**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.*

- 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