# Sample Lesson Plan: Intro to Satellite Altimetry

Ch. 5.2.7 Satellite Alitmetry – Torge, Wolfgang (2001) Geodesy. De Gruyter. Berlin, New York

Overview of Satellite Altimetry

Satellite Gravimetry v. Altimetry for Geoid Determination

OVERVIEW

• What is Satellite Altimetry?
• Active remote sensing
• Missions
• Application of Satellite Altimetry
• Mean Sea Surface
• Geoid Determination
• Dynamic Ocean Topography

OBJECTIVES

• Understand the basic principles of satellite altimetry (two-way measurement, footprint)
• The student should be able to break up the different components of the measurement and list the time variant and static variables.
• The assignment will help the student gain practice in creating GUIs and functions in Matlab as well as refresh their memory on satellite orbits (which should have been learned in a previous geodesy course)

WHAT IS SATELLITE ALTIMETRY?

• Satellite Altimetry is a form of active remote sensing.
• The most common form, and the one that this lecture is focused on, operates using microwave electromagnetic waves (RADAR), 13.5 GHz frequency (Ku Frequency). There are also laser altimetry which is used for polar missions that is not covered here.
• Two-way measurement (d = c/2*delta_t)
• Measures the instantaneous height of the sea surface
• If we break this up into the components we get d = r_s – r_p – (N + SST)
• Where d = measurement from satellite, r_s = the geocentric distance of the satellite, r_p = the geocentric distance to the point p where p lies on the surface of the reference ellipsoid and is contained by the vector of r_s, N is the geoid undulation at point p, and SST is the sea surface topography
• Satellite Altimetry operates by using a number of short pulses and taking the average of the pulses to filter out roughness in the sea surface.
• This translates into the footprint of the altimeter.
• Important Missions in Satellite Altimetry:
• NASA: Skylab, GOES-3, Seasat, Geosat, TOPEX/Poseidon, Jason 1,2,& 3
• ESA: ERS-1, ERS-2

POSSIBLE ASSIGNMENT

• Write a MatLab GUI that will calculate the time period of data to average from the Jason-2 satellite with an input footprint size. Use the orbital parameters for the Jason-2 satellite from NASA mission page. Assume a pulse frequency of 100 pulses/sec.

Marks (out of 10)

• Input (1 mark) the program asks for a footprint size
• Output (1 mark) the program displays a message with the resultant time
• Functions (4 marks) the student used functions/set variables rather than just one code (satellite parameters, calculate orbital speed, calculate single measurement footprint, calculate time)
• GUI (2 marks) the student produced a GUI program rather than using text-based operations
• Cycle code (2 marks) the program should be able to repeat calculations without having to restart
• Bonus (2 marks) the program has a drop down menu to select a different satellite to use and the results for those satellites are correct. (1 mark for 2 extra satellites, 2 marks for 4 extra satellites)

APPLICATION OF SATELLITE ALTIMETRY

• Limitation of satellite gravimetry in geoid determination is the resolution, satellite altimetry is of a higher resolution
• If we recall the equation d = r_s – r_p – (N + SST) we know that satellite altimetry measurements cannot directly measure the geoid as it contains the SST.
• If we set r_s – r_p to h_s which is the geodetic height of the satellite we can rearrange the equation to: h_s – d = N + SST
• we will then further simplify the equation by setting h_s – d = h which is the geodetic height of the sea surface at point p and represent SST by the symbol ζ, this gives us the equation: h = N + ζ
• Although this equation helps us visualize the situation it is not helpful for calculations
• Given this we must break N and ζ into their respective components.
• N can be broken into a long wavelength component (N_lw) and variations from the long wavelength (ΔN)
• Since satellite gravimetry measures the long wavelength of the geoid we can use a geoid model derived from such measurements as a value for N_lw in which we will call N_REF
• ζ can be broken into the mean  dynamic topography (ζ_MDT) and the temporal component of the sea surface topography from tides, atmospheric effects, wind, etc. (ζ(t))
• If we add the term e to represent the error in measured and modelled values we now have the equation: h = N_REF + ΔN + ζ_MDT + ζ(t) + e
• This equation still has 3 unknowns; ΔN, ζ_MDT, ζ(t);
• To reduce this to give us ΔN we can assume that the values of: ζ_MDT, ζ(t), and N_REF, do not fluctuate between observations so we can measure the slope of ΔN by measuring the change in the sea surface between observations: Δh_a-b = ΔN_a-b and combine these results with air- and shipborne gravity observations to define ΔN
• We can then calculate the geoid undulations using the equation N = N_REF + ΔN
• If we use the average of measurements over a long time span we may assume that the averaged value of ζ(t) = 0 and then can solve for ζ_MDT
• This then allows us to model ζ(t) based on each measurement of h from satellite altimetry.

