V Slope Methode

1. V Slope Method
2. V-slope-method Ventilatory Threshold
3. V Slope Method Excel

So your slope for this relationship is actually 0. Y is equal to 0x plus- and then, you could just realize that the equation of this is just that y is always equal to 2. So it's 0x plus 2, which is the same thing as y is equal to 2. From the the data gathered, I most likley don't have a very high Anerobic mucle precentage.I sorta knew this, but this was another test to solidify the Hypot.

1. For the planar method, the slope is measured as the maximum rate of change in value from a cell to its immediate neighbors. The calculation is performed on a projected flat plane using a 2D Cartesian coordinate system. The slope value is calculated using the.
2. Tangent Line Problem - Descartes vs Fermat Tangent Line.„, Is it possible to find the tangent line at any point x=a? Method Method Example 1 - Find the slope and then write an equation of the tangent line to the function y = x2 at the point (1,1) using Descartes' Method.i'2- n- Mxc u ' 1L -T- O ft.

Slope Algorithms

The slope is taken as a dZ value divided by a horizontal distance. The horizontal distance is the data spacing (east-west, north-south, or diagonal) for the one point and nine point methods, and twice that distance for the four and eight point methods. For DEMs like the USGS 10 m and 30, the NS and EW spacings are the same; they differ for data like the USGS NED, USGS 1:250K DEM, SRTM, and DTED with geographic spacing.

Aspect, the downhill direction, can be computed as a byproduct of the slope computation.

Slope and aspects, the magnitude and direction of the vector tangent to the topographic surface pointing downhill at a point, have been computed with a multitude of methods; Eyton (1991) and Carter (1992) list a number of references. The following discussion centers on the methods implemented in MICRODEM; the references point to clear expositions of the method, and not necessarily the first use of the method, many of which are in hard to location publications in the gray literature. The algorithms fall into several categories, depending on the number of neighboring points considered to compute the slope and aspect.

In general the choice of slope algorithm does not make much difference, and you should just stick with the default.

Two neighbors:

• Batson & others (1975): calculates component of slope in direction of one of the nearest neighbors. Strictly speaking this value of slope should be assigned to a spot halfway between the point and the neighbor used. This is the fastest algorithm, with no calculation for aspect, and has limited applicability. MICRODEM used it at one time for modeling radar reflectivity from the DEM, as the look direction for SLAR is typically to the east or west.

Three neighbors:

• O'Neill & Mark (1987): three points uniquely define a plane; this algorithm uses the point and its neighbors to the north and east (Z, Zn, and Ze). The simplest measure, it has the lowest correlations with all other methods. Other equally arbitrary choices of points to use would give different slopes and aspect directions. This method would be used for TINs, where the slope and aspect apply to the triangular facet and not the point at which the slope will change. Called the “simple method” by Jones (1998) who chose the points to the north and west. This method assigns the values from the triangle to the point in the center and assumes it covers a rectangular grid. A more sophisticated, but harder to implement, version would compute a different slope for each of the eight triangular regions about the point.
• Four contiguous right triangles (Onorati and others, 1992) Sub-pixel calculations: using two neighbors about the point, four subpixels can be computed. The four use points (Z, Zn, Zw), (Z, Zn, Ze), (Z, Zs, Zw), and (Z, Zs, Ze).

Four neighbors to N, S, E, and W (excluding point itself) (rook's case):

• Four neighbors (e.g. Ritter, 1987; Zevenbergen and Thorne, 1987; Band, 1989; Eyton, 1991; Carter, 1992) Called the Fleming & Hoffer (1979) method by Jones (1998) for a Purdue University technical report. Evans (1972) may have described this method, and amplified it in Durham University technical reports (Evans, 1979; Young, 1978); Shary (2002) calls this the Evans-Young method. This gets a dx from Ze-Zw, and a dy from Zn-Zs, and calculates the slope and aspect from them. The value is assigned to the central point, even though its elevation is not used in the calculation.

