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11.10: Tidal Forces In Other Planet Systems - Geosciences

11.10: Tidal Forces In Other Planet Systems - Geosciences


Tidal features have been observed in other planet systems. For instance, Jupiter’s moon, Europa, is covered with large cracks that are attributed to Jupiter’s enormous gravity pulling on the moon, causing the thick ice crust to fracture (Figure 11.22). Tidal forces release heat, enough to melt large quantities of ice below its surface, allowing the Solar System’s largest oceans to remain liquid.

Jupiter's moon, Io, is perhaps the most geologically active moon in the Solar System (Figure 11.23). Tidal forces between Jupiter and its other moons are generating heat within the moon that are driving volcanic activity, recycling the planets crust every few million years. Tidal forces also play a role in the heat generated within planet Earth, and may have a significant influence on plate-tectonics and magnetic reversals associated with the core.


Tidal disruption of inviscid planetesimals

A planetesimal passing within the Roche limit of a more massive body is subject to tidal forces that could result in disruption of the planetesimal into a spray of debris. The possible occurence of tidal disruption is of considerable importance for planetary accumulation in general and for the origin of the Moon in particular. Previous work has shown that strongly dissipative planetesimals are immune to tidal disruption. We have now examined the effects of tidal forces in the other extreme case, inviscid planetesimals that might arise from collisions energetic enough to result in total melting of the resultant planetesimal and debris. First, a simple analytical calculation implies that massive planetesimals avoid tidal disruption, with the critical mass for disruption being roughly a lunar mass. Second, in order to relax the assumptions inherent in this analysis, we have numerically simulated tidal disruption with the smoothed particle hydrodynamics (SPH) code previously used by two of us to model impacts between protoplanets. SPH models by Cameron and Benz (1991) show that relatively massive (Marssized), inviscid protoplanets do not undergo complete tidal disruption, even in near-grazing incidence collisions. Hence we have concentrated on studying tidal disruption of 0.01 M⦶ planetesimals passing by the Earth with variations in the impact parameter at perigee (rp) and velocity at infinity (ν). Even for these relatively small bodies, significant tidal disruption requires r p < 1.5R ⦶ and ν ∞ < 2 km sec −1 . The SPH models also show that tidal forces during a close encounter efficiently convert orbital angular momentum into spin angular momentum, initiating equatorial mass-shedding in inviscid planetesimals that have been spun up beyond the limit of rotational stability. This rotational disruption occurred for 1.5R ⦶ < r p < 1.9R ⦶ when ν ∞ = 2 km sec −1 but not at all for ν = 0, implying that rotational disruption may be more important than purely tidal disruption for planetary accumulation. Neither disruption process leads to capture of sufficient material in Earth orbit to permit lunar formation from the debris of a single encounter.


The Inner or Terrestrial Planets

Please review the properties of the Inner Planets at Bill Arnett's Nine 8 Planets website, an excellent resource.

Additionally, there is a great deal of information (with no annoying ads, like in the previous link) at NASA's Solar System Exploration page.

Our goal for this lesson is to come up with a list of the similarities between the planets, some trends in their properties, and an understanding of where some of the discrepancies in their properties may lie. To achieve this, we can begin with a simple bullet list that includes a summary of the properties of each of the three Terrestrial planets:

Mercury

  • Surface heavily cratered—in some ways very similar to Earth's Moon.
  • Very dark—only reflects a small amount of Sunlight (it has a low albedo).
  • Temperature on the surface varies from 100K to almost 700K.
  • In a 3:2 synchronous orbit with the Sun—three Mercury days (59 Earth days each) = two Mercury years (88 Earth days each).
  • The iron core of Mercury is very large, taking up most of the interior of the planet and creating a relatively strong magnetic field for a planet of its size.
  • No moon.
  • Less than the size of the Earth.
  • Visited by satellite Mariner 10—most of our knowledge comes from this one mission. The NASA Messenger mission studied Mercury in recent years it entered orbit around Mercury in March 2011 and studied it continuously before crashing to the planet's surface in 2015.
  • Messenger somewhat surprisingly revealed that Mercury has water ice in shadow in craters at its poles!

