# A quick guide to altimetry corrections

*Use links in the text to high-light relevant details in the illustration.*

Altimeters transmit a microwave pulse towards the surface of the Earth and record the time it takes for the echo to return.

From the return time and the speed of the microwaves through the atmosphere
we can calculate the distance between the satellite and the sea surface.
This is known as the altimeter
**range**.

We know the **orbit height** above the Earth's
reference ellipsoid, so we can calculate the
**sea surface height**:

SSH = orbit height - range.

To obtain a useful measure of sea surface height it is necessary to make a number of corrections. These fall into three broad groups:

- Atmospheric propagation corrections
- Tidal corrections:
- Sea surface corrections

### Atmospheric propagation corrections

The speed of electromagnetic waves through vacuum is well known. In air there is a delay, which has to be taken into account when retrieving altimetry range from the signal return time. These corrections depend on the properties of the atmosphere and vary in time and space.

The **ionosphere correction** accounts for the path delay due to charged particles
in the ionosphere. It is usually calculated by combining radar altimeter measurements acquired at two separate frequencies,
and ranges from 0 to 50cm.

There are two **troposphere corrections** - wet and dry.
The **dry** troposphere correction is well known and is calculated from meteorological models. Its order of magnitude is 2.3 m

The **wet** troposphere correction accounts for path delay due to liquid water in the atmosphere.
It can vary considerably at relatively short scales in time and space, and must be calculated from radiometer measurements
(when available), supplemented with meteorological models. Its magnitude ranges from 0 to 50 cm.

### Tidal corrections

Tides are caused by the gravitational attraction of the sun and moon on the Earth system. The effect of tides on the sea surface can be modelled, and this allows us to calculate the corrections we need to retrieve the mean sea level due to ocean currents.

**Ocean tides** make the sea surface rise and fall diurnally.
Tidal amplitudes can be up to 1m in mid-ocean and as much as 15-20 m near some coasts.
Global tidal models are used for standard altimetry data. Nearer the coast regional or local tidal models may be necessary.

The non-spherical shape of the Earth creates a torque from the sun and moon, which causes the Earth to wobble with a period of about 433 days.
One effect of this *Chandler Wobble* is to create periodic ocean currents known as the **pole tide**.
Corrections due to this tide have an amplitude of about 2cm.

**Solid earth tides** are a result of tidal forces acting on ground water
and the molten rock in Earth's interior.
The correction is calculated from models and can be up to 50 cm.

**Tidal loading**
is a correction for height variations due to longer term changes in tide-induced forces. Its order of magnitude is around 30cm.

### Surface corrections

The **inverse barometer**
correction accounts for variations in sea surface height due to changes in atmospheric pressure.
Sea surface height increases when atmospheric pressure is low. The correction is calculated from meteorological models, and is of the order of 15 cm, depending on atmospheric pressure.

** Sea state bias (SSB)**
is an error in the estimate of altimeter range caused by the presence of waves on the sea surface.
To understand why, it helps to know
how altimeter range is retrieved from the waveform
(shape) of the radar return echoes - along with significant wave height and wind speed.

SSB is a few % of significant wave height, which means it increases as waves become higher. Accurate SSB correction is critically important for calculating current speeds from sea surface height measurements.

SSB correction uses a model that relates the SSB to wave-height and wind speeds measured by the altimeter. The models in common use are based on theoretical studies and wave statistics from observations.

### Sea Level Anomaly (SLA)

The mean SSH after atmospheric, tidal and surface corrections is only an approximation to the dynamic topography
that relates to currents. This is because it is related to the reference ellipsoid, not to the geoid.
Until recently, therefore, oceanographers used the ** sea level anomaly (SLA)**
to study current dynamics.

The SLA relates the measures SSH to a sea level mean, an average of SSH measurements from decades of satellite altimetry:

SLA = SSH - mean sea level

The SLA is extremely useful for calculating the speed of currents in ocean eddies, and for studying variations in the speed of major currents over seasonal and inter-annual time scales. It does not however give the true dynamic the true dynamic topography need to calculate absolute current speeds.

### Dynamic Sea Surface Topography

To calculate true dynamic topography requires an accurate, high-resolution geoid, such as the one provided by GOCE.

** Dynamic topography**
(h_{d})
is calculated from the corrected
** altimeter range**
(H), the
** orbit height** ,
(H) and the
** geoid height ** ,
(h_{d})

## h_{d} =
H - h -
h_{g}

From the dynamic topography is is possible to calculate absolute geostrophic current speeds as explained in interactive 3.2.

The
** mean dynamic topography (MDT)** is the time average of the dynamic topography.
From this it is possible to obtain the average ocean circulation.

## Waveform retracking and retrieval of key parameters

Altimeter processing includes retracking. This means fitting the waveforms of the return echoes to a waveform model (green curve in the figure).

From the fitted waveform it is possible to retrieve three important parameters:

- altimeter range, which after corrections give sea surface height
- significant wave height: roughly speaking the average height of the highest third of the waves, and close to the wave height estimates that human observers would give.
- wind speed

Epoch (=delay) gives mean range

Slope of the leading edge is related to significant wave height. Steeper slopes means lower wave height.

Maximum amplitude is related to wind speed

Placing mean sea level at the midway point of the leading edge makes the assumption that the troughs and crests have the same shape and return the radar signal equally. For real waves this is not strictly true. Sea state bias is related to real waves having peakier crests and flatter troughs. It includes two types of bias:

Electromagnetic bias: the flatter wave troughs are better radar reflectors than the sharp crests at the nadir viewing angle of an altimeter. As a result the mean scattering level is below the mean sea level (bias towards the troughs).

Tracker bias: an instrument dependent error linked to how a particular altimeter tracks ocean echoes.
This includes a skewness bias resulting from the altimeter tracking the **median** scattering surface.
For real waves the median scattering level lies below the man scattering level (bias towards troughs).