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59-578: Lac Vieux Desert has a surface elevation 1,680 ft (510 m) above sea level , and a maximum depth of about 40 ft (12 m). The surface area is 4,260 acres (17.24 km; 6.66 sq mi), of which approximately two-thirds is in Wisconsin and one third in Michigan. Located in the Lake District of northern Wisconsin, the lake is a popular boating and fishing resort. The lake

118-459: A system of linear equations formulated via linearization of M {\displaystyle M} : where the partial derivatives are: Longer arcs with multiple intermediate-latitude determinations can completely determine the ellipsoid that best fits the surveyed region. In practice, multiple arc measurements are used to determine the ellipsoid parameters by the method of least squares adjustment . The parameters determined are usually

177-409: A defined barometric pressure . Generally, the pressure used to set the altimeter is the barometric pressure that would exist at MSL in the region being flown over. This pressure is referred to as either QNH or "altimeter" and is transmitted to the pilot by radio from air traffic control (ATC) or an automatic terminal information service (ATIS). Since the terrain elevation is also referenced to MSL,

236-423: A difference of the major and minor semi-axes of approximately 21 km (13 miles) (more precisely, 21.3846857548205 km). For comparison, Earth's Moon is even less elliptical, with a flattening of less than 1/825, while Jupiter is visibly oblate at about 1/15 and one of Saturn's triaxial moons, Telesto , is highly flattened, with f between 1/3 and 1/2 (meaning that the polar diameter is between 50% and 67% of

295-432: A few metres, in timeframes ranging from minutes to months: Between 1901 and 2018, the average sea level rose by 15–25 cm (6–10 in), with an increase of 2.3 mm (0.091 in) per year since the 1970s. This was faster than the sea level had ever risen over at least the past 3,000 years. The rate accelerated to 4.62 mm (0.182 in)/yr for the decade 2013–2022. Climate change due to human activities

354-507: A mathematical reference surface, this surface should have a similar curvature as the regional geoid; otherwise, reduction of the measurements will get small distortions. This is the reason for the "long life" of former reference ellipsoids like the Hayford or the Bessel ellipsoid , despite the fact that their main axes deviate by several hundred meters from the modern values. Another reason

413-438: A reference datum for mean sea level (MSL). It is also used in aviation, where some heights are recorded and reported with respect to mean sea level (contrast with flight level ), and in the atmospheric sciences , and in land surveying . An alternative is to base height measurements on a reference ellipsoid approximating the entire Earth, which is what systems such as GPS do. In aviation, the reference ellipsoid known as WGS84

472-406: A shape which he termed an oblate spheroid . In geophysics, geodesy , and related areas, the word 'ellipsoid' is understood to mean 'oblate ellipsoid of revolution', and the older term 'oblate spheroid' is hardly used. For bodies that cannot be well approximated by an ellipsoid of revolution a triaxial (or scalene) ellipsoid is used. The shape of an ellipsoid of revolution is determined by

531-470: A sharp reduction in greenhouse gas emissions, this may increase to hundreds of millions in the latter decades of the century. Local factors like tidal range or land subsidence will greatly affect the severity of impacts. For instance, sea level rise in the United States is likely to be two to three times greater than the global average by the end of the century. Yet, of the 20 countries with

590-428: A theoretical coherence between the geographic latitude and the meridional curvature of the geoid . The latter is close to the mean sea level , and therefore an ideal Earth ellipsoid has the same volume as the geoid. While the mean Earth ellipsoid is the ideal basis of global geodesy, for regional networks a so-called reference ellipsoid may be the better choice. When geodetic measurements have to be computed on

649-572: Is a judicial one: the coordinates of millions of boundary stones should remain fixed for a long period. If their reference surface changes, the coordinates themselves also change. However, for international networks, GPS positioning, or astronautics , these regional reasons are less relevant. As knowledge of the Earth's figure is increasingly accurate, the International Geoscientific Union IUGG usually adapts

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708-453: Is a mathematical figure approximating the Earth's form , used as a reference frame for computations in geodesy , astronomy , and the geosciences . Various different ellipsoids have been used as approximations. It is a spheroid (an ellipsoid of revolution ) whose minor axis (shorter diameter), which connects the geographical North Pole and South Pole , is approximately aligned with

767-405: Is a type of vertical datum  – a standardised geodetic datum  – that is used, for example, as a chart datum in cartography and marine navigation , or, in aviation, as the standard sea level at which atmospheric pressure is measured to calibrate altitude and, consequently, aircraft flight levels . A common and relatively straightforward mean sea-level standard