# Overview of Satellite Altimetry

Due to the complexity of the Continuous Vertical Datum for Canadian Waters (CVDCW) project, this will become clearer as I write more about it, it was necessary to form a (semi) multidisciplinary team to tackle the problems that the project faced. This team is comprised of the Canadian Hydrographic Service (CHS) and the Canadian Geodetic Survey (CGS).

The CVDCW combines the Canadian Gravimetric Geoid Model of 2013 (CGG2013), from the CGS, and ocean models of the sea surface topography, from the CHS, to obtain a Hydrographic Vertical Separation Surface (HyVSEP) that can reduce geodetic heights to a number of vertical water datums, primarily chart datum. This is done using eq. 1 where N is the geoid undulation from CGG2013, DOT is the Dynamic Ocean Topography, and sep(MSS-tidal level) is the difference between the Mean Sea Surface (MSS) and the tidal level or vertical water datum desired.

There are three measurements that therefore must be made to calculate the HyVSEP, all found on the right side of the equation. Each of these measurements, of course, have an uncertainty value associated with them, however, what Robin et al (2014) does not seem to address is that all three of these measurements for offshore modeling is reliant on satellite altimetry.

What then is satellite altimetry and how is it a key part in three separate measurements?

Typically, satellite altimetry is a form of microwave remote sensing (RADAR) that is found on special scientific satellites; SKYLAB, GEOS-3, SEASAT-1, Topex/Poseidon, GEOSAT, ERS-1&2, GFO, ENVISAT-1, Jason 1&2 (Jason 3 was launched January 2016 and has an altimeter); but is sometime found in the form of laser remote sensing particularly if polar regions are the target area; ICESat, Cryosat-2.

Unlike the InSAR method used in the RADARSAT missions, satellite altimetry is a nadir facing sensor which yields a much smaller footprint. Satellite altimetry can therefore achieve a much higher precision and accuracy because of the lower incident angle and through atmospheric corrections that are determined through measuring the atmospheric conditions with a Radiometer, which can only be utilized properly with a nadir sensor.

Image: JASON-2 satellite

There have been two different types of satellite altimetry missions, Geodetic Missions (GM) which has a non-repeating or very long repeat orbit, allowing for a more dense coverage and Exact Repeat Missions (ERM) which has a short repeat time and allows for the time modeling of time variant features. In total there has been 60 years of ERM data but less than 2.5 years of GM data as only ERS-1 and Geosat had GM orbits (Sanso and Sideris, 2013). This causes an issue for using satellite altimetry for geodetic purposes as the ERS-1 and Geosat altimetry data is not as accurate as modern altimetry data. The largest cause of this uncertainty is from the orbit determination of the satellite itself; Geosat used Doppler and ERS-1 used SLR (as well as PRARE until its malfunction early into the mission).

There has been effort to ‘re-process’ the data through remove-restore and crossover adjustment techniques which try to define a more accurate orbital model to the position of the satellite as well as combing the GM data with the more rigorous ERM data. I might have a future post that gets into more detail about this but for now it is just important to know that any surface solutions from satellite altimetry are a combination of GM and Interpolated ERM data.

There are many other considerations into determining accuracy that I won’t go into for this post but I do encourage you to check out the further readings if you are as interested in it as I am.

