Metric Definition and 1000 Threads

METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model developed by the University of Idaho, that uses Landsat satellite data to compute and map evapotranspiration (ET). METRIC calculates ET as a residual of the surface energy balance, where ET is estimated by keeping account of total net short wave and long wave radiation at the vegetation or soil surface, the amount of heat conducted into soil, and the amount of heat convected into the air above the surface. The difference in these three terms represents the amount of energy absorbed during the conversion of liquid water to vapor, which is ET. METRIC expresses near-surface temperature gradients used in heat convection as indexed functions of radiometric surface temperature, thereby eliminating the need for absolutely accurate surface temperature and the need for air-temperature measurements.

The surface energy balance is internally calibrated using ground-based reference ET that is based on local weather or gridded weather data sets to reduce computational biases inherent to remote sensing-based energy balance. Slope and aspect functions and temperature lapsing are used for application to mountainous terrain. METRIC algorithms are designed for relatively routine application by trained engineers and other technical professionals who possess a familiarity with energy balance and basic radiation physics. The primary inputs for the model are short-wave and long-wave thermal images from a satellite e.g., Landsat and MODIS, a digital elevation model, and ground-based weather data measured within or near the area of interest. ET “maps” i.e., images via METRIC provide the means to quantify ET on a field-by-field basis in terms of both the rate and spatial distribution. The use of surface energy balance can detect reduced ET caused by water shortage.
In the decade since Idaho introduced METRIC, it has been adopted for use in Montana, California, New Mexico, Utah, Wyoming, Texas, Nebraska, Colorado, Nevada, and Oregon. The mapping method has enabled these states to negotiate Native American water rights; assess agriculture to urban water transfers; manage aquifer depletion, monitor water right compliance; and protect endangered species.

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  1. J

    Contraction of the Riemann Tensor with the Weak Field Metric

    I have started with the space-time metric in a weak gravitational field (with the assumption of low velocity): ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2) Where \phi<<1 is the gravitational potential. Using the standard form for the Christoffel symbols have found: \Gamma^0_{00}=\phi_{,0}...
  2. J

    Contraction of the Riemann Tensor with the Weak Field Metric

    I have started with the space-time metric in a weak gravitational field (with the assumption of low velocity): ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2) Where \phi<<1 is the gravitational potential. Using the standard form for the Christoffel symbols have found...
  3. M

    Proving Convergence of Real Number Sequences with Metric Equations

    Homework Statement Prove that lim_{n} p_{n}= p iff the sequence of real numbers {d{p,p_{n}}} satisfies lim_{n}d(p,p_{n})=0 Homework Equations The Attempt at a Solution I think I can get the first implication. If lim_{n} p_{n}= p, then we know that d(p,p_{n}) = d(p_{n},p) <...
  4. Fredrik

    Generalizations (from metric to topological spaces)

    This is kind of a weird question. I like to think about how I would explain things to other people, and I realized that I don't know a great way to explain in general how terms defined in the context of metric spaces are generalized to the context of topological spaces. It's not at all difficult...
  5. B

    Dirac Gamma matrices in the (-+++) metric

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  6. Z

    Why can we use metric tensors to lower index of Christoffel symbol

    I haven't learned much of advanced mathematics. It seems that we can use metric tensors to lower or raise index of christoffel symbols. But isn't christoffel symbols made of metric tensors and derivatives of metric tensors? How can we contract indices of a derivative directly with metric tensors...
  7. M

    Question about metric spaces and convergence.

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  8. P

    Verifying the metric space e = d / (1 + d)

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  9. M

    Proving Openness in Metric Spaces

    Hi guys, two problems, first one I understand for the most part, the second one, I do not know how to set up and solve. Homework Statement Let X = R^{n} for x = (a_{1},...,a_{n}) and y = (b_{1},...,b_{n}), define d_{\infty}(x,y) = max {|a_{1}-b_{1}|,...,|a_{n}-b_{n}|}. Prove that this is a...
  10. A

    Is the Distance to a Closed Subset in a Metric Space Always Finite?

    Suppose (X,d) is a metric space and A, a subset of X, is closed and nonempty. For x in X, define d(x,A) = infa in A{d(x,a)} Show that d(x,A) < infinity. I really don't have much of an idea on how to show it must be finite. An obvious thought comes to mind, namely that a metric is...
  11. S

    Pressure depth problem in English units instead of metric

    Homework Statement Consider a submarine cruising 32 ft below the free surface of seawater whose density is 64 lbm / (ft^3). What is the increase in the pressure in psi exerted on the submarine when it dives to a depth of 172 ft below the free surface? Assume that the acceleration due to...
  12. U

    How Can Energy Density Be Calculated from a Metric in General Relativity?

    I have never been formally trained in GR, and have a question regarding the basics of how to calculate the energy density from a metric. This question arises from thought experiments involving a field with a negative energy density. This is important only because I expect the energy density...
  13. A

    Is every metric space a hausdorff space too?

