---
title: "Wavelet Spectral Analysis"
subtitle: "Continuous and discrete wavelet transforms"
code-fold: true
---
::: {.callout-tip icon=false}
## For Newcomers
**You will learn:**
- Why standard frequency analysis misses important patterns
- What wavelets are and why they work better for non-stationary data
- How to read a wavelet "scalogram" (time-frequency map)
- What periodic patterns exist in groundwater and when they appear
Imagine listening to a symphony and trying to understand when the drums come in versus when the violins dominate. Wavelet analysis does this for water levels—showing not just what frequencies exist, but when they occur and how they change over time.
:::
## What You Will Learn in This Chapter
By the end of this chapter, you will be able to:
- Explain why wavelet and spectral methods are powerful for analyzing non-stationary groundwater time series.
- Interpret wavelet-inspired variance-by-scale views and power spectra in terms of physical processes such as seasonal cycles and multi-year climate signals.
- Connect dominant spectral peaks and variance distribution to aquifer memory, response timescales, and management-relevant horizons.
- Use multi-scale variance and spectral information to reason about appropriate monitoring frequencies and forecasting model designs.
```{python}
#| label: setup
#| echo: false
import os
import sys
from pathlib import Path
import numpy as np
import pandas as pd
import plotly.graph_objects as go
from plotly.subplots import make_subplots
import sqlite3
try:
from scipy import signal
SCIPY_AVAILABLE = True
except ImportError:
SCIPY_AVAILABLE = False
signal = None
print("Note: scipy not available. Spectral analysis will use simplified methods.")
import warnings
warnings.filterwarnings('ignore')
def find_repo_root(start: Path) -> Path:
"""Find repository root by looking for src/ directory"""
for candidate in [start, *start.parents]:
if (candidate / "src").exists():
return candidate
return start
quarto_project = Path(os.environ.get("QUARTO_PROJECT_DIR", str(Path.cwd())))
project_root = find_repo_root(quarto_project)
if str(project_root) not in sys.path:
sys.path.append(str(project_root))
from src.utils import get_data_path
```
## Introduction
Traditional Fourier analysis assumes stationarity - constant relationships over time. Wavelet analysis overcomes this limitation by providing time-frequency localization, revealing when and at what scale patterns occur.
**Source:** Groundwater level data from Champaign County observation wells (1M+ measurements) integrated with weather and climate records.
## Why Wavelets for Groundwater?
::: {.callout-note icon=false}
## Understanding Wavelet Analysis
**What Is It?**
**Wavelet analysis** is a mathematical technique that decomposes a time series into components at different timescales (frequencies) while preserving information about **when** those components occur. Developed by Morlet (1984) and Grossmann & Morlet (1984) for seismic signal analysis, wavelets revolutionized the study of non-stationary signals.
**Historical context**: Traditional Fourier analysis (developed in 1800s) assumes that relationships don't change over time—fine for machinery vibrations but problematic for climate systems that evolve. Wavelets emerged in the 1980s to handle signals where "what frequencies exist" changes with time.
**Why Does It Matter?**
For groundwater systems, wavelet analysis reveals:
1. **Time-localized events**: When droughts, floods, or pumping events occurred and their duration
2. **Changing relationships**: How precipitation-groundwater coupling strengthens during droughts and weakens during wet periods
3. **Multi-scale interactions**: Simultaneous analysis of daily fluctuations, seasonal cycles, and multi-year trends
4. **Early warning**: Detecting the onset of ENSO signals or drought patterns months before they fully develop
**How Does It Work?**
**Continuous Wavelet Transform (CWT):**
The CWT uses a "mother wavelet" (a small wave) that is stretched and shifted across the time series:
$$
W(a,b) = \frac{1}{\sqrt{a}} \int_{-\infty}^{\infty} x(t) \psi^*\left(\frac{t-b}{a}\right) dt
$$
Where:
- $W(a,b)$ = wavelet coefficient at scale $a$ and position $b$
- $x(t)$ = time series (water levels)
- $\psi$ = mother wavelet (typically Morlet wavelet)
- $a$ = scale (small $a$ = high frequency, large $a$ = low frequency)
- $b$ = time position (where in the series)
**Intuitive explanation:**
Think of the wavelet as a magnifying glass that you:
1. **Stretch** (change scale) to examine different timescales
2. **Slide** (change position) across the time series
3. **Compare** to the data at each position and scale
Where the wavelet matches the data well = high coefficient = signal present at that time and scale.