Now to get onto the meat of this post; how satellite altimetry is (probably) used simultaneously for all three measurements.

Satellite Altimetry Diagram

The first and most obvious measurement is in the tide models. Satellite altimetry measures the distance between the satellite and the instantaneous sea surface (Δh), averaged over time this produces a Mean Sea Surface (MSS) and can be used to determine various water level definitions such as LLWLT which is used for chart datum in Canada. A distinct advantage of satellite altimetry over its more traditional counterpart, tide gauges, is that the measurements are not in relationship to a fixed point but rather in relationship to the geo-center, the center of mass of the earth, which allows the data to be free from the effects of crustal motion.

The second measurement is the geoid measurement. The geoid, an equipotential surface, is a lumpy reference surface, also known as a datum, that represents Mean Sea Level (MSL). If the oceans where static then the geoid would coincide with the MSS but as they are dynamic they differ by a value called the dynamic ocean topography (DOT), ζ. The geoid undulation, N, is the separation value of the Geoid and the Ellipsoid and is used for terrestrial applications to convert geodetic heights to orthometric heights. For marine applications the geoid undulation is not useful on its own as it has no relation to the sea surface. Please read my post on the difference between using satellite gravimetry and satellite altimetry for geoid determination here.

As can be seen in the figure, the difference between the Ellipsoid and MSS can be calculated using the geoid undulation, N, and the DOT, ζ. The DOT is the third measurement that is needed for the HyVSEP. The DOT value can be reduced through the removal of tidal corrections (the first measurement), 75% of signal variance, and dynamic atmospheric corrections, 10% of signal value. The rest of the signal is from wind and other high frequency effects which can be modeled. The primary method of determining DOT, that I have read, is through the reduction of MSS and N, with N also being a function of DOT and MSS.

It has been my observation that all three measurements are co-dependent on each other so it will be interesting to see how the CHS and CGS handled this, but significantly more reading is required to fully understand how they’ve approached the problem so expect more posts on the matter in weeks to come.

SANSÒ, F., & SIDERIS, M. G. (2013). Geoid determination: theory and methods.

SEEBER, G. (2003). Satellite geodesy: foundations, methods, and applications. New York, Walter de Gruyter.

JASON MISSIONS

# Re-Processing Old Altimetry Data

With satellite altimetry there have been two types of missions, geodetic missions (GM) and exact repeat missions (ERM). GM altimetry has a very high density of measurements that is due to its very long repeat orbit, this makes it particularly useful for deriving short wavelength features such as the variations in the geoid that are not measured through satellite gravimetry. ERM altimetry has a short repeat orbit which allows it to observe the temporal variations in the ocean topography, however this is at a cost of density.

Picture from the IAG 2006 geoid school presentation

The only two satellites to observe GM altimetry were ERS-1 and GEOSAT, which combined comprise of about two and a half years of observations. This is unfortunate as the technology difference between those two missions and what we have today is vastly different in many ways, particularly in the accuracy of altimetry measurements and orbits. This means that we only have precise measurements for ERM altimetry and have to interpolate the data within the areas without measurements. To perform the interpolation we can use the data from the GM altimetry, although dated, to provide a more rigorous solution then without it.

Now obviously this is not ideal due to the limited accuracy of the legacy data, however, we can apply the knowledge gained since then to reduce some of the errors from the estimated satellite position and the reduction of time elements such as tides. These are both parameters that are actually determined with the use of measurements from satellite gravimetry (which came at a later date) as well as satellite altimetry from more recent years. This means that the more recent observations are not only contributing to the established measurements of the geoid but also towards the interpolated values.

This just goes to show you that you should always keep legacy measurements because you might want to re-process with future knowledge.

SANSÒ, F., & SIDERIS, M. G. (2013). Geoid determination: theory and methods.

# Satellite Gravimetry v. Altimetry for Geoid Determination

The question posed in this post is; whether it is better to use satellite gravimetry or satellite altimetry to determine a marine geoid?

