Bill Menke's Research Notes 1977-present

These Research Notes address topics that I found of interest over the years. I do not make any claims that they are original or even that they are correct. The list below is not in any particular order, so a keyword search on this page is probably the most effective way of finding something.

• 278. Approximate Covariance of the Nonlinear Implicit Normal Inverse Problem
• 277. Appearance of the Data Resolution Matrix in Generalized Least Squares Formulas
• 276. Damped Least Squares Data and Model Resolution Equal for Symmetric Data Kernel
• 275. Successor PDF for the Metropolis-Hastings Algorithm on a Finite Interval
• 274. Error in Ensemble Mean and Covariance Follows Counting Statistics
• 273. Maximum Likelihood Applied to Binary Time Series
• 272. On Reconciling Ray Amplitudes and Energy Fluxes
• 271. For those who doubt the adjoint method can be applied to glacial isostatic problems
• 270. Identifying patches with different statistical properties
• 269. More on Accelerated Global Warming
• 268. Searching for Acceleration of Global Warming
• 267. Discussion of Units in Note 249 on Effective Diffusivity
• 266. Explicit Formula for Constrained Least Squares
• 265. Steepest Slope of a Plane by Measuring Slopes in Two Orthogonal Directions
• 264. Where the Delta Functions in the Adjoint Source Come From
• 263. Should Glacial Isostatic Calculation Consider Viscosity to be Advecting?
• 262. Thoughts on Frechet Derivatives
• 261. Sparsity Inversion for Vp/Vs ratio
• 260. Comparison of Several Ways to Compute Sensitivity Kernels
• 259. Generalized Factor Analysis for Time Series
• 258. Estimating horizontal shaking from observations of the aspect ratio of un-toppled glacial boulders
• 257. Second order correction to the linearized Resolution Matrix
• 256. Properties of the GLS Resolution Matrix
• 255. On the Linearized Resolution Matrix
• 254. Sensitivity kernel for uninvertible operator
• 253. Example of Rolling Gaussian Process Regression
• 252. Rolling Gaussian Process Regression
• 251. Real-time Gaussian Process Regression
• 250. Real-time Generalized Least Squares
• 249. Effective Pressure Diffusivity of a Long, Thin Pipe
• 248. Kalman Filters; an Application of Generalized Least Squares, Part 2
• 247. Kalman Filters; an Application of Generalized Least Squares
• 246. Autocorrelation Function Approximately Corresponding to a Second Derivative
• 245. Amplitude of a Rayleigh wave at constant energy flux
• 244. Further Thoughts on the Second Derivative in Amlitude Tomography
• 243. Forward Scattered P-waves Fully Characterize a Point Heterogeneity
• 242. Scattered P-waves Fully Characterize a Point Heterogeneity
• 241. An Iterated Integral Converted to a Single Integral
• 240. Making surface wave seismograms using the Brune(1962) formula
• 239. Sensitivity Kernel for Ray Amplitude for Initially Straight Rays
• 238. Late P-wave Traveltime Anomalies near Cape Yakutaga, Alaska
• 237. Time and Frequency Domain Inversions for a Bateman-type Attenuation Operator
• 236. Gaussian Process Regression equals Generalized Least Squares
• 235. P-Wave Undershooting the Alaskan Subduction Zone
• 234. Adjoint formulation for data kernel and error gradient when the data is a linear functional of the field
• 233. Thoughts on Differential Data
• 232. The Least Square Solution and its Covariance are Invariant under a Linear Transformation
• 231. Fictitious ISO Component of Moment Tensor Due to Erroneous Depth
• 230. Misfit Function for Bounding Data
• 229. Least Squares with a Data Kernel Involving an Unknown Parameter
• 228. Priors with Randomly-distributed Hyperparameters
• 227. Varying Parameters in the Autocovariance Matrix in GPE Problems
• 226. Differential Travel Time and T-star as Predicted by the Very Broadband Rheology Calculator
• 225. Notes on Attenuation
• 224. Critique of Nested Hierarchical Trans-Dimensional Models
• 223. The Bateman Attenuation Operator
• 222. Data Resolution matrix for Generalized Least Squares
• 221. Data Winnowing Via Importance
• 220. Simple Seismic Velocity - Melt Fration Derivatives Based on Clark and Lesher (2017)
• 219. The Derivative of Shear Velocity with Respect to Compressional Velocity at Constant Bulk Modulus and Density
• 218. Comparison of Receiver Function, Cross-Convolution and Transfer Function Methods
• 217. Moments of a Convolution
• 216. Is Cross-correlating Displacement or Velocity Better in Ambient Noise Studies?