Venus

  • Thick atmosphere and "runaway greenhouse effect" causes 730K surface temperatures.
  • Atmosphere consists mostly of CO2 (carbon dioxide), clouds of sulfuric acid—no water vapor or oxygen.
  • Huge volcanoes.
  • Clouds are highly reflective (high albedo)—Venus is one of the brightest objects in our night sky.
  • Rotates very slowly (243 Earth days) and it rotates in a retrograde manner (opposite of the Earth & Sun).
  • Not in a synchronous orbit—one Venus year = 225 Earth days.
  • No moon.
  • Almost identical in size to the Earth.
  • A number of missions to Venus have been sent by the USA and USSR. In 1994, Magellan mapped the surface of Venus.

  • Temperature always below the freezing point of water.
  • Giant volcanoes (largest of any in the entire Solar System).
  • Intense, large-scale dust storms.
  • Has an atmosphere that is much thinner than Earth or Venus, mostly CO 2 .
  • The surface is red because of the presence of iron oxide (rust).
  • One Mars day is only slightly longer than an Earth day (24.6 hours).
  • One Mars year is 1.88 Earth years.
  • Polar caps of dry ice and water ice that change with the seasons.
  • Frozen water in a permafrost layer (recently confirmed).
  • About half the size of Earth.
  • Two small, irregularly shaped moons—Phobos and Deimos.
  • The most visited planet in our Solar System, including several recent missions, Mars Exploration Rovers Spirit and Opportunity, the Phoenix Lander, Curiosity, and MAVEN.

Interior structure

There are several peculiarities about these planets, but we can understand the likely causes of these properties when we contrast them to the Earth. First, consider the interior structure of these planets. Just like the Earth, Mercury, Venus, and Mars are differentiated. Curiously, we find that, compared to the other planets, Mercury's core is relatively large. That is, it fills up most of the interior of this planet. Because the core of Mercury is probably mostly made up of iron and nickel and it fills so much of the planet's interior, Mercury is the densest inner planet. We expect that some of the core of each of the inner planets is molten, and we expect that this liquid metallic core must generate electric currents inside the planet as the core rotates with the rest of the planet. These electric currents should create magnetic fields, making the planets behave as if they have giant bar magnets inside them. Both Mercury and the Earth have measurable magnetic fields, but, contrary to our expectations, Venus and Mars do not. For Venus, the reason may be that it rotates too slowly to create the necessary conditions to create a magnetic field. The lack of magnetic field on Mars may be caused by its lack of a molten core.

Albedo

We observe the inner planets easily because they are nearby and shine by reflecting sunlight. You can see Mercury, Venus, and Mars at different times of the year without any instruments like a telescope or binoculars—all three of these planets are visible to your naked eye. Although all of the planets reflect sunlight, different materials reflect different amounts of sunlight. For example, look at the Earth at the NASA: The Blue Marble site. We see that the brightest parts of the Earth are the clouds and the ice near the North Pole. The darkest parts of the globe are the patches of deep ocean that are not covered by clouds. The amount of light reflected by a surface is called the albedo. For light colored clouds, the albedo is high, and much of the incoming sunlight gets reflected. Water and dark colored rock reflect much less light. Mercury is mostly covered by dark rock, so its albedo is very low. On the other hand, Venus is covered by a thick layer of clouds, so it reflects almost all of the Sunlight that hits it. Because of Venus' high albedo and its proximity to the Earth, it is the brightest object in the night sky after the Sun and the Moon.

Atmosphere

Observations of these planets tell us that all three have atmospheres. Mercury has a very thin atmosphere with a strange composition. Mercury is so small that gas easily escapes from its gravitational pull, so its atmosphere must be regenerated continually, perhaps by the bombardment of the surface of the planet by energetic particles from the Sun. Venus, on the other hand, has a thick, dense atmosphere. The greenhouse effect has "run away" on Venus, making the surface temperature hotter than Mercury, even though it is farther from the Sun. Mars' atmosphere is much thinner than Venus or Earth, and it consists primarily of carbon dioxide. The atmosphere of every inner planet originated when the planet was formed and has been supplemented by gas that has escaped from underneath the crust.