    I've encountered the term Hausdorff space in an introductory book about Topology. I was thinking how a topological space can be non-Hausdorff because I believe every metric space must be Hausdorff and metric spaces are the only topological spaces that I'm familiar with. my argument is, take two...
  14. J

    Deriving the Schwarzschild Metric

    I've worked through a common-sense argumenthttp://www.mathpages.com/rr/s8-09/8-09.htm" showing the time-time component of the Schwarzschild metric g_{tt} = \left (\frac{\partial \tau}{\partial r} \right )^2\approx 1-\frac{2 G M}{r c^2} On the other hand, I've not worked through any...
  15. A

    Is Continuity at Isolated Points Properly Defined in Metric Spaces?

    I'm currently reading Ross's Elementary Analysis, which presents the definition of continuity as such: (not verbatim) Let x be a point in the domain of f. If every sequence (xn) in the domain of f that converges to x has the property that: lim f(xn) = f(x) then we say that f is...
  16. A

    Bounded sequences and convergent subsequences in metric spaces

    Suppose we're in a general normed space, and we're considering a sequence \{x_n\} which is bounded in norm: \|x_n\| \leq M for some M > 0. Do we know that \{x_n\} has a convergent subsequence? Why or why not? I know this is true in \mathbb R^n, but is it true in an arbitrary normed space? In...
  17. A

    The General Relativity Metric and Flat Spacetime

    Let us consider the General Relativity metric: {ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{{dx}_{1}}^{2}{-}{g}_{22}{{dx}_{2}}^{2}{-}{{g}_{33}}{{dx}_{3}}^{2} ---------------- (1) Using the substitutions: {dT}{=}\sqrt{{g}_{00}}{dt} {dX}_{1}{=}\sqrt{{g}_{11}}{dx}_{1}...
  18. U

    Laplacian of the metric

    When reading about the ideas behind ricci flow, I've often read that the ricci tensor is proportional to the laplacian of the metric, but only in harmonic coordinates. Can someone explain this to me? What laplacian operator would one use to show this as there are many different laplacians in...
  19. U

    How Do You Express the Variation of a General Metric Tensor?

    I'm currently working through General Relativity and I'm wondering how you would express the variation of a general metric tensor, or similarly, how you would write the total differential of a metric tensor (analogous to how you would write the total derivative for a function)? Also, on a...
  20. A

    Time-Invariant Space: Metric ds^2 and Coordinates

    Question: have some sense that in a space time with metric ds^2 = g_{tt}dt^2+ g_{xx}dx^2+ g_{yy}dy^2+g_{zz}dz^2, the coordinates x,y,z \in ]-\infty, \infty[ , but t \in [0, \infty[ ?
  21. T

    Proving the completion of a metric space is complete

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  22. A

    Cartesian product of separable metric spaces

    Dear readers, Let X be the product space of a countable family \{X_n:n\in\mathbb{N}\} of separable metric spaces. If X is endowed with the product topology, we know that it is again separable. Are there other topologies for X such that is separable? Is there a natural metric on X such that X...
  23. J

    Time-dependent Riemannian metric

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  24. B

    How to convert the energy in Joules to mass in a metric done

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  25. snoopies622

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  26. H

    Special sequences in a product metric space

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  27. D

    Open and closed sets in metric spaces

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  28. jfy4

    Is a (-,-,-,-) metric signature meaningful?

    Hi, Is a metric signature of (-,-,-,-) unphysical? Thanks,
  29. apeiron

    CP violation explained by Kerr metric

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  30. 0

    Metric field and coordinate system

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  31. Philosophaie

    Planetary Orbits calculated from the Metric

    I am learning about General Relativity. The planetary orbits can be calculated with more precision especially Mercury. I am stuck on how to get from the Schwarzschild Metric: a four variable Differential Equation to a radius(r,theta,phi,t) and velocity(r,theta,phi,t) of a single planet...
  32. L

    Proving Continuity of g(x) on a Metric Space T with f(x)=x

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  33. alemsalem

    First countable spaces and metric spaces.

    Homework Statement Show That every metric space is first countable. Hence show that every SUBSET of a metric space is the intersection of a countable family of open sets. Homework Equations no equation The Attempt at a Solution its easy to show that it is first countable, because for every...
  34. B

    A compact, B closed Disjoint subsets of Metric Space then d(A,B)=0

    Hi, All: Let X be a metric space and let A be a compact subset of X, B a closed subset of X. I am trying to show this implies that d(A,B)=0. Please critique my proof: First, we define d(A,B) as inf{d(a,b): a in A, b in B}. We then show that compactness of A forces the existence of a in A...
  35. A

    Energy-momentum tensor: metric tensor or kronecker tensor appearing?