• 215. Harmonic Series for Products of Powers of Sines and Cosines
• 214. Non-uniqueness of Anisotropy Inversions
• 213. Note on the Algebraic Eigenvalue Problem
• 212. Finding P-wave Fast Directions in Anisotropic Material
• 211. Correcting Telesesimic Traveltimes for Moho Depth
• 210. Correcting Telesesimic Travel Times for Upper Crustal Structure
• 209. Uplift Geochronology
• 208. Pore and Throat of Tetrahedron
• 207. Sensitivity of Seismic Velocities to Temperature
• 206. Sensitivity of Seismic Velocities to Magnewsium Number
• 205. Northern Appalachian Anomaly Parallax
• 204. The Effect of Resampling on the Trade-Off of Resolution and Variance
• 203. Covariance Estimates in Linearized Least Squares
• 202. Heat Flow in the Lithosphere
• 201. Frechet Derivative for Photographic Recording
• 200. Degrees of Freedom of the Natural Solution to an Inverse Problem
• 199. On Joint Inversion
• 198. Covariance of Layered Models
• 197. Trade Off Curves Computed for Ensembles
• 196. Backus-Gilbert Resolution Computed from Ensembles
• 195. Resolution Matrix computed from Ensembles of Solutions
• 194. Moment Tensor of a Fault in a Transversely Isotropic Material
• 193. Thoughts on Ford et al. (2018)
• 192. Exemplary solution to the advection-diffusion equation
• 191. Clumping Together of Aerosol Paricles Deceases Optical Depth
• 190. Gradient of a parameter in a nonlinear differential equation
• 189. Notes on the Trans-Dimensional Metropolis-Hastings (MH) Algorithm
• 188. Damped Least Squares has a Symmetric Resolution Kernel
• 187. Effect of omitting element on factors
• 186. Derivative of Wavefield Error with respect to source parameters
• 185. Estimating attenuation and delay
• 184. Statistical test for a bivariate normal distribution
• 183. Nonuniqueness in Anisotropic Tomography
• 182. RC Circuits
• 181. Anisotropic phase velocity implies anisotropic group velocity
• 180. Duration Scaling Thoughts for Slow Earthquakes
• 179. Anisotropic Parameter eta for a Layered Material
• 178. Lithospheric Structure at the NAA from Ekstrom [2017] Rayleigh Wave Velocities
• 177. Special case of Wu's [1982] apparent attenuation formula
• 176. Attenuation can brighten an S-wave Receiver Function
• 175. Surface wave dispersion and lithospheric cooling
• 174. Time Needed for the Amplitude of an Upper Mantle Thermal Anomaly to Decay by 50%
• 173. Estimating the frequency-dependence of t-star from spectral ratios
• 172. Sensitivity kernel for finite-freqeuncy double difference travel times
• 171. Sensitivity kernel for the squared envelope
• 170. Differential travel time delay associated with attenuation
• 169. Finite Frequency Travel Times Example
• 168. Thoughts on Force Equivalents
• 167. Poles and Zeros Geophone Response
• 166. The Azimi Attenuation Model
• 165. Numerical Approximation for Geometrical Spreading using Ray Tangents
• 164. Effect of Density Perturbation on Vp/Vs
• 163. Area under the Envelope
• 162. Linear Interpolation with Excel
• 161. The Wave Field Error is Invariant Under Downward Continuation
• 160. Application of Adjoint Methods to Travel Time Tomography
• 159. Review of the Derivation of Adjoint Methods
• 158. Derivative of error with respect to a parameter in a differential equation
• 157. Definition of Power Spectral Density in 2D
• 156. Sampling Layered Models with the Metropolis Algorithm
• 155. Energy flux in an elastic wave
• 154. Confidence limits for the power spectral density of an AR1 stochastic process
• 153. Velocity and traveltime perturbations
• 152. Nonuniquenness of anisotropic teleseismic tomography
• 151. Fourier transforms of left and right halves on an interval
• 150. Covariance of differential delays
• 149. Test of Lee & Seung algorithm for non-negative factorization S = CF
• 148. Degrees of freedom of the generalized least squares error
• 147. Application of binomial distribution to an event counting problem