The laws of physics tell us that the hotter a gas is, the faster the particles inside that gas move. For Mercury and Mars, these planets are so small that they have weak gravitational fields, which means that much of the gas in their atmosphere can escape if it is warm enough (the average speed of the particles in the gas exceeds the escape velocity from the planet). For Venus and the Earth, which are more massive and have stronger gravitational fields, the lightest gases can escape, but gases like carbon dioxide and nitrogen do not.

Additionally, on the Earth, we know that plant and animal life process the gas in the atmosphere. For example, humans breathe oxygen and exhale carbon dioxide, while plants use carbon dioxide and produce oxygen. Because of these biological processes on the Earth, our atmosphere has evolved significantly from its original content. The Earth's atmosphere may have once been much like Venus' atmosphere is now, however, as plant life on the Earth arose, the process of photosynthesis contributed to the removal of carbon dioxide and the increase in oxygen content.

Rates of rotation

There is also some diversity in the rotation rates of the inner planets. We are most familiar with the Earth. It rotates in a counterclockwise direction in close to 24 hours. Mars is quite similar to the Earth. Its rotation axis is tilted by 24 degrees (Earth's axial tilt is 23.5 degrees), and its rotation period is just over 24 hours. So on Mars, the length of the day and the variations in the height of the Sun above the horizon over the course of one Mars year are similar to what we are used to on the Earth. Mercury has become tidally locked to the Sun, similar to how the Moon is locked to the Earth. However, in the case of Mercury, the planet rotates in 59 days and orbits the Sun in 88 days. What this means is that Mercury rotates completely on its axis (that is, 360 degrees) 3 times in the same amount of time it takes to orbit the Sun 2 times. This is referred to as a 3:2 resonance. In the case of the Moon, since it orbits the Earth 1 time in the same amount of time as it rotates 1 time, this is referred to as a 1:1 resonance. Although this seems a bit peculiar, Venus' rotation properties are much stranger. It takes 243 days to rotate once (and there is no tidal locking, so the Venus year has nothing to do with the Venus day), but it rotates in the opposite sense compared to Mercury, Earth, and Mars. All of these planets orbit the Sun in a counterclockwise direction, and Mercury, Earth, and Mars rotate counterclockwise around their axes. Venus rotates slowly in a clockwise manner around its axis. One theory is that a collision or collisions with Venus by other objects caused its rotation sense to reverse, but an alternative theory is that the tidal influence of the Sun and the other planets in the Solar System may also have caused this unique rotation.

Want to learn more?

Recall that Starry Night allows you to travel to the surface of each planet. If you would like to compare the rotation rates, orientation of the axis of rotation, and orbital periods for each planet, you can travel to Mercury's, Venus', or Mars' surface, set the time flow rate to 3000x (or the time step rate to say 1 Earth day), and as time goes by note the location, date, and time of sunrise and sunset, the height of the Sun above the horizon, and the length of 1 day on that planet.

You can also choose to hover over a point on the surface, and if you choose to hover over the north pole of each planet, you can watch them rotate below you. For Venus, you will have to turn up the time flow rate to a very high number in order to see it rotate at all.


Subtidal frequency estuary-shelf interaction: Observations near Delaware Bay

Interaction between a large estuary and the adjacent inner continental shelf at subtidal frequencies occurs through a variety of physical processes. I use recent field observations to study these processes near the weakly stratified estuary of Delaware Bay on the east coast of the United States. The primary observations were shipboard hydrography and long time series of current, temperature, and conductivity. The observations revealed three distinct spatial regions for the coupled circulation between estuary and shelf: the estuary mouth, the inner shelf where the mean flow is landward, and a buoyancy-driven coastal current. The coastal current is the principal discovery of the work. It begins near the estuary mouth as lighter water from Delaware Bay exits the mouth on the right side when viewed to seaward. Initially, the current is about one internal Rossby radius in width, but it broadens as it flows seaward to reach a width of about 20 km off the coast of Delaware. Observed mean currents were 3–5 cm s −1 there. In all three regions temporal variability in current and salinity was induced by variations in alongshore wind and river discharge into the estuary, the latter occurring mainly at very long periods of several weeks and longer.


