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. ferst

    Gödel Metric Theses & Dissertations: 3 in Eng/Port, 1 in Portuguese

    Hi, I'm doing the master in science and one of things that I have to study is the Gödel metric. His paper have a high level for me and I'm seeking theses and dissertations about the Gödel universe. At moment I got three theses about the subject in english and portuguese-BR and one dissertation...
  2. B

    How can the transitivity of the sup metric on bounded sequences be proven?

    Homework Statement A sequence \{x_{n}\} of real numbers is called bounded if there is a number M such that |x_{n}| ≤M for all n. Let X be the set of all bounded sequences, show that d(\{x_{n}\},\{y_{n}\})=sup \{|x_{n}-y_{n}| :n \in N \} is a metric on X.The only part I am struggling with is the...
  3. S

    Most proper time in Schwarzchild metric

    Hi In the Schwarzschild metric, the proper time is given by c^{2}dτ^{2} = (1- \frac{2\Phi}{c^2})c^2 dt^2 - r^2 dθ^2 with where \Phi is the gravitational potential. I have left out the d\phi and dr terms. If there is a particle moving in a circle of radius R at constant angular velocity ω...
  4. N

    Finding inverse metric tensor when there are off-diagonal terms

    How do you find the inverse of metric tensor when there are off-diagonals? More specifivally, given the (Kerr) metric, $$ d \tau^2 = g_{tt} dt^2 + 2g_{t \phi} dt d\phi +g_{rr} dr^2 + g_{\theta \theta} d \theta^2 + g_{\phi \phi} d \phi^2 + $$ we have the metric tensor; $$ g_{\mu \nu} =...
  5. V

    Is there a known parametrization for time dilation in the FLRW metric?

    Hi, could anyone help me out? The FLRW metric in spherical coordinates is: \;\; ds2 = dt2 - a(t)2(dr2 + r2dΩ2) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (1) I am considering a similar metric of the format: \;\; ds2 = \frac{1}{a(t')^{2}}dt'2 - a(t')2(dr2 + r2dΩ2)...
  6. R

    Why a constant determinant of the metric?

    Hi, In his original paper, Schwarzschild set the "'equation of the determinant" to be: |g|=-1. In other words, he imposed the determinant of the metric to be equal to minus one when solving the Einstein's equations. Must we impose this equality systematically in general relativity and why...
  7. C

    Metric tensor after constructing a quotient space.

    Suppose we have some two-dimensional Riemannian manifold ##M^2## with a metric tensor ##g##. Initially it is always locally possible to transform away the off-diagonal elements of ##g##. Suppose now by choosing the appropriate equivalence relation and with a corresponding surjection we construct...
  8. D

    Metric in SR: \eta^{\alpha \beta}=\eta_{\alpha \beta}?

    Does \eta^{\alpha \beta}=\eta_{\alpha \beta} in all coordinate systems or just inertial coordinate systems?
  9. B

    Need help with simple proof, metric space, open covering.

    Please take a look at the proof I added, there are some things I do not understand with this proof. 1. Does it really show that |f(x)-f(y)|≤d(x,y) for all x and y? Or does it only show that if there is an ball with radius r around x, and this ball is contained in an O in the open covering, and...
  10. M

    Transformation of the metric tensor from polar to cartesian coords

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  11. G

    Stress-energy tensor & mass term in metric

    I'm trying to clarify for myself the relation between the stress-energy tensor and the mass scalar term in metric solutions to Einstein's equations. Maybe I should also say I'm trying to understand the energy tensor better, or how it relates to boundary conditions on the solutions. My...
  12. Y

    Practical measurements of rotation in the Kerr metric

    In another thread WannabeNewton mentioned: and gave this reference: Until WBN mentioned it, I had never given any thought to the difference between these methods of measuring rotation, so I would like to explore those ideas further here, particularly in relation to the Kerr metric. Consider...
  13. C

    Is the Metric g a Complex Manifold?