Satellite Gravimetry

Major Missions: GOCE (ESA), GRACE (NASA)

How it works: GOCE uses a gradiometer which is comprised of six accelerometers that measure differences in the gravity field. GRACE measures gravity by measuring the distance between the two satellites which varies with fluctuations in the gravity field.

Satellite Altimetry

Major Missions: Skylab (NASA), GOES-3 (NASA), Seasat (NASA), Geosat (NASA),  ERS missions (ESA), TOPEX/Poseidon (NASA), Jason satellites (NASA),

How it works: An altimeter emits electromagnetic radiation in the microwave bandwidth at a near-nadir direction. The radiation is reflected off the earth’s surface (or the ocean surface) and back to the satellite. By measuring the travel time the distance between the satellite and the target can be calculated.

Gravity and Geometry

By reading the short introductions above, it would be reasonable to disregard satellite altimetry for geoid determination, and for the geoid over solid terrain you would be right, but over the oceans satellite altimetry becomes an important and necessary part in geoid determination. This is due to the dynamic nature of the oceans and the limitations of gravity missions.

The major problem with satellite gravimetry is that only the low frequency features of the geoid can be measured through space based techniques, for now anyway. Just with the shear surface volume of our oceans supplementing these satellite missions with higher frequency measurements through airborne and shipborne gravity measurements is just not practical. Luckily geodesists and hydrographers already had a way of geoid determination that can supplement the satellite based observations. To do this they used equation 1 below where h is the geodetic height, N is the geoid-ellipsoid separation, ζ is the dynamic sea topography and e is the error. This means that by measuring h with satellite altimetry and combining this data with tidal and dynamic sea models you can determine the geoid-ellipsoid separation value. This is why some of the earlier altimetry missions where designed to have geodetic missions, the newer missions all have exact repeat missions which have a lower density of measurement but can measure temporal data better.

Equation 1: h = N + ζ + e

This method of course depends on having accurate dynamic sea models. However as gravity missions started in the early 2000s it was determined that you could combine the low frequency data from the satellite gravimetry with the high frequency data from the altimetry missions. This is done using Equation 2 where N_REF is the long wavelength geoid from the gravimetry missions, ΔN is the high frequency residuals in the geoid-ellipsoid data, ζ _MDT is the mean dynamic topography,  and ζ(t) is the time varying sea surface topography (this includes tides, dynamic atmospheric effects, wind, etc.). By assuming that N_REF, ζ _MDT, and ζ(t) are all long wavelength features then the slope of ΔN can be measured through comparing h from one observation to another, given that they were subsequent measurements.

Equation 2: h = N_REF + ΔN + ζ _MDT + ζ(t) + e

Image from NOAA

# Continuous Vertical Datum for Canadian Waters (CVDCW)

When navigating a ship the number one objective is to not sink the ship. To achieve this, it of course helps if the ship does not run into anything above or below the water surface but it is not always possible to see submerged objects coming up in time to navigate around them and so the navigator of a ship will refer to a chart in order to avoid obstacles. These charts contain 2 pieces of information, the horizontal position of features, and their associated depth in relationship to what is called the chart datum.

Chart datum is defined in Canada as the Lower Low Water, Large Tide (LLWLT) which is to say that the depth is relative to the yearly average of the predicted lowest water level for a 19 year lunar cycle based on observations from tide gauges. The water depths are then determined through bathymetric surveys in relation to these tide gauges. With the advent of GNSS the ocean mapping community saw the advantage to perform their surveys in relationship to the ellipsoid and then linking these heights to the tide gauges, this in turn lead the Canadian Hydrographic Service (CHS) to create a seperation model (SEP) between the chart datum and the ellipsoid by measuring the separation with GNSS on the tide gauges.

Of course the ocean system is dynamic, meaning that the water levels measured at a tide gauge are only relevant to those waters nearby (less than 10 km) as water levels are not a linear function of distance. This is due to different factors that change over time such as the chemical composition of the water, the currents, etc.. This creates a problem as it would be unrealistic to have a complete network of permanent tide gauges covering the entire Canadian coastline (stretching over 200,000 km it is the longest in the world) and re-survey these waters, tieing them into the tide gauges.