3 Discussion

There is a relative paucity of tidally aligned grooves S and SE of the sub-Mars point. This might be attributed to the absence of a cohesive material in this location (e.g., coarser regolith [Scheeres et al., 2010 ]). Limited spectroscopy data show regional compositional differences that might correspond to varying regolith elastic properties [Witasse et al., 2014 Rivkin et al., 2002 ]. Nonaligned grooves are not random (cooler colors in Figure 3) but cluster in the equatorial zones between the sub-/anti-Mars tidal bulges they require an alternative or additional explanation for their formation. Considering that the predominant formation orientations are at high angles to the maximum compression direction, such grooves could represent contractional failure of surface materials. Alternatively, they may have formed under different stress conditions, such as the stresses produced when Phobos experienced nonsynchronous rotation before becoming tidally locked [Weidenschilling, 1979 ]. They are found predominantly in the orbital leading hemisphere, which might also allow for their formation by sweep up of coorbiting material derived from Phobos, Deimos, or even Mars [Murray and Heggie, 2014 ].

Morphologically, grooves resemble extensional tectonic features. The majority of grooves form remarkably linear, consistent topographic depressions in distinct sets. Groove widths and spacings are relatively homogeneous, as is typical of extensional fracture sets in a uniform thickness brittle layer. Scalloped edges along many grooves are consistent with pit chain morphologic evolution, whereby initially isolated pits merge along strike to create a continuous feature, as is described across the solar system [Horstman and Melosh, 1989 Ferrill et al., 2004 Wyrick et al., 2004 Nahm and Kattenhorn, 2015 ]. Initially isolated pits in loose regolith align above an underlying dilational crack, into which regolith progressively drains. The circular edges of the original pits create the scalloped margin effect, although linear margins may also develop if the underlying structure is a dilational fault or graben [e.g., Ferrill et al., 2004 Grant and Kattenhorn, 2004 ]. Bright margins along shadowed edges of some grooves imply upturned flanks, which can be produced by footwall uplift along normal faults. Inferred extensional deformation thus justifies comparisons to the principal extension direction to determine the viability of an orbital decay driving mechanism.

Where two or more crosscutting sets with disparate orientations exist (Figure 3), multiple stages of tectonic extension and direction are implied. In such cases, relative ages of groove sets can be determined either by direct evidence of crosscutting relationships or by their ages relative to craters that formed intermediate to the two groove sets. For example, some of the ENE oriented groove sets west of Stickney (cooler colors in Figure 3) have pit chain morphologies that imply that they formed in extension, despite currently experiencing compression in the contemporary orbital decay stress field. Moreover, they are older than the N-S grooves in the same region that show a strong correlation to the orbital decay stress field. These relationships are observed elsewhere across Phobos. Hence, orbital decay is likely to be the key driving force behind the most recent Phobos deformation. Older extensional features imply that ancient stress fields differed from that produced spatially by orbital decay today, perhaps explaining mismatches (purple grooves) in Figure 3. In rare cases (e.g., the NW oriented groove projecting NW away from Stickney crater), linear features show less resemblance to pit chains and may be catenae (aligned impact craters). In such cases, the pits (craters) vary in size and spacing and are typically larger than pits caused by the draining of regolith. The catena mentioned above almost encircles Phobos, creating a sinusoidal trace across the surface in the fracture map (Figure 3) and predates the youngest tectonic grooves.

Orbital decay from 3.0 to 2.77 RMars can produce

50 kPa of tensile stress at the sub-/anti-Mars point (Figure 4 the 0° longitude and 0° latitude point in Figure 2). While these stresses may seem low, experiments with Lunar simulant suggest that tensile failure occurs at stresses as low as 1 kPa [Arslan et al., 2008 ]. We find that stresses from orbital decay are up to 50 times larger, making deep groove formation plausible by our proposed mechanism and high enough to drive fractures to depths of up to 5 km, thus providing a mechanism to create dilational cracks below the surface layer into which regolith may have drained to form surface pit chains.