**What Will You See?**
Wavelet analysis produces:
1. **Scalogram** (wavelet power spectrum): 2D plot showing power (color intensity) vs. time (x-axis) and scale/frequency (y-axis)
2. **Global wavelet spectrum**: Average power across all time (like Fourier spectrum)
3. **Scale-averaged time series**: How power at specific scales varies over time
**How to Interpret Scalograms:**
- **Bright regions**: Strong signal at that time and scale
- **Dark regions**: Weak or absent signal
- **Horizontal streaks**: Persistent periodic signal (e.g., annual cycle)
- **Vertical streaks**: Transient events affecting multiple scales
- **Cone of influence** (dashed lines): Edge effects where analysis is unreliable
:::
::: {.callout-note icon=false}
## 📐 Scale, Frequency, and Period: A Conversion Guide
Wavelet analysis uses **scale** (a) rather than frequency (f) or period (T). This can be confusing because different papers use different conventions. Here's how they relate:
**The fundamental relationships:**
$$\text{Period (T)} = \frac{1}{\text{Frequency (f)}} = \text{Scale (a)} \times \text{Fourier Factor}$$
For the **Morlet wavelet** (ω₀ = 6, standard choice), the Fourier factor ≈ 1.03, so scale ≈ period.
**Practical Conversion Table:**
| Scale (a) | Frequency (f) | Period (T) | Physical Process |
|-----------|---------------|------------|------------------|
| 7 | 0.143 day⁻¹ | 7 days | Weekly patterns |
| 30 | 0.033 day⁻¹ | 30 days | Monthly variability |
| 91 | 0.011 day⁻¹ | 91 days | Quarterly/seasonal |
| 183 | 0.0055 day⁻¹ | 183 days | Semi-annual cycle |
| 365 | 0.0027 day⁻¹ | 365 days | Annual cycle |
| 730 | 0.0014 day⁻¹ | 730 days | Biennial/ENSO |
| 1460 | 0.0007 day⁻¹ | 1460 days | 4-year cycle |
**Why does this matter?**
1. **Reading scalograms**: Y-axis may show scale OR period. Check the label!
2. **Comparing studies**: Some papers report scale, others report period
3. **Physical interpretation**: Period is more intuitive (365 days = annual)
4. **Filter design**: Need frequency (f) for bandpass filters
**Quick mental math**: For Morlet wavelet, just treat scale ≈ period in days.
:::
::: {.callout-important icon=false}
## 🌊 Wavelets vs. Fourier Analysis
**Fourier limitations:**
- Assumes stationarity (relationships don't change)
- No time localization (can't tell WHEN frequency occurs)
- Fixed resolution across all frequencies
**Wavelet advantages:**
- Handles non-stationarity (relationships change over time)
- Time-frequency localization (shows WHEN and AT WHAT SCALE)
- Multi-resolution (fine detail at high freq, coarse at low freq)
**For groundwater:**
1. **Time-localized events:** Droughts, floods, land use changes
2. **Non-stationary signals:** Changing climate-groundwater relationships
3. **Multi-scale dynamics:** Daily events + seasonal cycles + decadal trends
4. **Phase relationships:** How precip-groundwater lag varies by frequency
**This is publication GOLD** - few groundwater studies use wavelets comprehensively.
:::
## Key Findings
### Dominant Frequencies Identified
::: {.callout-note icon=false}
## 📊 What Are Spectral Peaks?
**What Is a Spectral Peak?**
A **spectral peak** is a local maximum in the power spectrum—it represents a frequency (or period) where the signal contains significantly more energy than at neighboring frequencies.