Four neighbors to NW, SE,N E, and SW (excluding point itself) (bishop's case):

• Diagonal Ritters: proposed by Jones (1998), using the 4 diagonal neighbors. This gets a 'dx' from Znw-Zse, and a 'dy' from Zne-Zsw, and calculates the slope and aspect from them. This effectively averages slopes over a longer distance, since the diagonals are longer than the distances to the nearest neighbors. The value is assigned to the central point, even though its elevation is not used in the calculation.

Eight neighbors (excluding point itself) (Queen's case):

• Horn method (Horn, 1981): nearest points weighted more than diagonal neighbors. This method is also known as the Sobel operator (Richards, 1986). The point itself has no influence on the calculated slope (Guth, 1995). Jones (1998) discusses two variants, using 1/r and 1/r² weightings. The value is assigned to the central point, even though its elevation is not used in the calculation.
• Sharpnack & Akin (Sharpnack and Akin, 1969; also 'mean method' in ELAS, cited in Ritter, 1987): all neighbors weighted equally. The point itself has no influence on the calculated slope (Guth, 1995) The value is assigned to the central point, even though its elevation is not used in the calculation.
• Local Trend Surface (Heerdegen and Beran, 1982): this method is equivalent to fitting a second order trend surface (z = a + bx + cy + dx² + ey² + fxy) to the nine points using the algorithm in Davis (1973), but an order of magnitude faster. The algorithm was modified for MICRODEM to allow different x and y data spacing, but gives the same results as the Sharpnack & Akin algorithm. The point itself has no influence on the calculated slope (Guth, 1995).The value is assigned to the central point, even though its elevation is not used in the calculation.
• Constrained quadratic surface method (Wood, 1996 thesis, cited in Jones, 1998). The trend surface is constrained to go through the point itself. The value is assigned to the central point.

Nine points:

• Steepest Adjacent Neighbor (Collins, 1975; Travis et al. 1975, Dept of Agriculture Forest Service Technical Report cited in Jones 1998; ELAS software cited in Ritter, 1987): this method consistently gives steeper results than all others. It picks the steepest of the eight adjacent points, considering that the diagonal distance is longer and that the x and y spacings might be different. The aspect is only available to the nearest 45° (approximate if the x and y spacings differ), and if the steepest slope is uphill is assigned as the opposite of the slope direction, which may not be the steepest downhill slope. The value is assigned to the central point.
  • slope1 = (Zn-Z) / (NS Spacing)
  • slope2 = (Zs-Z) / (NS Spacing)
  • slope3 = (Ze-Z) / (EW Spacing)
  • slope4 = (Zw-Z) / (EW Spacing)
  • slope5 = (Znw-Z) / (Diagonal Spacing)
  • slope6 = (Zne-Z) / (Diagonal Spacing)
  • slope7 = (Zse-Z) / (Diagonal Spacing)
  • slope8 = (Zsw-Z) / (Diagonal Spacing)
  • Pick largest absolute value from eight candidates as the slope
• Steepest downhill neighbor (D8): similar to the steepest neighbor, but the point must be lower in elevation than the central point. The value is assigned to the central point. Most drainage basin algorithms use a variant of this algorithm, looking only for the neighbor into which water will flow.
• Average neighbor: averages slopes to the eight nearest neighbors. Its estimate is about 50% of the Steepest neighbor, because the slopes at 90° to the maximum will typically be nearly horizontal. The value is assigned to the central point.

Table. Selected Slope and Aspect Algorithms expanded and adapted from Guth (1995).

