11.1 Statistical methods in geophysical data analysis

Statistical methods are crucial in geophysical data analysis. They help scientists make sense of complex datasets, identify patterns, and quantify uncertainties. From basic descriptive statistics to advanced techniques like PCA and time series analysis, these tools are essential for interpreting Earth's physical properties.

Data quality evaluation ensures reliable results in geophysics. Outlier detection, hypothesis testing, and cross-validation help researchers assess the validity of their findings. Bootstrapping and confidence intervals provide ways to quantify uncertainty, which is vital when dealing with Earth's complex systems.

Statistical Analysis of Geophysical Data

Descriptive Statistics and Correlation Analysis

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• Descriptive statistics summarize the central tendency and dispersion of geophysical data
  • Mean represents the average value of a dataset
  • Median is the middle value when the data is sorted in ascending or descending order
  • Mode is the most frequently occurring value in the dataset
  • Standard deviation measures the spread of the data relative to the mean
  • Variance is the average of the squared differences from the mean
• Correlation analysis measures the strength and direction of the linear relationship between two geophysical variables
  • Pearson correlation coefficient ranges from -1 to +1
    • Values close to +1 indicate a strong positive correlation (variables increase together)
    • Values close to -1 indicate a strong negative correlation (one variable increases as the other decreases)
    • Values close to 0 indicate a weak or no linear correlation

Regression Analysis and Time Series Analysis

• Regression analysis models the relationship between a dependent variable and one or more independent variables in geophysical data
  • Linear regression fits a straight line to the data, assuming a linear relationship between variables
  • Non-linear regression models more complex relationships (exponential, logarithmic, polynomial)
  • Example: Modeling the relationship between seismic wave velocity and depth in the Earth's crust
• Time series analysis techniques decompose geophysical data into its constituent frequencies and analyze temporal patterns and trends
  • Fourier analysis represents a time series as a sum of sinusoidal functions with different frequencies and amplitudes
  • Wavelet analysis uses wavelets (localized oscillatory functions) to analyze non-stationary signals at different scales and locations
  • Example: Analyzing seasonal variations in Earth's gravitational field using time series from GRACE satellites

Principal Component Analysis (PCA)

• Principal component analysis (PCA) reduces the dimensionality of geophysical data by identifying the principal components that explain the most variance in the data
  • Principal components are linear combinations of the original variables that are uncorrelated and ordered by decreasing variance explained
  • The first principal component captures the largest amount of variance in the data, followed by the second component, and so on
• PCA is useful for data compression and visualization
  • Projecting high-dimensional data onto a lower-dimensional space (2D or 3D) for easier interpretation and visualization
  • Example: Identifying patterns in multi-sensor geophysical data (seismic, electromagnetic, and gravitational) using PCA

Data Quality Evaluation for Geophysics

Outlier Detection and Hypothesis Testing

• Outlier detection methods identify data points that significantly deviate from the rest of the dataset
  • Z-score measures the number of standard deviations a data point is from the mean
    • Data points with Z-scores greater than a threshold (e.g., 3) are considered outliers
  • Tukey's method identifies outliers based on the interquartile range (IQR)
    • Data points below Q1 - 1.5 × IQR or above Q3 + 1.5 × IQR are considered outliers
  • Outliers can be caused by measurement errors or rare geophysical events (earthquakes, volcanic eruptions)
• Hypothesis testing assesses the statistical significance of observed differences or relationships in geophysical data
  • t-test compares the means of two groups to determine if they are significantly different
  • ANOVA (analysis of variance) tests for differences among three or more group means
  • chi-square test evaluates the association between categorical variables
  • Example: Testing if the mean seismic wave velocity differs significantly between two geological formations

Confidence Intervals and Cross-Validation

• Confidence intervals provide a range of values within which the true population parameter is likely to fall, given a specified level of confidence
  • 95% confidence interval means that if the sampling process is repeated many times, 95% of the intervals will contain the true population parameter
  • Confidence intervals help quantify the uncertainty in geophysical estimates (mean, regression coefficients)
  • Example: Estimating the true mean magnetic susceptibility of a rock formation with a 95% confidence interval
• Cross-validation techniques assess the predictive performance of geophysical models by partitioning the data into training and testing sets
  • k-fold cross-validation divides the data into k subsets, using k-1 subsets for training and the remaining subset for testing, repeated k times
  • Leave-one-out cross-validation (LOOCV) uses a single data point for testing and the remaining data for training, repeated for each data point
  • Cross-validation helps prevent overfitting and provides a more robust estimate of model performance

Bootstrapping

• Bootstrapping is a resampling method that estimates the sampling distribution of a statistic by repeatedly sampling with replacement from the original dataset
  • Generates multiple bootstrap samples of the same size as the original dataset
  • Calculates the statistic of interest (mean, median, correlation coefficient) for each bootstrap sample
  • Constructs a bootstrap distribution of the statistic to estimate its variability and confidence intervals
• Bootstrapping helps quantify the uncertainty in geophysical estimates when the underlying distribution is unknown
  • Non-parametric alternative to traditional parametric methods that assume a specific distribution (normal, t-distribution)
  • Example: Estimating the uncertainty in the median seismic wave attenuation coefficient using bootstrapping