50 kPa. The gray dot represents the maximum stress that could be generated if Phobos migrated from the synchronous orbital radius, which results in >100 kPa of stress.

These stress levels were theoretically achievable early in the history of Phobos' ongoing orbital decay (Figure 4), so in principle, there could be generations of fractures and a range of evolutionary morphologies. Hence, grooves do not all have to be young according to our model, consistent with the apparent protracted history of their formation [Weidenschilling, 1979 ]. As successive generations of grooves form, their pattern of orientations should remain relatively consistent over time as long as Phobos' tidal alignment with Mars remains constant. Impact events might break the tidal alignment occasionally, but Phobos' irregular shape presumably helps to quickly reestablish a tidally spin-locked system.

While the bulk rigidity (μB) of Phobos is not constrained, the bulk rigidity for the nominal model presented here for Phobos is

10 6 Pa. This value is lower than the lower bound estimate for Phobos' rigidity of 5 × 10 8 Pa [Yoder, 1982 ]. Surveys of 1 km class asteroid binaries have observed asteroids with rigidities ranging from 10 5 Pa to 10 8 Pa with a median value of

5 × 10 6 Pa [Taylor and Margot, 2011 ], assuming the tidal quality factor for dissipation (Q) is similar to rocky bodies, Q

100. If Phobos behaves like these smaller asteroids, it is plausible that its bulk rigidity can be

10 6 Pa. However, larger 100 km class binaries were found to be more rigid, ranging from 10 7 Pa to 10 13 Pa with a median value of

10 9 Pa (again assuming Q = 100) [Taylor and Margot, 2011 ], but these bodies are gravity dominated and Phobos' size places it at transition point between strength and gravity-dominated bodies [Asphaug et al., 2015 ] therefore, we posit that its rigidity could be similar to the 1 km class asteroids.

A finding that Phobos' interior is weak is consistent with hypothesized formation scenarios. If Phobos was formed by the accretion of debris in Martian orbit, resulting from a large impact [Citron et al., 2015 ], then the process of moon building may have resulted in Phobos being inherently weak, with a core similar to a rubble pile. Predictions of the rigidity of a 10 km rubble pile yield values

10 7 Pa [Goldreich and Sari, 2009 ], which are only an order of magnitude stronger than the model in this study. Regardless, the tidal model for fracture formation can still produce large stresses on Phobos for other assumptions of layer rigidities (Figure 5).

6 RMars to the current orbital radius (gray dot in Figure 4). This model is not unique and a variety of combinations of models with inner layer rigidity (μi) and outer layer rigidity (μo) can produce the stress in excess of 100 kPa, when two-layer modes with the indicated rigidity values are evaluated (see Appendix B and Appendix C). Our nominal model produces a bulk rigidity (μB) of

Our assumption that the outer layer is 100 m thick is a conservative upper limit, as the estimated thickness of the regolith varies from 5 to 100 m [Basilevsky et al., 2014 ]. The effect of thinner shells is explored in Appendix D and in general results in higher levels of stress. Therefore, we find that our results are robust and are valid even if future investigations of Phobos can place different constraints on some of the values we assume for Phobos' interior properties. Once fractures have been formed on Phobos' surface, they can remain active through diurnal tidally driven motions. Phobos' orbital eccentricity produces a dynamic diurnal stress field, which changes throughout Phobos' 7.6 h orbit. While these stresses are smaller by an order of magnitude, they can still work to dilate the grooves similar to the tidal dilation of fractures on Enceladus [Hurford et al., 2007 ]. The reworking of fractures by these stresses may be natural seismic sources [Lee et al., 2003 ], which could be exploited by future missions to probe Phobos' interior. However, Phobos quakes are only possible if the interior is weak, allowing an enhancement of tidal diurnal stress. Thus, a seismometer could quickly determine the state of Phobos' interior it would be rigid if there are no quakes on the diurnal time scale, and weak if there are quakes. Furthermore, Phobos quakes would allow a natural way to probe the properties of Phobos' internal structure.