Name	Short Name	Neighbors Used	Neighbors	Comments	References
Steepest Down Hill	SDN	9	Also called D8; only 8 aspects	O'Callahan and Mark, 1987
Steepest Adjacent Neighbor	SAN	9	Only 8 aspects	(Sharpnack & Akin, 1969)
Average Neighbor	AVN	9	Only 8 aspects
Guth Hybrid (Steepest + 8 even for aspect)	HYB	9	Preferred in Guth, 1995	Guth, 1995
Four Neighbors	N4	4	zn,zs,ze,zw	'Rook's case': Preferred in Jones, 1998 Second order finite difference	Fleming and Hoffer, 1979; Ritter, 1987; Zevenbergen and Thorne, 1987; O'Neill & Mark (1987)
3 Neighbors	N3	3	z,zn,zne	In a grid, two choices for the triangles	O'Neill and Mark, 1987
8 neighbors, even weighting	N8E	8	'Queen's case': Equivalent to fitting a second order trend surface. Preferred by Evans, 1998; Florinsky, 1998. Identical weights. Third order finite difference	Sharpnack and Akin, 1969; Horn, 1981; Heerdegen and Beran, 1982; Wood, 1996
8 neighbors weighted	N82	8	Third order finite difference weighted by reciprocal of twice distance
8 neighbors weighted	N8R	8	Third order finite difference weighted by reciprocal of sqrt2 * distance	Unwin, 1981
Frame Finite Difference	FFD	4	Difference above, below, right, and left
Simple Difference	SD	3	The other choice of points from O'Neill and Mark, 1987
2	z,ze	No aspect	Batson & others (1975)

Assess the effect of region size on computed slope.

Slope units, in percent.

Three slope methods appear to have the broadest support:

• Four closest neighbors (FCU):
• Eight neighbors unweighted (ENU): present preferred method in MICRODEM. Users must still be aware that this method produces peaks that are too flat, but its smoothing providew a 'better' slope surface.

The Steepest Adjacent Neighbor (SAN): requires adjustments to get reliable aspect distribution, and does not give a smooth slope distribution. It was the preferred method in MICRODEM until the summer of 2003, and there are still reasons to prefer it. The Eight Neighbors Unweighted method became the default in MICRODEM the summer of 2003, primarily because it produces the smoothest, most realistic slope histograms, and reasonable aspect distributions It still has problems with smoothing of slopes in valleys, peaks, and ridges.

Guth (1995) reviewed all the independent slope and aspect algorithms he could find. He suggested that the steepest adjacent neighbor provided the best estimate of slope, and the 8 neighbors with even weights provided the best estimate of aspect. More recent comparisons of slope algorithms include Hodgson (1998) and Jones (1998), but neither cited Guth (1995) and their dismissal of the steepest adjacent neighbor does not appear completely valid.

Justification to use largest slope of 8 to adjacent elevations in grid:

1. Each point is surrounded by 8 neighbors. Four are to the N,S,E,& W, and 4 approximately NE,SE,SW,& NW. How close those are depend upon the type of DEM and latitude; it is exact at the equator where x and y spacing is equal or using a UTM based DEM. Withe DTED, the worst cases will be in the vicinity of 50°, where just to the south the spacing is 60 and 90 m, and just to the north the spacing is 90 and 120 m. There the diagonal values are at 53 and 34° instead of 45° (arc tan (120/90) and arc tan (60/90)).
2. The direction of maximum slope can thus occur no more than 28° away from the largest of the eight directional slopes to nearest neighbors in the DTED grid (half of 56, the complement of 34).
3. If we assume a plane for the surface in the vicinity of the greatest slope, we can use vector arithmetic to project the slope in various directions. The projection will be a function of the cosine. For the worst case, the estimated slope to the DTED value in the closest direction will be 87.9% of the true slope (cos(28) = 0.879), and that occurs only when the true maximum lies directly between two grid values. At the equator, with maximum error in estimating, the estimated slope is 92.4% of its true value.
4. Since maximum error is only about 10%, and given the overall accuracy of the data to start with, we do not feel it necessary to obtain the added rigor of fitting a surface to the data points in the vicinity and then computing the slope of the tangent surface.
5. The other methods effectively compute the slope are a region twice the size of the data spacing.
6. See Guth (1995) for additional details and a comparison on the effect of algorithm on calculated slope.