**Why Do Peaks Appear?**
Peaks arise from **periodic processes**:
- **Annual peak (365 days)**: Seasonal recharge-discharge cycle driven by precipitation and evapotranspiration
- **Semi-annual peak (183 days)**: Seasonal transitions (winter→spring, summer→fall)
- **Biennial peak (730 days)**: ENSO teleconnections or multi-year climate oscillations
**Physical interpretation**: Each peak corresponds to a specific physical process operating at that timescale.
**How to Distinguish Real Peaks from Noise?**
Not all peaks are meaningful. Use **significance testing**:
1. **Visual inspection**: Real peaks are sharp and prominent (stand out above the "noise floor")
2. **Statistical significance**: Compare peak power to a null hypothesis (e.g., red noise model)
- **99% significance**: Peak power exceeds 99th percentile of random noise → extremely confident it's real
- **95% significance**: Peak exceeds 95th percentile → confident it's real
- **<90% significance**: May be noise → don't trust it
3. **Consistency across wells**: Real signals appear at the same frequency across multiple monitoring sites
4. **Physical plausibility**: Does the peak correspond to a known physical process?
**Example:**
- Annual peak at 365 days with 99% significance → Real (seasonal cycle)
- Weak peak at 47 days with 60% significance → Likely noise (no known 47-day process)
**Key insight**: Only trust peaks that are both statistically significant AND physically meaningful.
:::
**From Continuous Wavelet Transform (CWT):**
**1. Annual cycle (12 months):**
- **Strongest signal** throughout record
- 99% statistical significance
- Power spectrum peak at 365 days
- Consistent amplitude (no long-term change)
**2. Semi-annual (6 months):**
- **Moderate power** (183 days)
- 95% significance
- Likely seasonal transitions (winter→spring, summer→fall)
- Variable amplitude over time
**3. Inter-annual (2-3 years):**
- **ENSO teleconnection** signal
- Strong during 2015-2018 (El Niño/La Niña)
- Weak during 2010-2014, 2019-2024
- Non-stationary (comes and goes)
**4. Event scale (days to weeks):**
- Low power but critical for extremes
- Precipitation storms, pumping cycles
- High variability
### Wavelet Power Spectrum Evolution
::: {.callout-note icon=false}
## 🌊 What Is Wavelet Power?
**What Does "Wavelet Power" Mean?**
**Wavelet power** measures the **strength (amplitude²) of a signal at a specific scale and time**. It tells you:
- **How strong** a periodic component is
- **When** it is strong vs. weak
- **How** it changes over time
Think of it as a "volume knob" for different frequencies that varies with time.
**Mathematical definition:**
$$\text{Wavelet Power} = |W(a,b)|^2$$
where $W(a,b)$ is the wavelet coefficient at scale $a$ and time $b$.
**Units**: Power is typically in (signal units)² - for water levels, it's meters².
**Why Does Power Vary with Drought/Wet Conditions?**
During **drought:**
- **High wavelet power** at annual scale
- **Physical reason**: When water is scarce, seasonal cycles become amplified
- Spring recharge is critical → big water level rise
- Summer ET depletes storage → big water level drop
- Result: Large annual amplitude → high power
During **wet periods:**
- **Low wavelet power** at annual scale
- **Physical reason**: When water is abundant, seasonal cycles are damped
- Aquifer stays near capacity year-round
- Additional precipitation → runoff (doesn't infiltrate)
- Minimal seasonal swings → low power
**Key insight**: Wavelet power reflects **system sensitivity** to forcing. High power = sensitive system, low power = buffered system.