The presence of tidally driven fracturing on Phobos does not imply its imminent catastrophic disruption, particularly given that a friction angle of only

3° is sufficient to prevent downslope movement today, even in the complete absence of cohesion [Holsapple, 2001 ]. Tidal disaggregation of Phobos will occur only when it is much closer to Mars and is suggested to occur in 20–40 Ma [Black and Mittal, 2015 ]. But meanwhile, Phobos is weak enough internally, on the time scale of orbital migration, to permit global deformation that results in stresses to build up within an elastic lunar-like outer layer. According to our model, its failure is opening up granular fissures, an interpretation that is consistent with the hypothesis that the pitted grooves are formed by regolith draining into fractures [Horstman and Melosh, 1989 ].


About the Author

Gerd Masselink is a Professor in Coastal Geomorphology and Associate Head of Marine Science in the School of Marine Science and Engineering at Plymouth University, UK.  Gerd specialises in nearshore sediment transport processes, surf zone hydrodynamics and beach morphodynamics.

Roland Gehrels
is a Professor in Physical Geography at the University of York, UK. He studies sea-level changes over various timescales, but has a particular interest in regional sea-level variability during past centuries. Roland is the President of the Commission on Coastal and Marine Processes of the International Quaternary Union (INQUA).

The Editors have published over 160 peer-reviewed articles in coastal and sea-level research.


IGBT Applications

B. Jayant Baliga , in The IGBT Device , 2015

15.5 Tidal Power

Tidal power is fundamentally different from wave power because tides are generated due to gravitational pull of the moon and the Sun [49] rather than energy delivered by the Sun to the Earth via solar radiation which fuels the production of waves. The first large power generation installation using tidal energy was the Rance Tidal Power Station set up in France in 1966 with an installed capacity of 240 MW. Other large tidal electricity generation installations are the 20 MW Annapolis Royal Generation Station set up in Nova Scotia, Canada, in 1984 the 3 MW Jiangxia Tidal Power Station in Hanzhou China in 1985 the 254 MW Sihwa Lake Tidal Power Plant set up in South Korea in 2011 and the 50 MW Tidal Power Station in Gujarat India in 2012. An advantage of tidal power when compared with wind or solar power is its consistency due to regular orbit of the moon around the Earth. Tides create energy loss that slows down Earth's rotation causing the period of Earth's rotation to increase from 21.9 to 24 h during the last 620 million years. Extraction of tidal energy has a negligible impact on this phenomenon. The 1.4 MW SeaGen tidal energy generation station is shown in Fig. 15.45 . It takes advantage of 400 millions gallons of water that flows in and out of Strangford Lough in the United Kingdom twice a day due to the tides [50] .

Figure 15.45 . SeaGen tidal energy generation station.

The fluctuating energy produced by the tidal energy generator can be delivered to the grid using the topology previously shown in Fig. 15.44 where the variable AC power is first rectified to create a DC bus. The regulated energy to the grid is then delivered using PWM control of an IGBT inverter stage. A demonstration of the “stream power” technology has been reported by using power electronic loads [51] .


4 Analysis of the Results

4.1 Comparison of NMT and CWU Solutions

The comparison of the velocity estimates from CWU and NMT analyses shows that the secular rate estimates from each analysis match at the level of 0.11 mm/yr in north and east and 0.40 mm/yr in height when all stations are included in the comparison (Table 4). When stations with velocity standard deviations less than the median standard deviations are compared, the WRMS differences between the velocity estimates reduces to <0.08 in north and east and 0.28 mm/yr in height (Table 5). Here we consider in more detail the nature of the comparison between the two analyses. First, we compare the differences in velocity estimates and then the differences in position estimates. Figure 4 shows histograms of the differences in velocity estimates from the two GAGE analysis centers. Overall, these differences appear to be Gaussian in shape with little skewness. The mean difference in the height velocity from the histogram (−0.15 mm/yr) differs slightly from that reported in Table 4 (−0.07 mm/yr) because the table reports the weighted mean of the height velocity differences.