• 146. Perturbation series for inverse of a linear operator
• 145. Data smoothing based on function minus its localized average is small
• 144. Data smoothing with exponential covariance of prior information
• 143. Effect of Choice of Covariance Matrix on Curve Fitting
• 142. d(slowness)/d(structure) at constant frequency from modes
• 141. Marquering et al.'s formula for dT/du
• 140. Notes on Anisotropic Tomography
• 139. Resolution of Generalized Least Squares
• 138.Computing Model Variance Without Computing the Complete Generalized Inverse
• 137. Waveform Fitting of Cross-Spectra to Determine Phase Velocity Using Aki s (1957) Formula
• 136. Resolution of the Data Smoothing Problem
• 135. Deviation of P-axes of earthquakes from the direction of maximum compression
• 134. Simple example of linearized generalized least squares
• 133. Matrix as row constant + vector + column vector + residual (see also 020)
• 132. Phase velocity of surface wave by waveform fitting
• 131. 2D traveltime tomography with anisotropy using analytic Fourier kernels
• 130. Magnetotelluric method, including Iceland, Ricotti equation, skin depth, propagator matrix formulation, and Monte Carlo inversion
• 129. Reconciling Saito's and Singh et al.'s seismic source discontinuity formulas
• 128. Transformation of variables to cluster grid spacing near origin
• 127. Decay of pressure away from a cylindrical volume
• 126. Effect of horizontal water movement on temperature profile
• 125. Flow of heat in wet sediments
• 124. Displacement and stress around a pressurized cylindrical borehole
• 123. Resolving thermal history from present-time temperature
• 122. Effect of carbonate dissolution on sediment porosity
• 121. Sediment grain size distributions
• 119. Damping of a horizontally oscillating heat flow by a sediment lid
• 120. Fluid flow in a non-consolidating rigid sediment
• 118. Error in S wave polarization angle due to the free surface
• 117. Error in P wave angle of incidence due to the free surface
• 116. Musings on seismic attenuation in magma due to bubbles
• 115. Cross-convolution method compared to minimum tangential energy method
• 114. Shear wave splitting due to a single anisotropic layer
• 113. Weighted L1 norm fitting of a straight line
• 112. Partial derivative pf reflected wave traveltime with interface position
• 111. Comparing two histograms
• 110. Musings on the quation d(i) = a(i) - b(i)T
• 109. Derivation of formula for weighted mean
• 108. Confidence intervals for events
• 107. Estimating thermal history from temperature-dependent degredation data
• 106. Relationship between covarariance, resolution and near-zero singular values
• 105. Eigenvectors of a block-constant matrix
• 104. Product of a gaussian is a gaussian
• 103. Gauge invariance
• 102. Investination of the equation t(i,j)=t0(i,j)+(sum k) d(i,j,k)s(k)
• 101. Differential traveltimes
• 100. Shear wave splitting for two anisotropic layers
• 99. Estimating Moho depths from receiver function traveltimes
• 98. Rotational invariance of cross-convolution
• 97. Traveltime along a circular ray path
• 96. Perturbation of a straight line ray path
• 95. Valid forms of the ray equation
• 94. The correlation coefficient cannot exceed unity
• 93. Statistics of clusters
• 92. Variance of a spectral peak
• 91. Varimax factors
• 90. Inverse problem Gm=d when both G and m are functions of a parameter p
• 89. Polarization direction estimated by minimizing the energy on other components
• 88. Absolute earthquake location with the double difference method
• 87. Adjoint method of computing the data kernel
• 86. Discriminating two beams from one using a tripartite array
• 85. Schrodinger equation conserves total probability
• 84. Speed of rocket with direct mass-to-light conversion drive
• 83. Pressure in permeable slab drven by sinusoidal surface pressure
• 82. Variance of an estimate must decline as one uses additional data
