by ECMWF — registration takes 2 minutes.

Go to the Copernicus CDS website and create a free account

Open cds.climate.copernicus.eu in a new tab. Click the Register button in the top-right corner of the page. Fill in your name, email and a password. You'll get a confirmation email — click the verification link inside it. The service is entirely free.

cds.climate.copernicus.eu

Copernicus CDS

Datasets

Applications

Docs

👆 Click here
Register

Already have an account? Log in →

After verifying your email, log back in. Click your name or avatar in the top-right corner, then select Profile from the dropdown that appears.

cds.climate.copernicus.eu

Copernicus CDS

Datasets

Applications

your.name@email.com ▾

👆 Click here
Profile

Settings

Log out

Scroll down your Profile page and copy your API key

On the profile page, scroll down until you see a section called "API key" or "Your token". You'll see a long string like a1b2c3d4-e5f6-7890-abcd-ef1234567890. Click Copy next to it (or select the whole string and press Ctrl+C / Cmd+C). Keep this key private — treat it like a password.

cds.climate.copernicus.eu/profile

API key

Use this key to access ERA5 data programmatically.

👆 Copy this

a1b2c3d4-e5f6-7890-abcd-ef1234567890

User UID: 123456

💡 Some accounts show two fields: a numeric User UID and the API key UUID. You only need the UUID — the longer string in the format shown above.

Accept the ERA5 dataset licence — this is required before any data downloads

Visit the ERA5 pressure levels dataset page. Scroll to the very bottom and click "Accept terms". You only do this once. Without it, API requests return a licence error even with a valid key.

cds.climate.copernicus.eu/datasets/reanalysis-era5-pressure-levels

ERA5 hourly data on pressure levels from 1940 to present

Licence to use Copernicus Products

By accepting, you agree to acknowledge ECMWF as the data source in any publications.

👆 Click to accept
Accept terms

⚠ This is the step most people miss! If you get a "licence not accepted" error when running the calculation, come back here and complete step 4.

Paste your key into the field above and run the calculation

Scroll back up to the API key field at the top of this section and paste your key. The green confirmation box will appear. Click Run Refraction Calculation below and the tool will request real ERA5 atmospheric data for your exact coordinates and time.

🕐 ERA5 data has a processing lag of about 5 days. For very recent observations, the ICAO standard atmosphere is used automatically.

Results appear on step 4

Initialising…

Results

—

Computed k

—

refraction coefficient

Required k for visibility

—

refraction coefficient

What does this mean?

Full Analysis Report

All detailed results, diagrams and methodology — delivered as a PDF.

$12

—Observation path

—Date / Time (UTC)

—Path distance

—Computed k

Unlock the full analysis

Ray path diagrams · Required vs computed k · Sensitivity analysis · Elevation angles · Full atmospheric data table · Methodology & data provenance

A — Visibility analysis: required k vs computed k

Minimum refraction needed for target to be geometrically visible, compared to the computed atmospheric value:

Required k

—

Computed k

—

Standard k

0.13

—

B — Ray height clearance along the path

Shows how far above (or below) the geometric horizon the refracted ray travels at each point along the path.

C — Ray path diagram

D — Sensitivity analysis: how k responds to atmospheric variation

—

T − 2°C

—

Computed (base)

—

T + 2°C

—

RH − 10%

E — Elevation angles: geometric vs apparent

At Observer (A)

—

geometric

—

apparent

→

Refraction lifts apparent position

→

At Target (B)

—

geometric

—

apparent

Patent Pending · SA 1020263556 · PCT/SA2026/050034

Computing atmospheric refraction…

Connecting to KAPPA backend…

Refraction CoefficientCalculator

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Refraction Coefficient
Calculator