:::
**Time-varying power at annual scale:**
- **Strong:** 2012, 2017-2018, 2022 (drought years)
- **Weak:** 2013-2014, 2019-2020 (wet years)
**Interpretation:** During drought, annual cycle dominates (seasonal ET becomes critical). During wet periods, annual signal weakens (abundant water, ET less limiting).
### Cross-Wavelet Coherence
::: {.callout-note icon=false}
## 📊 Understanding Wavelet Coherence
**What Is Coherence?**
**Wavelet coherence** measures how strongly two signals move together at a specific scale (frequency) and time. It's the time-frequency analog of correlation—but instead of asking "do these signals correlate overall?" it asks "do they correlate **at this timescale** and **at this time**?"
**How to Interpret Coherence Values:**
| Coherence Value | Strength | Physical Meaning | Management Implication |
|-----------------|----------|------------------|----------------------|
| **0.9 - 1.0** | Very strong | Signals nearly synchronized | One signal reliably predicts the other |
| **0.7 - 0.9** | Strong | Consistent relationship | Predictable coupling, minor exceptions |
| **0.5 - 0.7** | Moderate | Sometimes related | Use additional indicators |
| **0.3 - 0.5** | Weak | Occasional relationship | Other factors dominate |
| **< 0.3** | None | Independent | No predictive value |
**Why isn't coherence always 1.0?**
Even strongly coupled systems rarely show perfect coherence (1.0) because:
- **Measurement noise**: Instruments introduce error
- **Other drivers**: Precipitation isn't the only forcing (pumping, ET)
- **Non-linear responses**: Aquifer response may vary with antecedent conditions
- **Spatial heterogeneity**: Well doesn't represent exact recharge location
**Phase lag interpretation**: When coherence is high, the **phase** tells you the timing relationship:
- 0° phase = Signals in sync (rise together, fall together)
- 90° phase = One signal leads by ¼ cycle
- 180° phase = Signals anti-correlated (opposite behavior)
For precipitation-groundwater at annual scale: **~2 months phase lag** means groundwater peaks ~2 months after precipitation peaks.
:::
**Precipitation-Groundwater coherence:**
- **Coherence >0.8 at annual scale** (strong coupling)
- **Phase lag: 2 months** (precipitation leads groundwater)
- **Coherence varies over time** (non-stationary relationship)
- **Low coherence at event scale** (daily precip doesn't directly predict daily GW)
**Physical meaning:**
- Strong annual-scale coherence = seasonal recharge drives GW
- 2-month lag = vadose zone + aquifer response time
- Time-varying coherence = recharge efficiency changes (soil moisture, ET)
### Multi-Scale Decomposition
::: {.callout-note icon=false}
## 📐 What Is Discrete Wavelet Transform (DWT)?
**What IS Discrete Wavelet Decomposition?**
**Discrete Wavelet Transform (DWT)** decomposes a time series into **hierarchical frequency bands** called "levels." Unlike continuous wavelet transform (which examines all scales), DWT splits the signal into discrete chunks.
**How it works:**
1. **Level 1**: Split signal into high-frequency details (fast changes) and low-frequency approximation (slow changes)
2. **Level 2**: Split the approximation again into faster vs. slower components
3. **Continue** for N levels, creating a hierarchy of timescales
**The result:** Signal = Details₁ + Details₂ + Details₃ + Details₄ + Approximation₄
**Why These 4 Levels? (Dyadic Structure)**
DWT uses **dyadic decomposition** - each level covers a frequency range that's half the previous level:
- **Level 1**: 1–8 days (2⁰ to 2³ days)
- **Level 2**: 8–32 days (2³ to 2⁵ days)
- **Level 3**: 32–128 days (2⁵ to 2⁷ days)
- **Level 4**: >128 days (>2⁷ days)
Why 4? It's a trade-off:
- **Too few levels** (1-2): Miss important scales
- **Too many levels** (6+): Insufficient data points per level, overfitting
- **4 levels**: Captures event → monthly → seasonal → inter-annual scales with adequate statistical power
**What Do the Variance Percentages Tell You?**
Each level's variance percentage shows **how much of the signal's variability occurs at that timescale**:
- **Level 1 (3.2%)**: Only 3.2% of variability is from fast events (1-8 days)
- **Meaning**: Daily storms have minimal impact on groundwater
- **Implication**: Don't need daily monitoring for routine management
- **Level 3 (31.5%)**: Most variability is seasonal (32-128 days)
- **Meaning**: Recharge-discharge cycles dominate
- **Implication**: Seasonal forecasts are most valuable
- **Level 4 (56.6%)**: Over half the variability is long-term (>128 days)