We can also examine the differences in the daily position estimates. For each station, we differentiate the time series of the position estimates and find the weighted mean differences in north, east, and height. The histograms of the weighted mean of the differences are shown in Figure 5. In the horizontal components, the mean, and median differences are small and less than or equal to 0.10 mm. The mean height differences show a small bias of −0.65 and −0.81 mm for the mean and median differences. The observation that the horizontal components are unbiased is not unexpected, since each daily position estimate is aligned to the same reference frame we note that this alignment is made with

575 stations while the histograms are generated for all 2154 stations in the analysis.

Although the mean of height differences between the analyses by the two ACs are small, the temporal behavior of the height differences is complex as shown Figure 6. Here we show the time series of the estimates of the averages of the height differences at the reference frame stations for NMT, CWU, and combined analyses and the differences of the means between the AC analyses and the PBO combined solution. The curves have been offset in order to show the patterns more easily. Two aspects of the figure are very clear. The PBO combined mean height difference (MHD) (black curve) nearly tracks the CWU MHD estimates (blue curve) exactly. This tracking is very clear in the time series that shows the difference between the CWU and PBO MHD values (magenta, offset −20 mm). The other feature of Figure 6 is that the NMT analysis shows long-term systematic differences, which at times can exceed 10 mm for extended periods of time. Between 1999 and 2003, the mean difference for NMT is −5.9 mm compared to −0.4 mm for CWU. The period from 1999 to 2003 also covers the period of the sunspot maximum in solar cycle 23 [e.g., Nandy et al., 2011 ]. We initially inferred that that this correlation may arise because of the neglect of higher-order ionospheric delay corrections [Kedar et al., 2003 Hernández-Pajares et al., 2007 ] in the reprocessing. Trial re-processing of data over this time frame with higher-order ionospheric delay corrections applied shows that this neglect is not the direct cause of the offset. (Higher-order ionospheric delay corrections are applied in standard processing since 05 November 2014 for CWU and 14 June 2015 for NMT.). Analysis of the behavior of the NMT solution reveals that the offset arises because of scale-like correlations in the position estimates and strong correlations between network translations and scale. We believe that the bias in the NMT GAMIT solution arises because the GAGE network only covers one quadrant of the globe. The double difference operator in GAMIT, which effectively estimates all receiver and satellite clocks, results in scale-like correlations that ultimately manifest in the height estimate difference (i.e., common-mode errors in the clock estimates will project as common height offsets in all the stations). Explicitly constraining a scale estimate is one way of reducing the effects of these correlations but that would impose a condition that the mean height differences at the reference frame stations be zero. The solution that we are now testing reduces these correlations by combining the NMT GAMIT solution with overlapping stations from a global double-difference network analysis. We are evaluating the use of the MIT submissions to IGS operational and reprocessing campaigns for this purpose. Initial analyses do show that the NMT MHD are reduced to levels similar to the CWU PPP solutions when the covariance matrix and position estimates for the sites common between the GAGE analyses and MIT IGS analyses are included in the combined solution. For the CWU GIPSY PPP solutions, the satellite clock values are fixed based on a global analysis from the NASA Global Geodetic Network (GGN), and these fixed clocks reduce the effect of these correlations on individual station height position estimates.

As mentioned above, the impact of the correlations in the NMT solutions can be seen if scale change parameters are explicitly estimated. Although the typical standard deviation of the height estimates in the NMT and CWU solutions are similar (due to the reweighting factors discussed in section 2.5), the standard deviations of scale estimates differ by a factor of

5. For this reason, the mean height differences in the combined solution are dominated by the CWU contribution. The CWU solution dominates for parameter estimates that average over a large number of stations, such as scale. The coordinates of individual stations behave more like the simple average of the two solutions.

