What is Metric: Definition and 1000 Discussions

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. N

    How to caculate the inverse metric tensor

    Given a metric tensor gmn, how to calculate the inverse of it, gmn. For example, the metric g_{\mu \nu }= \left[ \begin{array}{cccc} f & 0 & 0 & -w \\ 0 & -e^m & 0 &0 \\0 & 0 & -e^m &0\\0 & 0 & 0 & -l \end{array} \right] From basic understanding, I would think of divided it, that is...
  2. E

    Non discrete metric space on infinite set

    Homework Statement let d be a metric on an infinite set m. Prove that there is an open set u in m such that both u amd its complements are infinite. Homework Equations If d is not a discrete metric, and M is an infinite set (uncountble), then we can always form an denumerable subset...
  3. P

    Prokhorov Metric - Understanding the Definition

    Homework Statement I am wondering if anyone understands why this metric is defined the way it is because i can't seem to make sennse of it. I get that way we use the underlying metric space to define the borel sigma field and then the set of all borel measures, but the actual definition...
  4. S

    Origins of Scale Factor of FRW Metric and Misc Questions of GR Equations

    In the context of Friedmann's time, 1922, how did he know to make the metric scale factor, a, a function of time when Hubble's redshifts were not yet published? I understand that he took the assumption that the universe is homogenous and isotropic, but does that naturally imply that the universe...
  5. J

    Are Metric Space Completions Topologically Equivalent?

    Hi all, Given a metric space (X,d), one can take its completion by doing the following: 1) Take all Cauchy sequences of (X,d) 2) Define a pseudo-metric on these sequences by defining the distance between two sequences to be the limit of the termwise distance of the terms 3) Make this a...
  6. S

    Simple quesion about metric on extended real line

    I was told the extended real \hat{R}=R\cup\{-\infty,\infty\} is homeomorphic to [0,1], I was wondering if the mapping h: [0,1]\rightarrow\hat{R}, h(x)=\cot^{-1}(\pi x), 0<x<1, h(0)=\infty, h(1)=-\infty is a valid homeomorphism, so that a metric may be defined by the metric on [0,1]? Thank...
  7. M

    Is this a homeomorphism that does not preserve metric completeness?

    I'm well aware and understand that homeomorphism do not need to preserve metric completeness, I'm just trying to work out a simple counterexample. I have tried searching around just for kicks, but only seem to find more complex ones. I'm wondering if the one I have works for it for sure? On...
  8. Thinkor

    Schwartzschild metric difficulty

    I am trying to understand how the Schwartzschild metric works and coming to the conclusion that if I drop an object that it will not fall to the Earth but just stay there when I open my hand. Therefore, I'm confident I have made a mistake, but I don't see where it is. Here is the metric...
  9. F

    How to calculate Riemannian Metric by distance function?

    How to calculate Riemannian Metric by distance function?? Dear Folks: Here is the problem: in |z|<1, we difine a distance between any two points z1 and z2 by d(z1 , z2) = ln((z1 - b)(z2 - a)/((z1 - a)(z2 - b))) ,where a is the intersection of line z1z2 and the circle which is nearer to...
  10. D

    Time Travel - Between two Kerr metric black holes w/detached event horizons

    So imagine your on Earth at a latitude of 30 to 45° N, between two rotating Kerr Metric Blackholes with detached event horizons (dual singularities) allowing you to be shielded from the crushing force of the black holes. Which way do the rotating black holes need to rotate for the past and...
  11. A

    Metric Connection from Geodesic Equation

    For the following two-dimensional metric ds^2 = a^2(d\theta^2 + \sin^2{\theta}d\phi^2) using the Euler-Lagrange equations reveal the following equations of motion \ddot{\phi} + 2\frac{\cos{\theta}}{\sin{\theta}}\dot{\theta}\dot{\phi} = 0 \ddot{\theta} -...
  12. A

    Another christoffel symbols from the metric question

    Another "christoffel symbols from the metric" question Homework Statement Find the Christoffel symbols from the metric: ds^2 = -A(r)dt^2 + B(r)dr^2 Homework Equations \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{x^a}} \right) = \frac{\partial L}{\partial x^a} The...
  13. S