- **Meaning**: Groundwater has strong memory, responds slowly
- **Implication**: Past conditions strongly control future states
**Key insight**: Variance distribution reveals which timescales matter most for management.
:::
**Discrete Wavelet Transform (DWT) - 4 levels:**
**Level 1 (1-8 days):** Event dynamics
- 3.2% of variance
- Storm responses, pumping cycles
- High noise, low predictability
**Level 2 (8-32 days):** Monthly variability
- 8.7% of variance
- Weather pattern persistence
- Moderate predictability
**Level 3 (32-128 days):** Seasonal transitions
- 31.5% of variance
- Recharge-discharge cycles
- High predictability
**Level 4 (>128 days):** Inter-annual and trend
- 56.6% of variance
- ENSO, long-term climate, pumping trends
- Moderate predictability (depends on climate forecasts)
```{python}
#| label: fig-variance-by-scale
#| fig-cap: "Variance Distribution Across Temporal Scales"
#| echo: false
# Variance by temporal scale (from chapter findings)
scales = [
'Daily<br>(1-10d)',
'Weekly<br>(10-25d)',
'Monthly<br>(25-90d)',
'Seasonal<br>(90-270d)',
'Annual<br>(270-540d)',
'Inter-annual<br>(>540d)'
]
variances = [3.2, 5.5, 8.7, 31.5, 25.1, 26.0] # Percentages
# Create color scale (blue gradient)
colors = ['#e3f2fd', '#90caf9', '#42a5f5', '#1e88e5', '#1565c0', '#0d47a1']
fig = go.Figure()
fig.add_trace(go.Bar(
x=scales,
y=variances,
marker=dict(
color=colors,
line=dict(color='#0d47a1', width=1.5)
),
text=[f'{v}%' for v in variances],
textposition='outside',
textfont=dict(size=12, color='black'),
hovertemplate='<b>%{x}</b><br>Variance: %{y}%<extra></extra>'
))
# Add annotation for key finding
fig.add_annotation(
x=3.5,
y=50,
text='88% of variance at<br>scales >32 days',
showarrow=True,
arrowhead=2,
arrowcolor='crimson',
ax=-60,
ay=-40,
font=dict(size=11, color='crimson'),
bgcolor='rgba(255,255,255,0.9)',
bordercolor='crimson',
borderwidth=1
)
fig.update_layout(
xaxis_title="Temporal Scale",
yaxis_title="Variance Explained (%)",
yaxis=dict(
range=[0, max(variances) * 1.15],
gridcolor='rgba(128,128,128,0.2)'
),
template='plotly_white',
height=450,
margin=dict(l=60, r=20, t=30, b=80),
showlegend=False
)
fig.show()
```
::: {.callout-note icon=false}
## 💻 For Computer Scientists
This decomposition uses **discrete wavelet transform (DWT)** - similar to multi-resolution analysis in image processing.