• 81. The Kronecker delta is the only isotropic 2nd order tensor
• 80. Inference using Bayes Theorem
• 79. Spectral smoothing vs. windowing
• 78. Maximum entropy derivation of standard inverse theory formula
• 77. Equivalent ways of thinking about receiver function times
• 76. Propagator method with a source term
• 75. Zero-D thermal model of a radioactive convecting earth
• 74. Cross-convolution method of determining shear wave splitting parameters
• 73. Variance of shear wave splitting measurements
• 72. Intersection of a circular arc with a plane
• 71. Earthquake source representation in propagator method
• 70. Quantum mechanical rule for probability
• 69. Covariation of slowness and traveltimes
• 68. Statistics of detects using cross-correlation
• 67. Distribution of the product of two random variables
• 66. Eingenvalues as a funtion of tensor invariants
• 65. 1D tomography with unknown origin time
• 64. Dice rolling thought experiment relevant to likelihood functions
• 63. Bayesian inference
• 62. Fisher distribution of directions on a sphere
• 61. Elliptical reflections on rippled lakes
• 60. Reordering a list of elements to diagonalize corresponding correlation matrix
• 59. Estimating parameters of Gutenberg-Richter distribution with a kink
• 58. Spatial response of fan filter
• 57. Definition of mean velocity
• 56. Interpolating unit vectors
• 55. Backus-Gilbert kernel for crossing rays of finite width
• 54. Backus-Gilbert kernel for elliptical sieve problem
• 53. Smallest rectangle that surrounds an ellipse
• 52. Radon transform on a family of curves on the plane
• 51. Comments to Singer on his 1990 Science paper
• 50. Density profile of a planetary atmosphere
• 49. Least-sqares with a singular weight matrix
• 48. Darcy's Law from the Navier-Stokes equation
• 47. Gutenberg-Richter distribution with a kink
• 46. Inferring displacement from stress
• 45. Pressure in spherical self-gravitating cloud of an ideal gas
• 44. Correlation coefficient invariant under linear transformations
• 43. Bending moment due to thermal stress of ridge
• 42. Ray equation from principle of wavefront advance
• 41. Least-squares fitting of navigational data
• 40. Relationship between 3D Fourier and cosine transform
• 39. Random tiling of a sphere
• 38. Polarization tomography applied to light rays in curved spacetime
• 37. Isotropic third order tensors do not exist
• 36. Alternative to Parker's seminorm method
• 35. Seismic coherence notation
• 34. Precession of a satellite
• 33. Moser's ray tracing ideas
• 32. Kalman filters
• 31. Derivation of Parsevel's theorem
• 30. Envelope and instataneous frequency of sinc function
• 29. Populations that evolve to evolve
• 28. Richter scale for Palisades NY seismometer
• 27. Reparameterization involving a midpoint
• 26. Kriging
• 25. Convolution theorem for the probability of y=x1+x2
• 24. Conditional probabilities
• 23. Energy efficiency
• 22. Partial factor reconstruction
• 21. Fitting a differential equation to data
• 20. Matrix as row vector + column vector + residual (see also 133)
• 19. Marine magnetic anomalies
• 18. Confidence intervals for rates
• 17. Global adjustment of cross-over errors
• 16. Modes of vibration of volcano tube
• 15. Test of great circle fitting algorithm
• 14. Perpendicular distaces on a sphere
• 13. New (lat, lon) when the north pole is moved
• 12. Algorithm to fit least sqaures great circle
• 11. Distances on a sphere
• 10. Error in small number approximations
• 9. Perturbation theory for algebraic eigenvalue problem
• 8. Ramsey's and Watson's methods of fitting great circles
• 7. Derivation of the Boltzman distribution
• 6. Effect of using wrong seismic velocity on epicenter location
• 5. Simplification of a surface interval involving torque
• 4. Response of a toroidal accelerometer
• 3. Energy in a wave on a string
• 2. Phase and group velocity formulas using wavelength
• 1. Plane wave incident on fluid-fluid interface