    Metric and completeness of real numbers

    So far I have learned a bit about topological spaces, there has been several occasions regarding metric spaces where I had to invoke completeness (or at least uncountability) of real numbers R, which is itself a property of the usual metric space of R. For example, to show that any countable...
  14. A

    Equations of motion from Lagrangian and metric

    Disregard. I done figured it out.Homework Statement Find equations of motion for the metric: ds^2 = dr^2 + r^2 d\phi^2Homework Equations L = g_{ab} \dot{x}^a \dot{x}^b \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{x}^a} \right) = \frac{\partial L}{\partial x^a} The Attempt at a...
  15. L

    Equivalence classes and Induced metric

    (X,\rho) is a pseudometric space Define: x~y if and only if ρ(x,y)=0 (It is shown that x~y is an equivalence relation) Ques: If X^{*} is a set of equivalence classes under this relation, then \rho(x,y) depends only on the equivalence classes of x and y and \rho induces a metric on...
  16. A

    Equivalent conditions on a metric space

    Homework Statement Let X be a metric space and A a subset of X. Prove that the following are equivalent: i. A is dense in X ii. The only closed set containing A is X iii. The only open set disjoint from A is the empty set Homework Equations N/A The Attempt at a Solution I can...
  17. L

    Proving the Validity of a New Metric Space

    Suppose that (X,d) is a metric Show \tilde{d}(x,y) = \frac{d(x,y)}{\sqrt{1+d(x,y)}} is also a metric I've proven the positivity and symmetry of it. Left to prove something like this Given a\leqb+c Show \frac{a}{\sqrt{1+a}}\leq\frac{b}{\sqrt{1+b}}+\frac{c}{\sqrt{1+c}} I try to...
  18. R

    A basic question about the use of a metric tensor in general relativity

    I have very little knowledge in general relativity, though I do have a decent understanding of the theory of special relativity. In special relativity, points in space-time can be represented in Minkowski space (or a hyperbolic space) so that the metric tensor (that is derived in order to...
  19. A

    How can we define the induced metric on a brane?

    Hello, I have a problem to understand what people say by "induced metric". In many papers, it is written that for brane models, if we consider the metric on the bulk as g_{\mu\nu} hence the one in the brane is h_{\mu\nu}=g_{\mu\nu}-n_\mu n_{\nu} where n_{\mu} is the normalized spacelike...
  20. A

    MHB Show a certain sequence in Q, with p-adict metric is cauchy

    I left this following question from my excercises last, hoping that solving the others will give me an insight onto how to proceed. But I still don't have a plan on how to start it: Consider the sequence s_n = Ʃ (k=0 to n) (t_k * p^k) in Q(rationals) with the p-adic metric (p is prime); where...
  21. A

    Show a certain sequence in Q, with p-adict metric is cauchy

    I left this following question from my excercises last, hoping that solving the others will give me an insight onto how to proceed. But I still don't have a plan on how to start it: Consider the sequence s_n = Ʃ (k=0 to n) (t_k * p^k) in Q(rationals) with the p-adic metric (p is prime); where...
  22. A

    MHB Surjectivity of an Isometry given the metric space is complete.

    Hello, the following is a post that was in progress and I am continuing it here after I received a message saying that most of the members had moved from mathhelpforum here. Me: I have a problem where I am asked to show that for a complete metric space X, the the natural Isometry F:X --> X* is...
  23. G

    Metric tensor transformations

    Hi, i was thinking about the metric tensor transformation law: g_{cd}(x) = \frac{{dx'}^a}{{dx}^c} \frac{{dx'}^b}{{dx}^d} g'_{ab}(x') and, in view of this definition, the differences between Poincare transformations and reparametrization-like transformation (f.e. various conformal...
  24. O

    Prove this function on metric space X is onto

    (1) (X,d) is a COMPACT metric space. (2) f:X->X is a function such that d(f(x),f(y))=d(x,y) for all x and y in (X,d) Prove f is onto. Things I know: (2) => f is one-one. (2) => f is uniformly continuous. I tried to proceed by assuming the existence of y in X such that y has no...
  25. A

    Sequence in Q with p-diatic metric. Show it converges to a rational

    This is the problem I'm trying to slove: Consider the sequence s_n = Sumation (from k=0 to n) p^k (i.e. s_n=p^0+p^1+p^2...+p^n) in Q(rationals) with the p-adic metric (p is prime). Show that s_n converges to a rational number.[/B] Now, I do get some intuition on showing that the...
  26. m4r35n357

    Conserved quantities in the Doran Metric?