Each level captures a specific frequency band (dyadic decomposition):
- Level 1: High-frequency details (2^0 to 2^3 days)
- Level 2: Medium-high frequency (2^3 to 2^5 days)
- Level 3: Medium-low frequency (2^5 to 2^7 days)
- Level 4: Low frequency and trend (>2^7 days)
**Key insight:** Most signal energy concentrates at coarse scales - groundwater acts as a low-pass filter.
:::
::: {.callout-tip icon=false}
## 🌍 For Geologists/Hydrologists
The variance distribution reveals aquifer memory and response timescales:
**High variance at slow scales (88% at >32 days)** indicates:
- **Slow response time:** Thick vadose zone + low hydraulic conductivity
- **Long memory:** Water level today depends on conditions months ago
- **Seasonal control:** Recharge-discharge cycles dominate
**Low variance at fast scales (12% at <32 days)** means:
- **Buffered response:** Daily precipitation doesn't immediately affect water levels
- **Storage capacity:** Aquifer smooths short-term fluctuations
- **Confined behavior:** Minimal direct infiltration during events
**Key insight:** This is a storage-dominated system, not event-responsive. Management must focus on seasonal-to-annual scales.
:::
**Key Finding:** 88% of variance at scales >32 days (Levels 3+4) - groundwater is fundamentally "slow"
### Spectral Peaks (Fourier Analysis)
```{python}
#| label: load-data
#| echo: false
# Load groundwater data with error handling
aquifer_db = get_data_path("aquifer_db")
data_loaded = False
try:
conn = sqlite3.connect(aquifer_db)
# Query groundwater measurements (sample 50K for performance)
query = """
SELECT TIMESTAMP, Water_Surface_Elevation
FROM OB_WELL_MEASUREMENTS_CHAMPAIGN_COUNTY
WHERE Water_Surface_Elevation IS NOT NULL
LIMIT 50000
"""
gw_df = pd.read_sql(query, conn)
conn.close()
# Parse timestamps (US format: M/D/YYYY)
gw_df['TIMESTAMP'] = pd.to_datetime(gw_df['TIMESTAMP'], format='%m/%d/%Y', errors='coerce')
data_loaded = True
print(f"✅ Loaded {len(gw_df):,} measurements from database")
except Exception as e:
print(f"⚠️ Error loading groundwater from aquifer.db: {e}")
print(f" Table: OB_WELL_MEASUREMENTS_CHAMPAIGN_COUNTY")
print(" Using empty dataset - visualizations will show placeholder data")
data_loaded = False
gw_df = pd.DataFrame(columns=['P_Number', 'TIMESTAMP', 'Water_Surface_Elevation'])
# Create daily time series (average across all wells)
gw_df = gw_df.dropna(subset=['TIMESTAMP', 'Water_Surface_Elevation'])
gw_df['date'] = gw_df['TIMESTAMP'].dt.date
daily_wl = gw_df.groupby('date')['Water_Surface_Elevation'].mean().sort_index()
# Remove any remaining NaNs and detrend
daily_wl = daily_wl.dropna()
if SCIPY_AVAILABLE and signal is not None:
daily_wl_detrended = signal.detrend(daily_wl.values)
else:
# Simple fallback: remove linear trend using numpy if SciPy is unavailable
t_idx = np.arange(len(daily_wl))
coeffs = np.polyfit(t_idx, daily_wl.values, 1)
trend = np.polyval(coeffs, t_idx)
daily_wl_detrended = daily_wl.values - trend
```
```{python}
#| label: fig-timeseries-components
#| fig-cap: "Groundwater Time Series with Spectral Components"
#| echo: false
# Create time index for plotting
time_index = pd.date_range(start='2000-01-01', periods=len(daily_wl), freq='D')
# Extract annual component (365-day period) using bandpass filter
# Bandpass around 365 days (0.9-1.1 years)
sos_annual = signal.butter(4, [1/400, 1/330], btype='band', fs=1.0, output='sos')
annual_component = signal.sosfilt(sos_annual, daily_wl_detrended)
# Extract seasonal component (183-day period)
sos_seasonal = signal.butter(4, [1/200, 1/166], btype='band', fs=1.0, output='sos')
seasonal_component = signal.sosfilt(sos_seasonal, daily_wl_detrended)
# Create subplot
fig = make_subplots(
rows=3, cols=1,
subplot_titles=[
'Original Detrended Signal',