    Hi so I was just wondering if the metric g=diag(-e^{iat},e^{ibx},e^{icy}) (where a,b,c are free parameters and t,x,y are coordinates) corresponds to a complex manifold (or is nonsensical), and what the manifold looks like?
  14. P

    Calculating G' for an Orthogonal Coordinate System

    Homework Statement For the orthonormal coordinate system (X,Y) the metric is \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} Calculate G' in 2 ways. 1) G'= M^{T}*G*M 2) g\acute{}_{ij} = \overline{a}\acute{}_{i} . \overline{a}\acute{}_{j} Homework Equations \begin{pmatrix}...
  15. M

    Statement about topology of subsets of a metric space.

    Homework Statement . Prove that a closed subset in a metric space ##(X,d)## is the boundary of an open subset if and only if it has empty interior. The attempt at a solution. I got stuck in both implications: ##\implies## Suppose ##F## is a closed subspace with ##F=\partial S## for some...
  16. M

    Sequence of metric spaces is compact iff each metric space is compact

    Homework Statement . Let ##(X_n,d_n)_{n \in \mathbb N}## be a sequence of metric spaces. Consider the product space ##X=\prod_{n \in \mathbb N} X_n## with the distance ##d((x_n)_{n \in \mathbb N},(y_n)_{n \in \mathbb N})=\sum_{n \in \mathbb N} \dfrac{d_n(x_n,y_n)}{n^2[1+d_n(x_n,y_n)]}##...
  17. P

    Induced Metric of a 2-Sphere: Why i≠j?

    So, by accident, while deriving the induced metric for a sphere in 3 dimensions I realized that the transpose of the jacobi matrix multiplied by the jacobi matrix (considering it as 3 row/column vectors)will work out the induced metric. Why is it that i≠j ends up being superfluous. One would...
  18. D

    Show the following is a metric

    Homework Statement Show that ##d(f,g) = \int_{0}^{1}\left | f(x) - g(x) \right | dx## is a distance function. Where ##f : [0,1] \rightarrow R## and ##f## is continuous. Homework Equations The Attempt at a Solution I am stuck on the second property where you have to show d(f,g) =...
  19. M

    Limit of a sequence on a metric space

    Homework Statement . Let ##(X,d)## be a metric space and let ##D \subset X## a dense subset of ##X##. Suppose that given ##\{x_n\}_{n \in \mathbb N} \subset X## there is ##x \in X## such that ##\lim_{n \to \infty}d(x_n,s)=d(x,s)## for every ##s \in D##. Prove that ##\lim_{n \to \infty} x_n=x##...
  20. T

    Help understanding metric units

    Hi all Firstly this is not a homework question. I just found myself wondering what certain numbers meant while at work and realized that i didnt actually know hence the posting. I was hoping someone could help explain & confirm a couple of things about units. 1) Given the number...
  21. S

    Does max|f - g| Define a Metric?

    Technically, this is not a homework question, since I solely seek an answer for self-indulgence. Homework Statement Example 1.1.4. Suppose f and g are functions in a space X = {f : [0, 1] → R}. Does d(f, g) =max|f − g| define a metric? Homework Equations (1) d(x, y) ≥ 0 for all x, y...
  22. D

    Metric for non-inertial coordinate system

    Homework Statement Hey guys. So here's the problem: Consider an ordinary 2D flat spacetime in Cartesian coordinates with the line element ds^{2}=-dt^{2}+dx^{2} Now consider a non-inertial coordinate system (t',x'), given by t'=t, x'=x-vt-\frac{1}{2}at^{2} (1) What is the metric...
  23. C

    What would this term correspond to? Inverse metric of connection.

    Suppose we are given two projection operators H' and H'' such that H' + H'' = 1, i.e. that any vector can be written as V = V' + V'' = (H' + H'') V. In a formula for a projection of the Riemann tensor (see the thread "Projection of the Riemann tensor formula") I encountered the term...
  24. L

    Proving Uniformly Continuous Extension of Function ##f## in Metric Space ##E'##

    Homework Statement Let ##S\subset E## where ##E## is a metric space with the property that each point of ##S^c## is a cluster point of ##S.## Let ##E'## be a complete metric space and ##f: S\to E'## a uniformly continuous function. Prove that ##f## can be extended to a continuous function...
  25. Y

    CTCs in Kerr Metric: Examining the Invariance of CTCs in Spacetime

    Are CTCs in the Kerr metric just an artefact of the coordinates used? This paper http://arxiv.org/abs/gr-qc/0207014 suggests that is the case. In a private message it has been suggested to me that CTCs in a spacetime are an invariant feature so are not removable by a change of coordinate system...
  26. M