826-488: Is because the sea is in constant motion, affected by the tides, wind , atmospheric pressure, local gravitational differences, temperature, salinity , and so forth. The mean sea level at a particular location may be calculated over an extended time period and used as a datum . For example, hourly measurements may be averaged over a full Metonic 19-year lunar cycle to determine the mean sea level at an official tide gauge . Still-water level or still-water sea level (SWL)

885-427: Is due to change in either the volume of water in the world's oceans or the volume of the oceanic basins . Two major mechanisms are currently causing eustatic sea level rise. First, shrinking land ice, such as mountain glaciers and polar ice sheets, is releasing water into the oceans. Second, as ocean temperatures rise, the warmer water expands. Many factors can produce short-term changes in sea level, typically within

944-483: Is increasingly used to define heights; however, differences up to 100 metres (328 feet) exist between this ellipsoid height and local mean sea level. Another alternative is to use a geoid -based vertical datum such as NAVD88 and the global EGM96 (part of WGS84). Details vary in different countries. When referring to geographic features such as mountains, on a topographic map variations in elevation are shown by contour lines . A mountain's highest point or summit

1003-408: Is instead a long-term average of tide gauge readings at a particular reference location. Sea levels can be affected by many factors and are known to have varied greatly over geological time scales . Current sea level rise is mainly caused by human-induced climate change . When temperatures rise, mountain glaciers and polar ice sheets melt, increasing the amount of water in the oceans, while

1062-507: Is part of a more encompassing geodetic datum . For example, the older ED-50 ( European Datum 1950 ) is based on the Hayford or International Ellipsoid . WGS-84 is peculiar in that the same name is used for both the complete geodetic reference system and its component ellipsoidal model. Nevertheless, the two concepts—ellipsoidal model and geodetic reference system—remain distinct. Note that the same ellipsoid may be known by different names. It

1121-450: Is the level of the sea with motions such as wind waves averaged out. Then MSL implies the SWL further averaged over a period of time such that changes due to, e.g., the tides , also have zero mean. Global MSL refers to a spatial average over the entire ocean area, typically using large sets of tide gauges and/or satellite measurements. One often measures the values of MSL with respect to

1180-727: Is the main cause. Between 1993 and 2018, melting ice sheets and glaciers accounted for 44% of sea level rise , with another 42% resulting from thermal expansion of water . Sea level rise lags behind changes in the Earth 's temperature by many decades, and sea level rise will therefore continue to accelerate between now and 2050 in response to warming that has already happened. What happens after that depends on human greenhouse gas emissions . If there are very deep cuts in emissions, sea level rise would slow between 2050 and 2100. It could then reach by 2100 slightly over 30 cm (1 ft) from now and approximately 60 cm (2 ft) from

1239-531: Is typically illustrated with the AMSL height in metres, feet or both. In unusual cases where a land location is below sea level, such as Death Valley, California , the elevation AMSL is negative. It is often necessary to compare the local height of the mean sea surface with a "level" reference surface, or geodetic datum, called the geoid . In the absence of external forces, the local mean sea level would coincide with this geoid surface, being an equipotential surface of

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1298-667: The International Game Fish Association as the all-tackle world record was caught in Lac Vieux Desert. It was 54.0 inches (137 cm) long and weighed 23.21 kilograms (51 lb 3 oz). The record has stood longer than any other fishing record in either state. Sea level Mean sea level ( MSL , often shortened to sea level ) is an average surface level of one or more among Earth 's coastal bodies of water from which heights such as elevation may be measured. The global MSL

1357-529: The Upper Peninsula of Michigan , and two areas along the Indiana border with Berrien County in southern Michigan. There are multiple resorts and campgrounds on the lake. Most of the shoreline on the Wisconsin side of the lake is developed, however dense forest surrounds many of the properties. On July 16, 1919, a tiger muskellunge claimed by both Wisconsin and Michigan as a state record and by

1416-410: The flattening f , defined as: That is, f is the amount of flattening at each pole, relative to the radius at the equator. This is often expressed as a fraction 1/ m ; m = 1/ f then being the "inverse flattening". A great many other ellipse parameters are used in geodesy but they can all be related to one or two of the set a , b and f . A great many ellipsoids have been used to model

1475-570: The geoid of the Earth, which approximates the local mean sea level for locations in the open ocean. The geoid includes a significant depression in the Indian Ocean , whose surface dips as much as 106 m (348 ft) below the global mean sea level (excluding minor effects such as tides and currents). Precise determination of a "mean sea level" is difficult because of the many factors that affect sea level. Instantaneous sea level varies substantially on several scales of time and space. This

