Walter H. F. Smith NOAA Lab for Satellite Altimetry Silver Spring, Maryland

1. Vertical Deflection of Gravity& Bathymetry: global and littoral applications for a geodetic Delay-Doppler altimeter mission Walter H. F. Smith NOAA Lab for Satellite Altimetry Silver Spring, Maryland

2. What is a Vertical Deflection? Ocean floor topography causes variations in the magnitude (“gravity anomaly”, Dg) and direction (“deflection of the vertical”, VD) of the acceleration of gravity. Altimetry “sees” VD as a slope induced in the sea surface.

3. 2 products from 1 mission Gravity Bathymetry

4. Many uses (many partners) • VD and Gravity (Navy, USAF, NIMA, NOAA, NASA) • Inertial Navigation, positioning, geodesy • Gravity & Bathymetry combined (NSF, USGS, MMS, State, oil) • Exploring sub-seafloor geology & resources • Evaluating territorial claims under the UN Law of the Sea • Bathymetry (Navy, NOAA, NASA, NSF, State, industry, others) • Modeling ocean currents, tides, mixing, climate, tsunami hazard • Routing undersea cables and pipelines • Managing habitat, biodiversity, and food resources What does Navy need?

5. Bathymetry for Battlespace Characterization • Depth available for operations • Modeling of global or local ocean 3-d acoustic structure • submarine and anti-submarine warfare • Modeling of local currents and tides • mine warfare and avoidance • personnel swimming ashore Bottom topography controls currents and mixing rates, thus also 3-d acoustic structure.

6. Bathymetry for Modeling Currents Navy ocean models require correct global & coastal bathymetry Model Bathymetry Changed Only Here Approximates nature Intrudes unnaturally A single feature as small as 20 km across can steer a major current (Kuroshio mean flow in NLOM at 1/16° [Metzger & Hurlburt, 2001]). A new VD mission will get bathymetry needed for ocean models.

7. Inertial Navigation Systems need VD INS measure accelerations, integrate to get velocity and position. VD, if uncompensated or unknown, appears as a fictitious acceleration of the vehicle, producing error. Error is cumulative in position and velocity, due to integrations. Thus after traveling through a poorly-mapped area, errors persist, even if one stays in a well-known area. Flying from ship at sea to target on land, need VD on the entire path, including littoral zone, to accurately hit target.

8. Mapping VD the hard way Survey gravity anomaly Dg with ship or airborne gravimeter; convert Dg to VD. Note: this is an indirect method. Conversion requires “convolution” calculation in a ~100 km neighborhood; data gap or quality change at shoreline degrades littoral solution from coast to ~50 km out to sea. Global coverage will need > 100 ship-years and > 1 billion $. • Navy’s Ocean Survey Program did this for Trident VD. • Data exist only in limited and cold-war-focused areas. • No littoral data: surveys lie seaward of 100 fathom line. • Anisotropic resolution: OSP tracks run mostly East-West.

9. Mapping VD the easy way Measure VD directly by satellite radar altimetry of the sea surface slope. No conversion required. No “neighborhood problem” in littoral zone. Global coverage in ~1.5 years for $100 M. Can optimize coverage and minimize anisotropy through judicious choice of orbit geometry. Examples: USN Geosat Geodetic Mission (1985-6); ABYSS and ABYSS-Lite mission concepts.

10. Altimetry yields sea height + error A satellite altimeter measures the ocean surface height plus random and systematic errors. The ocean surface height is not on the “geoid” (gravity equipotential surface) due to dynamical displacements caused by tides, currents, and weather.

11. Most errors don’t affect VD The geoid height is irrelevant to VD; only the geoid gradient matters. Dynamical ocean displacements and altimetry errors are mostly long-wavelength, so their slope is negligible.

12. Sea surface slope nearly equals VD VD and sea surface slope angles are nearly always within 1 mrad VD Dh Dh/Ds=tan~=q h in mm / s in km q in mrad; slope 1 mrad ~ 0.2 arcsec Ds Exceptions: littoral currents & tides, and western boundary currents & eddies can yield a few mrad error.

13. Assessing current knowledge • Repeatability of altimetric sea surface slope: • Looks at along-track component of VD only. • Won’t find error due to time-invariant currents. • Comparison of altimetric gravity with ship gravity: • Requires “neighborhood” conversion of VD to Dg. • Lumps in errors in ship Dg data also. • Scaling Dg error to VD error uncertain by √2 • Comparison of altimetric vs. ship bathymetry: • Requires “neighborhood” calculations also. • Limited by errors in ship data, “upward continuation”, and limits on correlation between Dg and depth.

14. Error analysis: Geosat ERM Geosat Exact Repeat Mission data furnish a test: how repeatable is sea surface slope along altimeter profiles? Slope errors are typically 5 mrad (1 arc-sec), varying from 4 to 8 mrad, (0.8 to 1.7 arc-sec). Map pattern does not resemble ocean dynamics, ionosphere or troposphere, confirming that these are not important sources of error.