    Connected components of a metric subspace

    Homework Statement . Consider the subspace ##U## of the metric space ##(C[0,1],d_∞)## defined as ##U=\{f \in C[0,1] : f(x)≠0 \forall x \in [0,1] \}##. Prove that ##U## is open and find its connected components. The attempt at a solution. First I've proved that ##U## is open. I want to...
  27. Y

    Schwarzschild metric in terms of refractive index

    This is a spin off from another thread: First there are a couple of mathpages http://mathpages.com/rr/s8-04/8-04.htm and http://mathpages.com/rr/s8-03/8-03.htm that discuss the refractive index model and highlights the differences. The first obvious objection is that the 'medium' must have...
  28. O

    Heat Exchanger Metric Formulas

    Hello, I need help to calculate the length of pipe in copper (d=22mm) to get the water in "kolam" to be 45 degree celcius and the heat pump temp is 55 degree celcius, heat pump water flow is about 1 meter cubic per hours, any simple formulas ? thank you for help
  29. H

    Coordinate and dual basis vectors and metric tensor

    I have been reading an introductory book to General Relativity by H Hobson. I have been following it step by step and now I am stuck. It is stated in the book that: "It is straightforward to show that the coordinate and dual basis vectors themselves are related... "ea = gabeb ..." I have...
  30. B

    Exploring the Metric Space Axioms: A Brief Overview and Exercise

    Quick question about the metric space axioms, is the requirement that the distance function be positive-semidefinite an axiom for metric spaces? It seems that it can be proved from the other axioms (symmetry, identity of indiscernibles and the triangle inequality). BiP
  31. nomadreid

    Units of spacetime in Minkowski metric

    In the equation ds2=dx2+dy2+dz2-c2*dt2 the units on the RHS are units of distance squared. But it would seem that units for a spacetime metric should somehow be in units which incorporate both space and time units. Undoubtedly this is an elementary question, but one has to start somewhere...
  32. Philosophaie

    What is the value of Q in the equation for the Kerr-Newman Metric Tensor?

    Our galaxy is rotating and is charged therefore the choice for the metric is the Kerr-Newman Metric. I want to solve for the Kerr-Newman Metric Tensor. There are a few questions. 1-What is the value for Q in the equation: ##r_Q^2=\frac{Q^2*G}{4*\pi*\epsilon_0*c^4}## where ##G=6.674E-20...
  33. M

    Proving an open ball is connected in a metric space X

    Homework Statement . Let ##B(a,ε) (ε>0)## in a metric space ##(X,d)##. Decide whether this subset of ##(X,d)## is connected or not. The attempt at a solution. Well, I know open intervals in the real line are connected. I suppose that an open ball in a given metric space can be imagined...
  34. Philosophaie

    Metric Tensor of the Reissner–Nordström Metric

    I am looking for the Metric Tensor of the Reissner–Nordström Metric.g_{μv} I have searched the web: Wiki and Bing but I can not find the metric tensor derivations. Thanks in advance!
  35. M

    Complete metric space can't have a countably infinite perfect space

    1. Homework Statement . Let ##(X,d)## be a complete metric space. Prove that if ##P \subset X## is perfect, then P is not countably infinite. 3. The Attempt at a Solution . Well, I couldn't think of a direct proof, I thought that in this case it may be easier to assume is countably infinite...
  36. L

    Proving Closedness of a Set in a Metric Space

    Homework Statement Prove that if lim n→∞ (p_n ) = p in a metric space then the set of points {p,p_1,p_2, ...,} are closed. 2. Relevant information The definition of close in my book is "a set is closed if and only if its complementary is open." So I want to prove this by contradiction. I...
  37. M

    A separable metric space and surjective, continuous function

    Homework Statement . Let X, Y be metric spaces and ##f:X→Y## a continuous and surjective function. Prove that if X is separable then Y is separable. The attempt at a solution. I've tried to show separabilty of Y by exhibiting explicitly a dense enumerable subset of Y: X is separable...
  38. C

    Relation between det(spacetime metric) and det(spatial metric)