1534-418: The height above mean sea level (AMSL). The term APSL means above present sea level, comparing sea levels in the past with the level today. Earth's radius at sea level is 6,378.137 km (3,963.191 mi) at the equator. It is 6,356.752 km (3,949.903 mi) at the poles and 6,371.001 km (3,958.756 mi) on average. This flattened spheroid , combined with local gravity anomalies , defines

1593-405: The interior , as well as the subsequent flattening caused by the centrifugal force from the rotation of these massive objects (for planetary bodies that do rotate). Because of their relative simplicity, reference ellipsoids are used as a preferred surface on which geodetic network computations are performed and point coordinates such as latitude , longitude , and elevation are defined. In

1652-562: The 1690s. Satellite altimeters have been making precise measurements of sea level since the launch of TOPEX/Poseidon in 1992. A joint mission of NASA and CNES , TOPEX/Poseidon was followed by Jason-1 in 2001 and the Ocean Surface Topography Mission on the Jason-2 satellite in 2008. Height above mean sea level ( AMSL ) is the elevation (on the ground) or altitude (in the air) of an object, relative to

1711-646: The 1967 meeting of the IUGG held in Lucerne, Switzerland, the ellipsoid called GRS-67 ( Geodetic Reference System 1967) in the listing was recommended for adoption. The new ellipsoid was not recommended to replace the International Ellipsoid (1924), but was advocated for use where a greater degree of accuracy is required. It became a part of the GRS-67 which was approved and adopted at the 1971 meeting of

1770-973: The 19th century. With high emissions it would instead accelerate further, and could rise by 1.0 m ( 3 + 1 ⁄ 3  ft) or even 1.6 m ( 5 + 1 ⁄ 3  ft) by 2100. In the long run, sea level rise would amount to 2–3 m (7–10 ft) over the next 2000 years if warming stays to its current 1.5 °C (2.7 °F) over the pre-industrial past. It would be 19–22 metres (62–72 ft) if warming peaks at 5 °C (9.0 °F). Rising seas affect every coastal and island population on Earth. This can be through flooding, higher storm surges , king tides , and tsunamis . There are many knock-on effects. They lead to loss of coastal ecosystems like mangroves . Crop yields may reduce because of increasing salt levels in irrigation water. Damage to ports disrupts sea trade. The sea level rise projected by 2050 will expose places currently inhabited by tens of millions of people to annual flooding. Without

1829-641: The Earth in the past, with different assumed values of a and b as well as different assumed positions of the center and different axis orientations relative to the solid Earth. Starting in the late twentieth century, improved measurements of satellite orbits and star positions have provided extremely accurate determinations of the Earth's center of mass and of its axis of revolution; and those parameters have been adopted also for all modern reference ellipsoids. The ellipsoid WGS-84 , widely used for mapping and satellite navigation has f close to 1/300 (more precisely, 1/298.257223563, by definition), corresponding to

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1888-448: The Earth's gravitational field which, in itself, does not conform to a simple sphere or ellipsoid and exhibits gravity anomalies such as those measured by NASA's GRACE satellites . In reality, the geoid surface is not directly observed, even as a long-term average, due to ocean currents, air pressure variations, temperature and salinity variations, etc. The location-dependent but time-persistent separation between local mean sea level and

1947-440: The Earth's axis of rotation. The ellipsoid is defined by the equatorial axis ( a ) and the polar axis ( b ); their radial difference is slightly more than 21 km, or 0.335% of a (which is not quite 6,400 km). Many methods exist for determination of the axes of an Earth ellipsoid, ranging from meridian arcs up to modern satellite geodesy or the analysis and interconnection of continental geodetic networks . Amongst

2006-478: The GRS-80 flattening because the normalized second degree zonal harmonic gravitational coefficient, that was derived from the GRS-80 value for J 2 {\displaystyle J_{2}} , was truncated to eight significant digits in the normalization process. An ellipsoidal model describes only the ellipsoid's geometry and a normal gravity field formula to go with it. Commonly an ellipsoidal model

2065-873: The IUGG held in Moscow. It is used in Australia for the Australian Geodetic Datum and in the South American Datum 1969. The GRS-80 (Geodetic Reference System 1980) as approved and adopted by the IUGG at its Canberra, Australia meeting of 1979 is based on the equatorial radius (semi-major axis of Earth ellipsoid) a {\displaystyle a} , total mass G M {\displaystyle GM} , dynamic form factor J 2 {\displaystyle J_{2}} and angular velocity of rotation ω {\displaystyle \omega } , making