15. Error analysis: Geosat ERM The slope error map looks like a map of seasonally-averaged wave height. Conclusion: random errors due to ocean waves are the dominant error in VD from altimetry.(Argument for Delay-Doppler altimetry.)

16. Ship versus Altimeter Gravity

17. Ship vs Alt grav: Admittance Admittance = ratio of in-phase amplitudes (ship:altimetric) g. Altimetry has full amplitude at l > 100 km, falls off at shorter wavelengths, due to attenuation in filters used in processing and also increasing out-of-phase noise in either data type.

18. Ship vs Alt. grav: Coherence Coherency = correlation as a function of wavelength. Transition from correlated (good) to uncorrelated (bad) is around 25 to 30 km full wavelength: features 12 to 15 km across are resolved by both ship g and altimetric VD. Resolution requires signal strength above noise level.

19. Altimetric Bathymetry (So Pac)

20. South Pac. Bathymetry Profiles (km)

21. Profile Correlation by Wavelength Poor at all l Correlation This band is not resolved yet, but could be with a new mission

22. Assessing current knowledge • Repeatability of altimetric sea surface slope: • 4 to 8 mrad (0.8 to 1.6 arc-sec) along-track. • Comparison of altimetric gravity with ship gravity: • Dg error ~ 5 mGal implies VD error 3 to 5 mrad. • Errors begin to creep in at l < 100 km. • Coherency good at l > 25 km. • Comparison of altimetric vs. ship bathymetry: • Coherency good at l > 25 km. • Implies features > ~12 km across are resolved.

23. Littoral Approach problems Data are often “lost” near coastlines. Reprocessing can rescue data approaching shore, but data remain lost on retreat from coasts. (NNW paths shown.) Need d-D instrument for better littoral tracking.

24. A note about littoral VD Let’s remember that altimetry measures VD directly, whereas surveying Dg and then converting Dg to VD involves spreading error through a neighborhood. The Dg method for getting VD has problems within ~50 km of the coast, due to neighborhood calculation. The current altimeter data usually get within 20-30 km of coast in at least one direction. A Delay-Doppler altimeter should be able to get within 3-5 km of the coast in both directions. This would be a direct VD measurement.

25. Another VD Error: Anisotropy So far we have looked at altimeter measurement repeatability, which estimates error in the along-track component of VD, and ship gravity comparisons*, which estimate error in the combination of VD components into gravity through a neighborhood calculation. Now let’s look at how along- and across-track altimeter error project into north and east VD components. *I have shown open data. My understanding is that comparisons with gravity anomalies derived from classified OSP data show similar results in typical areas.

26. Another error issue: Anisotropy Repeatability analysis applies to along-track errors. Across-track errors are worse. Orbit geometry (inclination) blends errors differently into North-South and East-West components of VD. Over low to mid latitudes (most of the world’s ocean area), errors are anisotropic:E-W error is bigger than the N-S error, by as much as a factor of 3 at the Equator.

27. Along- vs Across-Track Errors Each pass gives sea height + error, not geoid. Along-track slope is ~ VD, but how to get across-track component of VD? “Leveling” height profiles demands extreme accuracy (1 mm per km) and so won’t work.

28. Track Crossing Angle: Error Anisotropy • Error propagation q - local inclination of track s - error in along-track slope sx - error in east slope sy - error in north slope North slope  East slope Orthogonal tracks are optimal

29. Nearly orthogonal in a small area of ocean Nearly orthogonal over a large area of ocean Angle vs Latitude & Inclination ABYSS proposal: new mission, less polar, better angle over majority of ocean area

30. Anisotropic error by Latitude Existing data were collected in orbits passing fairly close to the Earth’s poles, and so at low latitudes (most ocean area) they resolve E-W component of VD much worse than N-S.

31. Summary: Where are we now? • Accuracy: • 4 to 8 mrad (0.8 to 1.6 arc-sec) along-track • 3 to 9 mrad (0.6 to 1.8 arc-sec) in N and E • Anisotropic by a factor of 3 at low latitudes. • Spatial sampling (at Equator, widest point): • 5 km (Geosat GM) • 8 km (ERS-1 GM) • Spatial Resolution (1/2 l @ 1/2 coherency): • 13 to 20 km (7 to 11 nautical miles). (Resolution depends on signal strength as well as measurement noise.)

32. Goals: ABYSS >= NIMA-USAF ABYSS Goal: 1 mrad (0.2 arc-sec) for Oceanography, Geophysics, Climatology (NOAA, NASA, NIMA, NSF, Oil Industry) NIMA-USAF Goal: 0.5 arc-sec (2.4 mrad) on a 1 nautical mile (1.8 km) grid for Advanced Integrated Navigation Systems on F-117, B-2, B-52H. Does Navy have similar goals? (F/A-18E/F and WSN-7 navigator on Los Angeles class subs? Joint Strike Fighter?) What is the best way to achieve these common goals?

33. Our goals require altimetry Altimetric sea surface slopes measure VD at sea level and so capture the full signal. Gravimetry in orbit (CHAMP, GRACE, GOCE) measures gravity at satellite altitude. Upward continuation that far wipes out the signal.