Uncertainty Quantification in Geophysics

Probability Density Functions (PDFs) and Cumulative Distribution Functions (CDFs)

• Probability density functions (PDFs) describe the likelihood of a continuous random variable taking on a specific value
  • Normal (Gaussian) distribution is symmetric and bell-shaped, characterized by its mean and standard deviation
  • Log-normal distribution is skewed, with a long right tail, often used for positive-valued geophysical quantities (permeability, conductivity)
  • Exponential distribution models the time between events in a Poisson process (earthquake occurrences)
• Cumulative distribution functions (CDFs) give the probability that a random variable takes a value less than or equal to a given value
  • CDF is the integral of the PDF from negative infinity to the given value
  • Useful for determining percentiles and probability thresholds in geophysical data
  • Example: Calculating the probability that the magnitude of an earthquake exceeds a certain value using the Gutenberg-Richter law CDF

Bayes' Theorem and Monte Carlo Simulation

• Bayes' theorem updates the probability of a hypothesis (geophysical model) based on new evidence (additional data)
  • Prior probability represents the initial belief in the hypothesis before considering the evidence
  • Likelihood quantifies the probability of observing the evidence given the hypothesis
  • Posterior probability is the updated probability of the hypothesis after incorporating the evidence
  • Bayesian inference is widely used in geophysical inverse problems and uncertainty quantification
• Monte Carlo simulation generates random samples from a probability distribution to estimate the distribution of a geophysical quantity or the uncertainty in a geophysical model
  • Generates a large number of random samples from the input probability distributions
  • Evaluates the geophysical model or quantity of interest for each sample
  • Constructs an empirical distribution of the model outputs or quantity of interest
  • Monte Carlo methods are useful when analytical solutions are intractable
  • Example: Estimating the uncertainty in ground motion predictions using Monte Carlo simulation with random samples of earthquake source parameters

Markov Chain Monte Carlo (MCMC) Methods

• Markov chain Monte Carlo (MCMC) methods generate samples from a target probability distribution by constructing a Markov chain that converges to the desired distribution
  • Metropolis-Hastings algorithm proposes a new sample based on the current sample and accepts or rejects it based on a probability ratio
  • Gibbs sampling updates each component of the sample sequentially by sampling from its conditional distribution given the other components
  • MCMC methods are commonly used in Bayesian inference for geophysical inverse problems
  • Example: Sampling from the posterior distribution of Earth's mantle viscosity using MCMC with geophysical observations (gravity, topography, plate motions)

Data Reduction and Compression for Geophysics

Decimation and Filtering

• Decimation reduces the sampling rate of geophysical time series data by keeping only every nth sample
  • Reduces data volume while preserving the essential features of the signal
  • Requires an appropriate decimation factor to avoid aliasing (loss of high-frequency information)
  • Example: Decimating high-frequency seismic data for long-term storage or transmission
• Filtering removes unwanted frequency components from geophysical data
  • Low-pass filters remove high-frequency noise (instrumental noise, cultural noise)
  • High-pass filters remove low-frequency trends (tidal effects, temperature variations)
  • Band-pass filters retain a specific range of frequencies (seismic waves, electromagnetic signals)
  • Example: Applying a low-pass filter to remove high-frequency noise from gravitational data

Wavelet Compression

• Wavelet compression decomposes geophysical data into a set of wavelet coefficients and discards the coefficients below a certain threshold
  • Wavelets are localized oscillatory functions that capture both frequency and location information
  • Wavelet transform represents the data as a sum of wavelets with different scales and positions
  • Thresholding the wavelet coefficients removes the less significant details while preserving the essential features
  • Wavelet compression is effective for compressing non-stationary signals with localized features (seismic traces, satellite images)
  • Example: Compressing high-resolution airborne magnetic data using wavelet compression for efficient storage and transmission

Lossless and Lossy Compression

• Lossless compression techniques reduce data size without losing information
  • Run-length encoding replaces repeated sequences of identical values with a single value and a count
  • Huffman coding assigns shorter bit sequences to more frequently occurring values based on their probability distribution
  • Lossless compression is suitable for archiving geophysical data or when exact reconstruction is required
  • Example: Applying run-length encoding to compress geophysical well log data containing long sequences of identical values
• Lossy compression techniques achieve higher compression ratios by allowing some loss of information
  • Discrete cosine transform (DCT) represents the data as a sum of cosine functions with different frequencies and discards the high-frequency components
  • Fractal compression exploits the self-similarity of geophysical data at different scales and represents the data using a set of fractal parameters
  • Lossy compression is appropriate when some loss of detail is acceptable (geophysical visualization, rapid data transmission)
  • Example: Compressing satellite imagery of Earth's surface using DCT-based compression for efficient storage and transmission