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.

  • 307. Single layer Splitting Revisited
  • 306. When an Inner Product Involves a Component, the Adjoint Field is Parallel to the Component
  • 305. Adjoint Sources for Time and Space Sensitivities are the Same
  • 304. Adding Linear Prior Information to the Natural Solution of an Inverse Problem
  • 303. Addendum on How to Make Synthetic Micro-Seismic Noise
  • 302. Sensitivity Derivative for Two-Station Phase Velocity Measurements
    • 301. Accounting for Micro-Seismic Correlation Reduces Variance of Least Squares Estimates
  • 300. How to Make Synthetic Micro-Seismic Noise
  • 299. Three Station Waveform Method of Computing Phase Velocity and its Confidence
  • 298. Parameterized inversion for frequency-dependent time delay between two dispersed signals
  • 297. Addendum to Research Note 293, Differential Phase Method for Estimating Surface Wave Phase Velocity
  • 296. Estimated Phase Velocity for a Very Simple Rayleigh Wave Scattering Model
  • 295. Error in Phase Slowness Estimated by the Two-Station Method
  • 294. Python Function to Read a Mixed Numeric/String Table from a Text File
  • 293. The Differential Phase Method for Estimating Surface Wave Phase Velocity
  • 292. Filling in Missing Gaps in Heading Data
  • 291. On the Statistics of Correlated Time Series
  • 290. Station Amplitude and Attenuation can be Uniquely Determined
  • 289. Factors and Loadings Common to Two or More Datasets
  • 288. Derivation of the Frechet Derivative for the Martinson et al. (1982) signal correlation method
  • 287. The Set of All Real, Monotonically-Increasing Nth Order Polynomials that Map the (0,1) Interval onto Itself, Together with Sensitivities
  • 286. More on Computing the Derivative of the Acoustic Wave-function with Respect to Wavenumber
  • 285. Derivative of the Surface Wave Wave-function with Respect to Horizontal Wavenumber
  • 284. Raylegh Wave Amplitude Sensitivity
  • 283. Isotropic and Vertical Compensated Linear Vector Dipole Component of Moment Tensor Trade Off in Inversions that use Surface Wave Data
  • 282. Properties of modes of the possibly-anisotropic seismic wave equation
  • 281. Perturbation Theory Analysis of Rayleigh Wave Modes
  • 280. Ray Theoretical Transport Equation for Two Seismically-Relevant Scalar Wave Equations
  • 279. In a Horizontally-Propagating Wave, Energy Flux is Energy Density Times Velocity
  • 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