34. Limit on achievable resolution? Current noise Signal Future noise: d-D altimeter, 4 cycles (6 yrs) The VD signal appears to fall off rapidly with decreasing wavelength, reaching a point of diminishing return. We aren’t there yet, but we probably can’t do much better than l ~= 10 to 12 km, which requires noise reduction by 4 in amplitude (d-D altimeter and 4x redundant sampling)

35. Random errors limit resolution sslope = sheight2 / Ds Slope error grows large as the desired length scale, Ds, grows small. This limits resolution. Conventional altimeters have sheight ~= 25 mm in a one-second average (6.8 km along-track) giving sslope ~= 5.2mrad at a half-wavelength of 6.8 km (1.1 arc-sec at a half-wavelength of 3.7 n.m).

36. How to beat random errors? Method 1: use conventional (Geosat or Topex class) altimeter and get lots of redundant data for averaging. 0.5 a.s. (NIMA-USAF) goal requires factor of >2 drop in noise, factor of >4 in redundancy, or >6 year mission. 1 mrad (ABYSS) goal requires factor of 4 to 5 drop in noise, factor of 16-25 in redundancy, or 24-38 years! Method 2: use a better altimeter (delay-Doppler) 0.5 a.s. goal met with no need for redundancy! 1 mrad goal met by factor of 4 redundancy (6 year mission). This is the ABYSS concept. Some redundancy is good to assess & reduce time-varying errors (coastal tides, strong currents, etc.)

37. Conventional Height 3 Science * precision Requirement (cm) 2 1 Delay Doppler SWH PDF, Summer (after Lefevre and Cotton, 2001) 0 3 0 2 4 6 8 Significant wave height (m) Delay-Doppler best in wave noise ABYSS proposal: Use d-D altimeter to achieve 1.8 mrad @ 6 km half-wavelength in sea state of 3 m SWH. Collect 4 x redundant data (6 year mission) for 1 mrad final error. Note d-D altimeter meets NIMA-USAF precision goal without redundant mapping; 1.8 mrad = 0.375 arc-sec.

38. New mission can choose best inclination ABYSS proposal: More precise slopes, and better coastal tracking, using d-D instrument. More nearly equalize N and E error components, over latitudes where existing data are poor (80% of ocean area), with a less polar inclination.

39. WSOA won’t meet the goal Track spacing 15 km (8 n.m.) at best, worse in “yaw steering mode”, and leaves gaps. Precision 2.7 mrad (0.6 arc-sec) after 4 years of averaging on 15 x 15 km (8 x 8 n.m.) grid, but resolves only to 30 km full-wavelength: no net improvement WSOA tracks (no yaw)

40. Current altimeters have poor E-W control, high noise (ERS/GM), and uneven track spacing(Geosat/GM). The above design has a track spacing similar to this from ABYSS proposal. Solution for NIMA-USAF goal d-D altimeter in non-repeat orbit for 4 years yields 0.375 arc-sec (1.8 mrad) VD on tracks 1 n.m. apart. Cost ~$60M plus ~$30M for Pegasus Launch. Can use existing “ABYSS-Lite” design.

41. Something for Everyone? Traditionally, altimeter missions are of two kinds: For VD, a “geodetic orbit” gives spatially dense tracks without temporal repeat. An “exact repeat” orbit with tracks too wide for VD allows monitoring of ocean currents, eddies, tides. Can a mission tailored to VD and bathymetry also monitor ocean variability?

42. Gulf Stream Eddy movie If the animation works, the next slide will play a movie showing how a “geodetic” altimeter (ERS-1, phases E and F) can resolve the mesoscale ocean variability as well as a traditional “exact-repeat” ocean altimeter (Topex/Poseidon). To further prove this, we did not apply ionosphere and troposphere corrections to the ERS-1 data, in order to simulate a low-cost VD mission (ABYSS). The effect works well because the mean sea surface is already well-known at the length scales needed here, and because a “geodetic” orbit has “near repeats”.

44. Thank you for your attention! I mean no disrespect to the WSOA. It is fine for what it was designed to do, which is not VD. Additional slides follow, in case they are useful in later discussions or as background material.

45. VD errors from Satellite Altimetry From “ABYSS” proposal (peer-reviewed by NASA)

46. Scripps Workshop Report A draft of this report is in the background materials on this meeting’s CD. It addresses many new science topics one could do with a bathymetry & VD mission. Sponsorship could be shared among DoD, NSF, NOAA, NASA, and the oil industry.

47. VD gradient: 0.5 arc-sec per nm? 0.5 as/nm implies VGG of 13 Eotvos Gradient exceeds this by at least 5x over rough bottom.

48. d Precision needed in coastal tide models tides are shallow water waves tide model error for 1mrad slope error (T=1/2 day) wavelength ocean surface slope tide height

49. Most of Earth’s Ocean Area Is at Low to Mid- Latitudes Covered by Moderate Inclinations 0 10 20 30 40 50 60 70 80 90 100 % of Total Ocean Area

50. Ocean Area Coverage vs Latitude Coverage