Vertical Deflection of Gravity & its Correlation with Bathymetry

1. Vertical Deflection of Gravity& its Correlation with Bathymetry Walter H. F. Smith NOAA Lab for Satellite Altimetry Silver Spring, Maryland

2. Something for Everyone? Do Ocean Modelers and Inertial Navigators have common interests/requirements? Inertial Nav needs Vertical Deflections. VDs also show us bathymetry in poorly charted areas. Ocean models are sensitive to bathymetry. This synergy is the “ABYSS” mission concept. Can a mission tailored to VD and bathymetry also monitor mesoscale variability?

3. 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?

4. What is a Vertical Deflection? Geologic structure 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.

5. 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 But why, since altimetry measures instantaneous sea surface height plus errors, and not “geoid” height?

6. What size are VD signals? Typical global average value: ~25mrad (~5 arc-sec) Typical large value: 200 to 400mrad (40 to 80 arc-sec) at trenches, 75 to 150 (15 to 30 arc-sec) at seamounts Extreme values: near 500mrad (100 arc-sec) on flanks of Hawaii What is the smallest interesting value? Signals as small as a fewmrad (~1 a-s), perhaps as small as 1mrad (0.2 a-s), may be correlated with small seafloor geology signals. NIMA-USAF goal is 2.4 mrad (0.5 arc-sec). What is significant for Navy needs?

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

8. What size are VD errors? Dominant error, present world-wide, is a ~5mrad (~1 arc-sec) random noise due to ocean surface waves. Local errors of 3 to 10 mrad (0.6 to 2 arc-sec) caused by very fast currents (Gulf Stream, Kuroshio) and poorly known shallow water tides (Java Sea, Yellow Sea). Negligible errors (< 1mrad, or 0.2 arc-sec) come from large scale oceanography (El Niño, seasonal & inter-annual variability, planetary waves, general circulation) and from systematic altimetry errors (orbit error, ionosphere & troposphere delays, sea state bias). A VD mission needs ONLY a simple, low-cost altimeter (one frequency, no radiometer, e.g. GEOSAT, ABYSS or ABYSS-Lite).

9. 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.

10. 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.

11. 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).

12. 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.)

13. 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.

14. Can we improve existing data? David T. Sandwell (Scripps) and I are “retracking” existing altimeter data. This is experimental. We started with ERS-1 data because these are noisier than GEOSAT. Following are some preliminary results. We don’t know yet whether we can achieve similar things with GEOSAT. Perhaps not, as GEOSAT seems to have been a better-behaved instrument in the first place.

15. ERS-1 retracking: 1 “Retracking” means re-examining raw radar waveforms to estimate the sea surface height better than the on-board algorithm did. This can recover data that were “lost” by the on-board algorithm, near shore or in rain.

16. Retracking, 2: Coastal Approach Retracking saves “lost” data, especially approaching coastlines. Data remain lost on retreat from coasts, however. (True of all existing altimeters.) Need d-D instrument for better littoral tracking.

17. Retracking, 3: repeat profiles rad rad ERS-1, descending arcs in the South Pacific, low signal, high SWH, with some rain cells

18. Spatial Resolution Assessment Method: Examine “coherency” (correlation versus wavelength) of exact- repeat tracks. ERS-1 shown here before and after “retracking” radar waveforms. Result: Full-wavelengths down to ~40 km are resolved in poor conditions (high sea state, low signal) and down to ~25 km in good conditions (calm seas, large signal). Resolution depends on both signal and noise spectra.

19. Another error issue: Anisotropy The analysis so far 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.

20. 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.

21. 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

22. 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

23. Anisotropic error by Latitude 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.

24. 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)

25. Compare w/ NIMA-USAF goal • Accuracy: • 0.5 arc-sec (2.4 mrad) • Spatial sampling • 1 nautical mile (1.8 km) grid 0.5 arc-sec is not achievable with existing data. Can be achieved by one d-D altimeter in orbit for four years (or constellation of d-D altimeters in shorter time) Cannot be achieved by Wide Swath Ocean Altimeter, as currently designed, for several reasons.

26. 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)

27. 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.

28. 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.

29. 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.

30. Example: Navy’s Operational Model NLOM Forecast models require correct global 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.

31. Bottom Roughness Controls Mixing Less mixed water More mixed water Smoother bottom Rougher bottom Spatial variations in bottom roughness change mixing rates by order of magnitude (vertical diffusivity < 105at left and > 104 at right; actual in situ data shown). 10–30 km l bathymetry controls mixing Seafloor spreading shapes bathymetry at these scales.

32. Mixing and Ocean Circulation Changing the distribution of ocean mixing leads to changes in the modeled ocean circulation

33. I promised to talk about bathymetry • A proven technique • Needs only simple altimetry (Geosat, w/ no troposphere or ionosphere measurement, did just fine.) • Has resolved many interesting tectonic features • 1st order plate tectonics confirmed • 2nd order mysteries found Altimetric Bathymetry A new VD mission will do better! Abyssal hill scales still to come.

34. “Foundations” Seamounts

35. “Foundations” Bathymetry Profile (km)

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

37. A VD mission to ~1 mrad should do this well Correlation This limit is physical, not instrumental

38. What new science is in this band? Physical limit due to upward cont’n. What critical problems lie at these scales? Current Altimetry Full-wavelength (km) Is this resolution improvement sufficient to characterize nearly all the interesting bottom roughness properties? Can it capture the transition to fractal topography?

39. 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)

40. 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.

41. VD requires 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.

42. 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

43. Anisotropy in Error & Resolution Gravity and Bathymetry from VD flow chart Anisotropy Here

44. Gravity and bathymetry can be correlated Topography generated by ocean crustal processes is related to ocean surface gravity anomalies through a simple filter. Exploitation of satellite gravity can thus yield filtered depths, if geologic conditions are right: Ocean crust w/ thin sediment. Continental margin basins are different; gravity there shows sub-surface structure. Flexure Flexure

45. Upper limit bounded by physical law: upward continuation Plate thickness affects long l Topography to gravity bandpass filter “Isostatic compensation” attenuates topographic gravity at full-wavelengths longer than ~160 km. “Upward continuation” limits resolution when full-wavelength << 2 p x distance from sea floor to gravity measurement (sea surface, shown here, or in space).

46. Recipe: satellite gravity interpolates sparse ship bathymetry surveys

47. Example: SW Indian Ocean

48. Equatorial Atlantic tectonic fabric

49. mbes

50. Histogram of horizontal differences between 2500m contours from ETOPO5, Predicted, GEBCO and multibeam survey.Multibeam is considered true and the displacement of the other three measured seawards (+) or landward (-). S44 Uncertainty