2124-705: The Marégraphe in Marseilles measures continuously the sea level since 1883 and offers the longest collated data about the sea level. It is used for a part of continental Europe and the main part of Africa as the official sea level. Spain uses the reference to measure heights below or above sea level at Alicante , while the European Vertical Reference System is calibrated to the Amsterdam Peil elevation, which dates back to

2183-409: The axes of the Earth ellipsoid to the best available data. In geodesy , a reference ellipsoid is a mathematically defined surface that approximates the geoid , which is the truer, imperfect figure of the Earth , or other planetary body, as opposed to a perfect, smooth, and unaltered sphere, which factors in the undulations of the bodies' gravity due to variations in the composition and density of

2242-524: The context of standardization and geographic applications, a geodesic reference ellipsoid is the mathematical model used as foundation by spatial reference system or geodetic datum definitions. In 1687 Isaac Newton published the Principia in which he included a proof that a rotating self-gravitating fluid body in equilibrium takes the form of a flattened ("oblate") ellipsoid of revolution, generated by an ellipse rotated around its minor diameter;

2301-458: The different set of data used in national surveys are several of special importance: the Bessel ellipsoid of 1841, the international Hayford ellipsoid of 1924, and (for GPS positioning) the WGS84 ellipsoid. There are two types of ellipsoid: mean and reference. A data set which describes the global average of the Earth's surface curvature is called the mean Earth Ellipsoid . It refers to

2360-637: The equatorial radius a 0 {\displaystyle a_{0}} and for the flattening f 0 {\displaystyle f_{0}} . The theoretical Earth's meridional radius of curvature M 0 ( φ i ) {\displaystyle M_{0}(\varphi _{i})} can be calculated at the latitude of each arc measurement as: where e 0 2 = 2 f 0 − f 0 2 {\displaystyle e_{0}^{2}=2f_{0}-f_{0}^{2}} . Then discrepancies between empirical and theoretical values of

2419-413: The equatorial. Arc measurement is the historical method of determining the ellipsoid. Two meridian arc measurements will allow the derivation of two parameters required to specify a reference ellipsoid. For example, if the measurements were hypothetically performed exactly over the equator plane and either geographical pole, the radii of curvature so obtained would be related to the equatorial radius and

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2478-415: The existing seawater also expands with heat. Because most of human settlement and infrastructure was built in response to a more-normalized sea level with limited expected change, populations affected by sea level rise will need to invest in climate adaptation to mitigate the worst effects or, when populations are at extreme risk, a process of managed retreat . The term above sea level generally refers to

2537-482: The geoid is referred to as (mean) ocean surface topography . It varies globally in a typical range of ±1 m (3 ft). Several terms are used to describe the changing relationships between sea level and dry land. The melting of glaciers at the end of ice ages results in isostatic post-glacial rebound , when land rises after the weight of ice is removed. Conversely, older volcanic islands experience relative sea level rise, due to isostatic subsidence from

2596-634: The greatest exposure to sea level rise, twelve are in Asia , including Indonesia , Bangladesh and the Philippines. The resilience and adaptive capacity of ecosystems and countries also varies, which will result in more or less pronounced impacts. The greatest impact on human populations in the near term will occur in the low-lying Caribbean and Pacific islands . Sea level rise will make many of them uninhabitable later this century. Pilots can estimate height above sea level with an altimeter set to

2655-555: The height of planetary features. Local mean sea level (LMSL) is defined as the height of the sea with respect to a land benchmark, averaged over a period of time long enough that fluctuations caused by waves and tides are smoothed out, typically a year or more. One must adjust perceived changes in LMSL to account for vertical movements of the land, which can occur at rates similar to sea level changes (millimetres per year). Some land movements occur because of isostatic adjustment to

2714-696: The individual who derived them and the year of development is given. In 1887 the English surveyor Colonel Alexander Ross Clarke CB FRS RE was awarded the Gold Medal of the Royal Society for his work in determining the figure of the Earth. The international ellipsoid was developed by John Fillmore Hayford in 1910 and adopted by the International Union of Geodesy and Geophysics (IUGG) in 1924, which recommended it for international use. At

2773-494: The inverse flattening 1 / f {\displaystyle 1/f} a derived quantity. The minute difference in 1 / f {\displaystyle 1/f} seen between GRS-80 and WGS-84 results from an unintentional truncation in the latter's defining constants: while the WGS-84 was designed to adhere closely to the GRS-80, incidentally the WGS-84 derived flattening turned out to differ slightly from