    I've been doing some amateur simulations of particle trajectories in a few well-known metrics (using GNU Octave and Maxima), and in the case of the Schwarzschild and Gullstrand-Painleve metrics I have the ability to check my results using freely available equations for conservation of energy and...
  27. 6

    Exploring Closed Sets in Metric Spaces through Infinite Intersections

    Homework Statement Find (X,d) a metric space, and a countable collection of open sets U\subsetX for i \in Z^{+} for which \bigcap^{∞}_{i=1} U_i is not open Homework Equations A set is U subset of X is closed w.r.t X if its complement X\U ={ x\inX, x\notinU} The Attempt at a Solution Well...
  28. D

    Calculating proper time using schwartzchild metric

    I am using the schwartzchild metric given as ds^2 = (1 - \frac{2M}{r})dt^2 - (1 - \frac{2M}{r})^{-1} dr^2 , where I assume the angular coordinates are constant for simplicity. So if a beam of light travels from radius r0 to smaller radius r1, hits a mirror, and travels back to r0, I am...
  29. Fantini

    MHB Should I study metric spaces topology before general topology?

    Hello everyone. I want to study topology ahead of time (it begins next semester only) and I have two options: I could go straight for general topology (among the books I searched I found Munkres to be the one I felt most comfortable with) or go for a thorough study of metric spaces (in which...
  30. J

    Is the Empty Set Considered a Metric Space?

    Homework Statement Is empty set a metric space? Homework Equations None. The Attempt at a Solution It seems so because all the metric properties are vacuously satisfied. Mabe the question had better be put like this: Does mathematicians tend to think empty set as a metric space...
  31. W

    Can changing the metric in GR result in different spacetimes?

    Are there different kinds of metric in GR? For instance. I read in http://www.astronomy.ohio-state.edu/~dhw/A682/notes3.pdf there the FRW Metric is about: "In 1917, Einstein introduced the first modern cosmological model, based on GR, in which the spatial metric is that of a 3-sphere:"...
  32. W

    Does the expansion of space only occur in unbound systems?

    I'm coming with a good background in Big Bang expansion as the following sci-am article shows (which I've mastered): http://space.mit.edu/~kcooksey/teaching/AY5/MisconceptionsabouttheBigBang_ScientificAmerican.pdf What I'd like to understand is this. Expansion can only be felt in unbound...
  33. michael879

    CPT (M?) symmetries in Kerr-Newman metric

    So the confusion I'm having here really has to do with parity inversion in spherical (or boyer-linquist) coordinates. I've been looking at the discrete symmetries of the Kerr-Newman metric, and I've noticed that depending on how you define parity-inversion, you can get very different results...
  34. 8

    Metric spaces and convergent sequences

    Homework Statement let {xi} be a sequence of distinct elements in a metric space, and suppose that xi→x. Let f be a one-to-one map of the set of xis into itself. prove that f(xi)→x Homework Equations by convergence of xi, i know that for all ε>0, there exists some n0 such that if i≥n0...
  35. D

    Metric Perturbation: Finding Info for Einstein's Field Equation

    Hello guys. I was told to prepare a presentation on perturbed Einstein's field equation by my advisor. I got some of the things I needed to start with in the Weinberg's Cosmology book but it was not enough. Can anyone please tell me a book or anything with information on metric perturbation? Thanks
  36. D

    Vector difference metric that considers the variance of the components

    I am trying to match little square patches in an image. You can imagine that these patches have been "vectorized" in that the values are reordered consistently into a 1D array. At first glance, it seems reasonable to simply do a Euclidean distance style comparison of two of these arrays to get a...
  37. B

    General Relativity - Schwarzschild Metric

    Homework Statement A spaceship is moving without power in a circular orbit about an object with mass M. The radius of the orbit is R = 7GM/c^2 (1) Find the relation between the rate of change of angular position of the spaceship and the proper time and radius of the orbit. Homework...
  38. T

    Connectedness of a subset of a metric space.