Histograms of the weighted RMS scatter of the differences in position estimates between the CWU and NMT analyses are shown in Figure 7. The median WRMS scatter of the differences is less than 1 mm in north and east and less that 5.5 mm in height. The stations with the largest WRMS differences and NRMS differences are given in Table 6. Some of the stations with large WRMS differences have values of the NRMS scatter below unity, indicating that the overall noise level at these stations is high. In some cases, these large differences are associated with poor sky view at the station that arises because of obstructions from vegetation, landforms (e.g., hills and cliffs), and structures. In other cases, partial antenna failures have occurred yielding incomplete raw GPS observations. In these cases, enough data are collected to allow position estimates to be obtained by the ACs, but these estimates are corrupted. In some cases, these antenna failures show large seasonal deviations in the horizontal coordinates.

Mean Difference WRMS Scatter of Differences
Station No. of Days (mm) (mm) NRMS Scatter of Differences
Sorted by North WRMS
LONG 7110 1.20 3.59 0.83
BLYN 4555 −0.04 3.61 0.78
WDCB 1595 1.90 3.99 0.63
LOZ1 2680 −3.68 7.16 0.95
EISL 2673 −0.68 8.57 0.89
Sorted by East WRMS
LOZ1 2680 −2.09 4.76 0.75
MHMS 5749 3.00 4.83 1.46
HCES 4767 2.35 4.90 1.51
P561 3807 9.76 5.08 0.95
EISL 2641 −1.10 12.33 1.32
Sorted by Height WRMS
COUP 3297 −13.03 26.22 2.46
NJCM 3151 17.19 31.68 2.88
NJOC 3390 14.04 33.06 3.29
SGU1 1337 8.48 35.89 3.66
LOZ1 2676 22.43 36.04 0.87
Sorted by North NRMS
AC33 2704 −1.05 3.22 0.94
QHTP 4905 0.67 2.63 0.94
LOZ1 2680 −3.68 7.16 0.95
P656 1195 −0.89 3.25 0.97
LJRN 5376 −2.71 3.53 1.08
Sorted by East NRMS
AV04 3347 0.23 3.58 1.14
EISL 2641 −1.10 12.33 1.32
LJRN 5374 −0.63 3.85 1.35
MHMS 5749 3.00 4.83 1.46
HCES 4767 2.35 4.90 1.51
Sorted by Height NRMS
PTAL 4876 13.37 18.20 1.95
COUP 3297 −13.03 26.22 2.46
NJCM 3151 17.19 31.68 2.88
NJOC 3390 14.04 33.06 3.29
SGU1 1337 8.48 35.89 3.66
  • a Values are sorted for the largest WRMS differences (mm) and NRMS differences. Only stations with more 1000 measurements are included.

We can also compare the WRMS scatters of the position time series fits for the CWU, NMT, and combined PBO results. The medians of the WRMS scatters of the position NEU time series are given in Table 7. We see in the table that the combined solution has WRMS scatters that are less than or equal to each AC showing that even with just two ACs, the combination has reduced or equal scatter over the two contributing solutions. We also note that despite the large scatter in the mean height estimates (Figure 6) from the NMT solution compared to the CWU solution, the WRMS scatter in height of individual stations is slightly smaller for the NMT solution (and smallest for the combined solution).

Analysis No. of Stations Median N (mm) Median E (mm) Median U (mm)
CWU 2160 1.32 1.28 6.02
NMT 2169 1.11 1.18 5.83
PBO 2170 1.11 1.13 5.38
  • a Data spanning 1999 to 14 November 2015 are used in these statistics. The numbers of stations differ slightly because of small differences in the list of stations processed by each AC.

4.2 Spatial Distribution of the Quality of Position Estimates and Phase Data Noise

There are multiple statistics that we could use to assess how the quality of the position estimates of the stations used in the GAGE analysis depends on where the stations are located. As a general trend, stations in regions with little vegetation and low humidity (exemplified by the Basin and Range province) have smaller WRMS scatters of position estimates than stations in regions with large amounts of vegetation and high humidity (such as the Caribbean), as first noted by Mao et al. [ 1999 ]. Different metrics for assessing the quality of the station position estimates are shown in Figures 8-12. An overall spatial pattern of performance metrics of the stations, as judged by RMS scatter of different geodetic quantities, is similar for all of the metrics. Figure 8 shows the station averages of the phase residual RMS scatter over