2832-538: The jurisdictions of northern Michigan, Wisconsin and Minnesota in the United States, and northern Ontario through southeastern Manitoba in Canada. The Michigan shore of Lac Vieux Desert is in the largest of the few small areas of Michigan that are part of the drainage or watershed of the Mississippi River . There are three such areas along the Wisconsin border with Gogebic County and a bit of Iron County in

2891-591: The land; hence a change in relative MSL or ( relative sea level ) can result from a real change in sea level, or from a change in the height of the land on which the tide gauge operates, or both. In the UK, the ordnance datum (the 0 metres height on UK maps) is the mean sea level measured at Newlyn in Cornwall between 1915 and 1921. Before 1921, the vertical datum was MSL at the Victoria Dock, Liverpool . Since

2950-481: The melting of ice sheets at the end of the last ice age . The weight of the ice sheet depresses the underlying land, and when the ice melts away the land slowly rebounds . Changes in ground-based ice volume also affect local and regional sea levels by the readjustment of the geoid and true polar wander . Atmospheric pressure , ocean currents and local ocean temperature changes can affect LMSL as well. Eustatic sea level change (global as opposed to local change)

3009-493: The methods of satellite geodesy , especially satellite gravimetry . Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a reference ellipsoid . They include geodetic latitude (north/south) ϕ , longitude (east/west) λ , and ellipsoidal height h (also known as geodetic height ). The reference ellipsoid models listed below have had utility in geodetic work and many are still in use. The older ellipsoids are named for

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3068-453: The pilot can estimate height above ground by subtracting the terrain altitude from the altimeter reading. Aviation charts are divided into boxes and the maximum terrain altitude from MSL in each box is clearly indicated. Once above the transition altitude, the altimeter is set to the international standard atmosphere (ISA) pressure at MSL which is 1013.25 hPa or 29.92 inHg. Reference ellipsoid An Earth ellipsoid or Earth spheroid

3127-435: The polar radius, respectively a and b (see: Earth polar and equatorial radius of curvature ). Then, the flattening would readily follow from its definition: For two arc measurements each at arbitrary average latitudes φ i {\displaystyle \varphi _{i}} , i = 1 , 2 {\displaystyle i=1,\,2} , the solution starts from an initial approximation for

3186-472: The radius of curvature can be formed as δ M i = M i − M 0 ( φ i ) {\displaystyle \delta M_{i}=M_{i}-M_{0}(\varphi _{i})} . Finally, corrections for the initial equatorial radius δ a {\displaystyle \delta a} and the flattening δ f {\displaystyle \delta f} can be solved by means of

3245-568: The semi-major axis, a {\displaystyle a} , and any of the semi-minor axis, b {\displaystyle b} , flattening , or eccentricity. Regional-scale systematic effects observed in the radius of curvature measurements reflect the geoid undulation and the deflection of the vertical , as explored in astrogeodetic leveling . Gravimetry is another technique for determining Earth's flattening, as per Clairaut's theorem . Modern geodesy no longer uses simple meridian arcs or ground triangulation networks, but

3304-399: The shape parameters of that ellipse . The semi-major axis of the ellipse, a , becomes the equatorial radius of the ellipsoid: the semi-minor axis of the ellipse, b , becomes the distance from the centre to either pole. These two lengths completely specify the shape of the ellipsoid. In geodesy publications, however, it is common to specify the semi-major axis (equatorial radius) a and

3363-518: The times of the Russian Empire , in Russia and its other former parts, now independent states, the sea level is measured from the zero level of Kronstadt Sea-Gauge. In Hong Kong, "mPD" is a surveying term meaning "metres above Principal Datum" and refers to height of 0.146 m (5.7 in) above chart datum and 1.304 m (4 ft 3.3 in) below the average sea level. In France,

3422-411: The weight of cooling volcanos. The subsidence of land due to the withdrawal of groundwater is another isostatic cause of relative sea level rise. On planets that lack a liquid ocean, planetologists can calculate a "mean altitude" by averaging the heights of all points on the surface. This altitude, sometimes referred to as a "sea level" or zero-level elevation , serves equivalently as a reference for

3481-619: Was named by French fur trappers, who were some of the first Europeans in the region. They translated the name into French from the term used by the Ojibwe of the area: Gete-gitigaani-zaaga'igan, meaning "Lake of the Old Clearing", or "Old Garden." At the time of European colonization, the Ojibwe (also known in the US as Chippewa) occupied extensive territory around Lake Superior , in what are now

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