And so in 2010 the CHS and the Canadian Geodetic Survey (CGS) to examine the possibility of creating a model that would represent the continuously changing water levels which would be called the Continuous Vertical Datum for Canadian Waters. Such a model will aid in defining; coastlines, inter-tidal zones, maritime boundaries, marine cadastres, claims to sovereignty. As well as serving as a baseline for sea level rise & related climate change adaptation strategies, and improve coastal infrastructure maintenance & development.

To get an idea of just how complex such a model is, the input data for the model includes; the Canadian Gravimetric Geoid of 2013 (CGG2013), GNSS observations, Satellite Altimetry, Water Level Observations, Ocean Models, and Coastline Surveys.

As you are starting to gather by now, this is an extremely complex project that encompasses the most challenging of problems over multiple fields and disciplines. Aka a perfect topic for the Geodesy Corner and I look forward to exploring this further in the weeks to come.

Catherine Robin, Shannon Nudds, Phillip MacAulay, Andre Godin, Bodo De Lange Boom & Jason Bartlett (2016) Hydrographic Vertical Seperation Surfaces (HyVSEPs) for the Tidal Waters of Canada. Marine Geodesy, 39:2, 195-222, DIO:10.1080/01490419.2016.1160011

# Right-Handed/Left-Handed Rules

For any kind of vector mathematics it is common to find cases in which you want to find a vector that is perpendicular to two other vectors, e.x. current/magnetic field/force, the moment of force (the cross-product of two force vectors), coordinates, etc.. We know from linear algebra that we can calculate this perpendicular vector by taking the cross-product of the two vectors. The problem is that the cross product can yield an answer with two possible orientations depending on the order of the vectors within the cross-product equation.

To establish a way of properly orientating the vectors we can use the mnemonic of the right-handed or left-handed rule. This is where you ‘assign’ your thumb as the the third vector, or ‘Z’, your fingers as the first vector, or ‘X’, and then your palm is pointed in the direction of the second vector, or ‘Y’.

Left-Handed Rule

Right-Handed Rule

# Describing a Coordinate System

One of the fundamentals of any spatial project is establishing a coordinate system to describe the data. This coordinate system allows us to pass our data along to another person and they are able to understand it, or enter the data into a computer system to process the data.

A coordinate system is described by an origin, the orientation of three axes, and the parameters of which describe the position. (Krakiwsky and Wells, 1971)

The origin of a system is an arbitrary point of which all measurements will be in relation to, this point is often a stable, non-moving, point at the center or corner of the measurements. The origin must be a clearly defined point so that it maybe related to other coordinate systems, for example, we describe the origin of a geocentric system as being the center of mass of the earth.

Often the next part of the coordinate system we describe is the tertiary axis. In most topocentric terrestrial coordinate systems we define this as being inline with the plumb-line, i.e. the direction of gravity at that point, at the point of origin with the positive direction being in the zenith direction though it could run in the nadir direction. In some cases it is to our advantage to define the axis in terms of a primary pole which is an axis of symmetry for the system. Often for geocentric systems we assign the primary pole as the spin axis of the earth.

After the tertiary axis is defined we may set the primary plane as a plane that contains the origin and is perpendicular to the tertiary axis. Often for geocentric systems the primary plane is often the equator. Contained within the primary plane is the primary axis, and the secondary axis.

We can establish the primary axis through defining a secondary plane which is a plane that is perpendicular to the primary plane and contains the origin as well as another point, whether it be of interest or an arbitrary point. The intersection of the primary and secondary planes is then the primary axis with the values running positive in the direction of the defining point, this axis is also sometimes referred to as the secondary pole. For most earth systems Greenwich, UK, is the arbitrary point used to define the secondary plane.

The secondary axis is then defined as a line that runs perpendicular to the primary and tertiary axes. The direction of this axis is defined by the right/left-handed orientation. Figure (V) uses the left-handed rule.

The parameters that describe a system will be covered more in-depth in a later post.