    Hi This might be a stupid question, so I hope you are patient with me. When I look for the definition of the energy-momentum tensor in terms of the Lagrangian density, I find two different (?) definitions: {T^\mu}_\nu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)}\partial_\nu...
  36. G

    What metric from given manifold?

    Given a manifold as algebraic variety, say sphere, how do we obtain possible metrics? how do we classify them? If spcaetime manifold is n-sphere, Einstein's vacuum (for now) equation would be some special metric among many other possible metrics? i'm curious what role Einstein's equation...
  37. T

    Transformation to get a metric to diagonal form

    Hi, If you have a spherically symmetric spacetime metric in a set of spherical coordinates t,r,theta,phi: [P,Q,0,0;Q,R,0,0;0,0,S,0;0,0,0,Ssin^(theta)]. Here P,Q,R,S are functions of t and r. Now, if I want to choose cooridnates to get the metric in the generic diagonal form (that is by...
  38. T

    Get Lorentzian Spherically Symmetric Metric to Sylvester Form

    Hi, I'm trying to determine the exact transformation that brings a spherically symmetric spacetime metric in spherical coordinates to the Sylvester normal form (that is, with just 1 or -1 on its main diagonal, with all other elements equal to zero.) Assuming that the metric has Lorentzian...
  39. T

    Metric Tensors for 2-Dimensional Spheres and Hyperbolas

    Does anyone know what the metric tensor looks like for a 2 dimensional sphere (surface of the sphere)? I know that it's coordinate dependent, so suppose you have two coordinates: with one being like "latitude", 0 at the bottom pole, and 2R at the northern pole, and the other being like...
  40. T

    Hermitian Metric - Calculating Christoffel Symbols

    Hello, I am trying to understand what the differences would be in replacing the symmetry equation: g_mn = g_nm with the Hermitian version: g_mn = (g_nm)* In essence, what would happen if we allowed the metric to contain complex elements but be hermitian? I am not talking about...
  41. Y

    If already known the Action and unkown the Metric, how to get geodesic equation?

    If already known the form of Action and unkown the Metric, how to derive the geodesic equation?
  42. R

    Godel's metric in cylindrical coordinates

    Hello, In Godel's paper: an example of a new type of cosmological solutions of einstein's field equations of gravitation, he passes from his original metric to cylindrical coordinates by giving some transformation formulas. Can someone tell me how is this transformation obtained, or at least...
  43. L

    Schwarzschild Metric - Need help understanding

    Alright, first things first, I'm a grade 12 student residing in Ontario, Canada and I'm relatively new to these forums and to the world of physics. I'm doing my grade 12 ISU and I've taught myself how to work around spherical coordinate systems, however, the schwarzschild metric confuses me...
  44. M

    Gaussian curvature for a given metric

    Homework Statement Assume that we have a metric like: ds^{2}=f dr^{2}+ g d\theta^{2}+ h d\varphi^{2} where r,\theta , \varphi are spherical coordinates. f,g and h are some functions of r and theta but not phi. Homework Equations How can I calculate Gaussian curvature in r-theta...
  45. S

    Metric for an observer in free fall two schwarzchild radii from black hole.

    Hi all, I have a GR exam on tuesday and getting a bit confused as to how to find the metric for an observer in free fall a distance two schwarzchild radii from a black hole. I know this is a bit of a basic question but I am just wondering if I am correct to substitute r=2rs and dt=d(tau)...
  46. G

    Robertson-Walker metric and time expansion

    The Robertson-Walker metric applies a time-dependent scale factor to model the expansion of the universe. The scale factor is only applied to the spatial coordinates (in the frame of the "comoving observer"). That is not covariant and it is hard to see how c could remain constant if the same...
  47. M

    Proving The Hamming Metric: Open Subsets and Basis of X

    Homework Statement I'm stuck on how to start this. The Hammin metric is define: http://s1038.photobucket.com/albums/a467/kanye_brown/?action=view&current=hamming_metric.jpg and I'm asked to: http://i1038.photobucket.com/albums/a467/kanye_brown/analysis_1.jpg?t=1306280360 a) prove...
  48. M

    Proving or Disproving X+Y as Open in Metric Spaces | Homework Help

    Homework Statement Let X and Y be subsets of R^2, both non-empty. If X is open, the the sum X+Y is open. This is either supposed to be proved or disproved. Homework Equations The Attempt at a Solution This strikes me as false since we are only given the X is open...
  49. M

    Open Subsets in Metric Space A with Discrete Metric d

    Homework Statement Let A be a non-empty set and let d be the discrete metric on X. Describe what the open subsets of X, wrt distance look like. Homework Equations The Attempt at a Solution I think that the closed sets are the subsets of A that are the complement of a union of...
  50. haushofer

    Complex metric solutions in GR

    A friend of mine had the following funny question: Imagine I have a metric ansatz with two unknown functions. The Einstein equations give both real and complex solutions for the unknown functions. Question: Is there a decent interpretation of these complex solutions in GR? We know about...
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