Discussions of slope on arc-second DEMs

• https://gis.stackexchange.com/questions/14750/using-srtm-global-dem-for-slope-calculation/229186#229186),
• Equi-7

Last revised 8/29/2018

Available with Spatial Analyst license.

Available with 3D Analyst license.

The Slope tool identifies the steepness at each cell of a raster surface. The lower the slope value, the flatter the terrain; the higher the slope value, the steeper the terrain.

The output slope raster can be calculated in two types of units, degrees or percent (percent rise). The percent rise can be better understood if you consider it as the rise divided by the run, multiplied by 100. Consider triangle B below. When the angle is 45 degrees, the rise is equal to the run, and the percent rise is 100 percent. As the slope angle approaches vertical (90 degrees), as in triangle C, the percent rise begins to approach infinity.

The Slope tool is most frequently run on an elevation dataset as the following images show. Steeper slopes are shaded darker brown on the output slope raster.

The tool can also be used with other types of continuous data, such as population, to identify sharp changes in value.

Calculation methods and the edge effect

Two methods are available for slope computation. You can choose between performing Planar or Geodesic calculations with the Method parameter.

For the planar method, the slope is measured as the maximum rate of change in value from a cell to its immediate neighbors. The calculation is performed on a projected flat plane using a 2D Cartesian coordinate system. The slope value is calculated using the average maximum technique (Burrough, 1998).

With the geodesic method, the calculation will be performed in a 3D Cartesian coordinate system by considering the shape of earth as an ellipsoid. The slope value is calculated by measuring the angle between topographic surface and the referenced datum.

Both planar and geodesic computations are performed using a 3 by 3 cell neighborhood (moving window). For each neighborhood, if the processing (center) cell is NoData, the output is NoData. The computation also requires at least seven cells neighboring the processing cell have valid values. If there are fewer than seven valid cells, the calculation will not be performed, and the output at that processing cell will be NoData.

The cells in the outermost rows and columns of the output raster will be NoData. This is because along the boundary of the input dataset, those cells do not have enough valid neighbors.

Planar method

V Slope Method

For each cell, the tool calculates the maximum rate of change in value from that cell to its neighbors. Basically, the maximum change in elevation over the distance between the cell and its eight neighbors identifies the steepest downhill descent from the cell.

Planar slope algorithm

V-slope-method Ventilatory Threshold

The rates of change (delta) of the surface in the horizontal (dz/dx) and vertical (dz/dy) directions from the center cell determine the slope. The basic algorithm used to calculate the slope is as follows:

Slope is commonly measured in units of degrees, which uses the following algorithm:

Note:

The value 57.29578 shown here is a truncated version of the result from 180/pi.

The slope algorithm can also be interpreted as follows:

• where:

The values of the center cell and its eight neighbors determine the horizontal and vertical deltas. The neighbors are identified as letters from a to i, with e representing the cell for which the aspect is being calculated.

The rate of change in the x direction for cell e is calculated with the following algorithm:

• where:

  wght1 and wght2 are the horizontal weighted counts of valid cells.

  For instance, if:

  • c, f, and i all have valid values, wght1 = (1+2*1+1) = 4.
  • i is NoData, wght1 = (1+2*1+0) = 3.
  • f is NoData, wght1 = (1+2*0+1) = 2.

  Similar logic applies to wght2, except the neighbor locations are a, d, and g.

The rate of change in the y direction for cell e is calculated with the following algorithm:

• where:

  wght3 and wght4 are the same concept as in the [dz/dx] computation.

Planar slope calculation example

As an example, the slope value of the center cell of the moving window shown below will be calculated.

The rate of change in the x direction for the center cell e is:

The rate of change in the y direction for cell e is:

Taking the rate of change in the x and y direction, the slope for the center cell e is calculated using the following:

The integer slope value for cell e is 75 degrees.