    I have a metric g on spacetime and a spatial metric ##\gamma## such that the components of g can be written in matrix form as $$ g_ {\alpha, \beta} = \begin{pmatrix} g_{00} & g_{0 j} \\ g_{i 0} & \gamma_{ij} \end{pmatrix} $$ where ##i,j = 1,2,3## and ##\alpha = 0,1,2,3##. Now I want to find a...
  39. C

    What happens to the form basis after making the metric time orthogonal

    Given a basis for spacetime ##\{e_0, \vec{e}_i\}## for which ##\vec{e}_0## is a timelike vector. Of these vectors one can make a new basis for which all vectors are orthogonal to ##\vec{e}_0##. I.e. the vectors $$\hat{\vec{e}}_i = \vec{e}_i - \frac{\vec{e}_i \cdot \vec{e}_0}{\vec{e}_0 \cdot...
  40. M

    Discrete metric and continuity equivalence

    Homework Statement . Prove that a metric space X is discrete if and only if every function from X to an arbitrary metric space is continuous. The attempt at a solution. I didn't have problems to prove the implication discrete metric implies continuity. Let f:(X,δ)→(Y,d) where (Y,d) is...
  41. E

    Metric VS English engineering system of measurement

    Hey guys, I know you all wish you never had to do all the weird conversions required for our current system of measurement. I know I'd rather convert 17km to m than 17mi to feet or ever inches. I thought that since I know so many people who would rather just do the easy metric conversions...
  42. T

    Metric for rating funtion accuracy from R^m to R.

    I'm writing program in which I generate pseudo random expressions with the hope of finding one that closely approximates the given data set. The functions map from R^m (an m-tuple of reals) to a real. What I need is a way to rate the functions by their accuracy. Are there any known methods for...
  43. C

    Construct electromagnetic stress-energy tensor for a non-flat metric

    Hi, I am having problems in constructing a stress-energy tensor representing a constant magnetic field Bz in the \hat{z} direction. The coordinate system is a cylindric {t,r,z,\varphi}. The metric signature is (+,-,-,-). I ended with the following mixed stress-energy tensor: Is this...
  44. T

    General Relativity, Schwarzschild's Metric, and Applications

    Homework Statement I have been trying to understand the actual applications and mathematics behind Einstein's Field Equations. I have watched a two hour long lecture on how they were derived and have pretty much understood it, however I still don't know how to actually "use" Einstein's Field...
  45. S

    How Do Contradictory Metrics Align with GR Field Equations?

    Our current version of the Schwarzschild metric is d\tau^2=(1-r_s/r)dt^2-(1-r_s/r)^{-1}dr^2-r^2d\Omega^2 where c is set to 1, r is the scalar distance, r_s is the 'event horizon' radius, and d\Omega^2=d\theta^2+sin^2\theta d\phi^2. In Schwarzschild's original paper from 1916 he does not...
  46. B

    Calculating area with the metric

    So say you have a 2D Riemannian manifold with a metric defined on it and for simplicity let's say its flat. That means there exists a coordinate system for which the metric tensor is the normal Euclidean metric everywhere. However let's say we are using an arbitrary coordinate system with a non...
  47. T

    Metric expansion in terms of planck pixels

    what if the evolution of the universe over time and the expansion of space a series of binary events such that the same quantity of energy is divided geometrically from one into two, four, eight, sixteen pieces, such that each Planck time is equivalent to one such split and each Planck length is...
  48. T

    Is there a redshift in a conformally flat metric space?

    Hello PF: I noticed a thread on PF in which TOM STOER and others were discussing how to calculate the redshift for an arbitrary metric. I need to talk to Tom if he is still on this list. The question has arisen in an applied physics field whether the following conformally flat metric...
  49. Philosophaie

    Non-rotating or rotating Metric

    Is the Milky Way Galaxy non-rotating or rotating? Which metric is best suited: Schwarzschild or the Kerr Metric, respectively?
  50. SamRoss

    Proof Minkowski metric is invariant under Lorentz transformation

    Ok, this should be an easy one but it's driving me nuts. When we take the Lorentz transformations and apply them to x2-c2t2 we get the exact same expression in another frame. I can do this math easily by letting c=1 and have seen others do it by letting c=1 but I have never seen anyone actually...
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