    Homework Statement Show that an open (closed) subset of a metric space E is connected if and only if it is not the disjoint union of two nonempty open (closed) subsets of E. Homework Equations The definition of connectedness that we are using is as follows: A metric space E is...
  39. J

    Calculating a metric from a norm and inner product.

    I typed the problem in latex and will add comments below each image. The supremum of |1 - x| seems dependent on the interval [a, b]. For instance, if [a, b] = [-500, 1], then 501 is the supremum of |1 - x|. But if [a, b] = [-1, 500], then 499 is the supremum of [1 - x]. So what should I...
  40. D

    Showing that a given metric is Riemannian

    So I have the hyperboloid in \mathbb{R}^3 given by x^2 + y^2 - z^2 = -1 . I define the scalar product of 2 tangent vectors (a1, b1, c1) and (a2, b2, c2) at a point as a1a2 + b1b2 - c1c2. I want to show that this defines a Riemannian metric on the hyperboloid. I know that a Riemannian...
  41. Alesak

    Why do manifolds require a Riemannian metric?

    When reading other threads, following question crept into my mind: When given a manifold, why shouldn't I give it distance function by giving it a simple metric function, that is MxM→ℝ with the usual axioms? I could happily measure distances in coordinate-independent way for ever after...
  42. J

    Is U an Open Subset in the Space of Continuous Functions?

    Homework Statement Let a,b \in \mathbb{R} and we define I = [a,b] ([] means closed set). Let \mathcal{C}_{\mathbb{K}}(I) be the space of all continuous functions I \to \mathbb{K} with the norm f \mapsto ||f||_I = \displaystyle \sup _{x \in I} |f(x)|. Let U be the set of all continuous functions...
  43. T

    Triangle Inequality for a Metric

    Homework Statement Prove the triangle inequality for the following metric d d\big((x_1, x_2), (y_1, y_2)\big) = \begin{cases} |x_2| + |y_2| + |x_1 - y_1| & \text{if } x_1 \neq y_1 \\ |x_2 - y_2| & \text{if } x_1 = y_1 \end{cases}, where x_1, x_2, y_1, y_2 \in \mathbb{R}...
  44. N

    [topology] The metric topology is the coarsest that makes the metric continuous

    [topology] "The metric topology is the coarsest that makes the metric continuous" Homework Statement Let (X,d) be a metric space. Show that the topology on X induced by the metric d is the coarsest topology on X such that d: X \times X \to \mathbb R is continuous (for the product topology on X...
  45. G

    Tachyonic Energies in a Minkowski Metric

    So I'm working on a problem (Hartle problem 6, chapter 6) dealing with tachyons. So far, I have determined the four-velocity and the four-momentum (up to a sign) of a tachyon. I have, with the four-velocity being a unit spacelike four-vector, u^{\alpha}=\frac{\pm...
  46. 8

    Diameter of a union of metric spaces

    Homework Statement suppose that a metric space A is a union A = B U C of two subsets of finite diameter. Prove A has finite diameter. Homework Equations The Diameter of a metric space M is sup D(a,b) for all a,b in M. The Attempt at a Solution Really, no idea where to begin. I just...
  47. C

    Convergence of {fn} wrt to C(X) metric

    Let C(X) be the set of continuous complex-valued bounded functions with domain X. Let {fn} be a sequence of functions on C. We say, that fn converges to f wrt to the metric of C(X) iff fn converges to f uniformly. By definition of the uniform convergence, for any ε>0 there exists integer N...
  48. G

    A metric space of equivalent sequence classes

    A metric space of equivalent Cauchy sequence classes (Z, rho) is defined using a metric of the sequence elements in the space (X,d), where d is from XX to R (real numbers). The metric of the sequence classes is rho = lim d(S, T), where S and T are the elements of the respective sequences. To...
  49. S

    Extending Bounded metric spaces to compact spaces

    Hi Suppose (X,d) is a bounded metric space. Can we extend (X,d) into (X',d') such that (X',d') is compact and d and d' agree on X? ( The reason for asking the question: To prove a theorem in Euclidean space, I found it convenient to first extend the bounded set in question to a compact one (...
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