Geodesic method

The geodesic method measures slope in a geocentric 3D coordinate system—also called the Earth Centered, Earth Fixed (ECEF) coordinate system—by considering the shape of the earth as an ellipsoid. The computation result will not be affected by how the dataset is projected. It will use the z-units of the input raster if they are defined in the spatial reference. If the spatial reference of the input does not define the z-units, you will need to do so with the z-unit parameter. The geodesic method produces a more accurate slope than the planar method.

Geodesic coordinate transformation

The ECEF coordinate system is a 3D right-handed Cartesian coordinate system with the center of the earth as the origin, where any location is represented by X, Y and Z coordinates. See the following figure for an example of a target location T expressed with geocentric coordinates.

The geodesic computation uses an X, Y, Z coordinate that is calculated based on its geodetic coordinates (latitude φ, longitude λ, height h). If the coordinate system of the input surface raster is a projected coordinate system (PCS), the raster is first re-projected to a geographical coordinate system (GCS) where each location has a geodetic coordinate, and then transformed to the ECEF coordinate system. The height h (z-value) is the ellipsoid height referenced to the ellipsoid surface. See the illustration graphic below.

To transform to ECEF coordinates from a geodetic coordinate (latitude φ, longitude λ, height h), use the following formulas:

• where:
  • N( φ ) = a²/ √(a²cosφ²+b²sinφ²)
  • φ = latitude
  • λ = longitude
  • h = ellipsoid height
  • a = major axis of the ellipsoid
  • b = minor axis of the ellipsoid

The ellipsoid height h is in meters in the above formulas. If your input raster's z-unit is specified in any other unit, it will be internally transformed to meter.

Slope computation

The geodesic slope is the angle formed between the topographic surface and the ellipsoid surface. Any surface parallel to the ellipsoid surface has a slope of 0. To calculate the slope at each location, a 3 by 3 cell neighborhood plane is fitted around each processing cell using the Least Squares Method (LSM). The best fit in the LSM minimizes the sum of squared difference (dz_i) between the actual z-value and the fitted z-value. See the illustration below for an example.

Here, the plane is represented as z = Ax + By + C. For each cell center, dz_i is the difference between the actual z-value and the fitted z-value.

V Slope Method Excel

The plane is best fitted when ∑⁹_i=1dz_i² is minimized.

After the plane is fitted, a surface normal is calculated at the cell location. At the same location, an ellipsoid normal perpendicular to the tangent plane of the ellipsoid surface is also calculated.

The slope, in degrees, is calculated from the angle between the ellipsoid normal and the topographic surface normal, represented as β here. From the illustration above, the angle α is the geodesic slope, which is the same as angle β, pursuant to the law of congruent geometry.

To calculate slope in percent rise, the following formula is used:

Should I use the Surface Parameters tool?

If the Input raster parameter value (in_raster in Python) is high resolution with a cell size of less than a few meters, or particularly noisy, consider using the Surface Parameters tool and its user-defined neighborhood distance option instead of the immediate 3 x 3 neighborhood of this tool. Using a larger neighborhood can minimize the effect of noisy surfaces. Using a larger neighborhood can also better represent landforms and surface characteristics when using high resolution surfaces.

Use of a GPU

For the Geodesic method, this tool is capable of delivering increased performance if you have certain GPU hardware installed on your system. See the GPU Processing with Spatial Analyst section for details on how this is supported, how to configure it, and how to enable it.

References

Burrough, P. A., and McDonell, R. A., 1998. Principles of Geographical Information Systems (Oxford University Press, New York), 190 pp.

Marcin Ligas, and Piotr Banasik, 2011. Conversion between Cartesian and geodetic coordinates on a rotational ellipsoid by solving a system of nonlinear equations (GEODESY AND CARTOGRAPHY), Vol. 60, No 2, 2011, pp. 145-159

B. Hofmann-Wellenhof, H. Lichtenegger and J. Collins, 2001. GPS - theory and practice. Section 10.2.1. p. 282.

David Eberly 1999. Least Squares Fitting of Data (Geometric Tools, LLC), pp. 3.

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