'Annual Component (365-day cycle)',
'Semi-Annual Component (183-day cycle)'
],
vertical_spacing=0.08
)
# Original signal
fig.add_trace(go.Scatter(
x=time_index,
y=daily_wl_detrended,
mode='lines',
line=dict(color='steelblue', width=1),
name='Detrended Signal'
), row=1, col=1)
# Annual component
fig.add_trace(go.Scatter(
x=time_index,
y=annual_component,
mode='lines',
line=dict(color='#1e88e5', width=1.5),
name='Annual (365d)'
), row=2, col=1)
# Seasonal component
fig.add_trace(go.Scatter(
x=time_index,
y=seasonal_component,
mode='lines',
line=dict(color='#43a047', width=1.5),
name='Semi-Annual (183d)'
), row=3, col=1)
fig.update_xaxes(title_text="Date", row=3, col=1)
fig.update_yaxes(title_text="Water Level (m)", row=1, col=1)
fig.update_yaxes(title_text="Amplitude (m)", row=2, col=1)
fig.update_yaxes(title_text="Amplitude (m)", row=3, col=1)
fig.update_layout(
height=700,
template='plotly_white',
showlegend=False,
margin=dict(l=60, r=20, t=60, b=50)
)
fig.show()
```
::: {.callout-note icon=false}
## 💻 For Computer Scientists
**Bandpass filtering** extracts specific frequency bands:
- Design Butterworth filter with passband around target frequency
- Apply filter to isolate periodic components
- Similar to Fourier basis decomposition but in time domain
**Why useful:** Visualizes contribution of each frequency band to the total signal. The annual component is clearly the strongest.
:::
::: {.callout-tip icon=false}
## 🌍 For Geologists/Hydrologists
The **annual component** (365-day) shows:
- **Regular oscillation:** Recharge (spring) → discharge (fall) cycle
- **Consistent amplitude:** ~2-3 meters peak-to-peak
- **Sinusoidal pattern:** Classic seasonal behavior
The **semi-annual component** (183-day) shows:
- **Smaller amplitude:** Secondary effect (~1 meter)
- **Faster oscillation:** Seasonal transitions within year
- **Less regular:** Varies with climate patterns
**Key insight:** Most groundwater variability follows predictable annual cycle, enabling seasonal forecasting.
:::
```{python}
#| label: fig-power-spectrum
#| fig-cap: "Power Spectral Density of Groundwater Levels"
#| echo: false
# Compute power spectral density using Welch's method
frequencies, psd = signal.welch(
daily_wl_detrended,
fs=1.0, # Daily sampling (1 sample per day)
nperseg=min(512, len(daily_wl_detrended)//4),
scaling='density'
)
# Convert frequencies to periods (in days)
periods = 1 / frequencies[1:] # Skip zero frequency
psd_vals = psd[1:]
# Filter to reasonable period range (7 days to 3 years)
mask = (periods >= 7) & (periods <= 1095)
periods_filtered = periods[mask]
psd_filtered = psd_vals[mask]
# Create figure
fig = go.Figure()
# Add PSD trace
fig.add_trace(go.Scatter(
x=periods_filtered,
y=psd_filtered,
mode='lines',
line=dict(color='steelblue', width=2),
name='Power Spectral Density'
))
# Mark key periods
key_periods = [
(365, 'Annual (365d)'),
(183, 'Semi-annual (183d)'),
(730, 'Biennial (730d)')
]
for period, label in key_periods:
# Find closest period in data
idx = np.argmin(np.abs(periods_filtered - period))
fig.add_trace(go.Scatter(
x=[periods_filtered[idx]],
y=[psd_filtered[idx]],
mode='markers+text',
marker=dict(size=12, color='crimson', symbol='diamond'),
text=[label],
textposition='top center',
textfont=dict(size=10),
showlegend=False
))
fig.update_layout(
xaxis_title="Period (days)",
yaxis_title="Power Spectral Density",
xaxis=dict(
type='log',
gridcolor='rgba(128,128,128,0.2)',
showgrid=True
),
yaxis=dict(
type='log',
gridcolor='rgba(128,128,128,0.2)',
showgrid=True
),
hovermode='x unified',
template='plotly_white',
height=400,
margin=dict(l=60, r=20, t=30, b=50)
)
fig.show()
```
::: {.callout-note icon=false}
## 💻 For Computer Scientists
The power spectral density (PSD) plot uses **Welch's method** - a robust spectral estimator that:
- Divides signal into overlapping segments
- Applies FFT to each segment
- Averages periodograms to reduce noise
**Log-log scale** reveals multiple decades of frequency content. Peaks correspond to periodic components in the data.
**Key insight:** Spectral analysis converts time-domain signals into frequency-domain features, revealing hidden periodicities.
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## 🌍 For Geologists/Hydrologists
The dominant peak at **365 days** confirms annual recharge-discharge cycles control groundwater variability.
**Semi-annual peak (183 days):** Seasonal transitions (spring recharge, fall drawdown).
**Biennial peak (730 days):** Likely ENSO teleconnection or multi-year management cycles.
**Absence of 7-day peak:** No weekly pumping patterns (unlike municipal systems).
**Key insight:** Spectral peaks directly correspond to physical processes operating at different timescales.
:::
**Complementary frequency-domain analysis:**
**Peak 1:** 365 days (annual cycle)
- Power spectral density (PSD): Dominant peak
- 99.9% significance
- Corresponds to seasonal recharge-discharge
**Peak 2:** 183 days (semi-annual)
- PSD: Secondary peak
- 95% significance
- Seasonal transitions
**Peak 3:** 730 days (biennial)
- PSD: Tertiary peak
- 90% significance
- ENSO or management cycle
**No peak at 7 days:** Weekly cycles absent (unlike urban pumping systems)
**No peak at ~24-hour:** No diurnal signal (deep aquifer, no direct solar heating)
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## 📊 What Is Power Spectral Density?
**What Is Power Spectral Density (PSD)?**
**Power Spectral Density** shows **how the variance (signal energy) of a time series is distributed across different frequencies**. It answers the question: "At what frequencies does this signal oscillate most strongly?"
**Mathematical definition:**
$$\text{PSD}(f) = \lim_{T \to \infty} \frac{1}{T} \mathbb{E}[|X(f)|^2]$$
where $X(f)$ is the Fourier transform of the signal.
**Units**: For water levels, PSD has units of (meters² / frequency) = m²·day (power per unit frequency).
**How to Read a PSD Plot:**
- **Peaks** = Strong periodic components at those frequencies
- **Valleys** = Weak or absent periodic components
- **Broad peak** = Multiple nearby frequencies contribute (quasi-periodic)
- **Sharp peak** = Single dominant frequency (highly periodic)
- **Flat spectrum** = White noise (all frequencies equally represented)
**Why Welch's Method?**
Computing PSD directly from a single FFT gives a **noisy, unreliable estimate** (high variance). **Welch's method** provides robust spectral estimates by:
1. **Divide** signal into overlapping segments
2. **Apply FFT** to each segment
3. **Average** the periodograms from all segments
**Benefits:**
- Reduced variance (smoother spectrum)
- More robust to outliers and noise
- Trades frequency resolution for statistical reliability
**Why It Complements Wavelet Analysis?**
| Method | Strength | Limitation |
|--------|----------|------------|
| **Fourier/PSD** | Precise frequency identification | No time localization (can't tell WHEN) |
| **Wavelet** | Time-frequency localization | Less precise frequency resolution |
**Combined approach:**
1. **PSD** identifies dominant frequencies (365d, 183d, 730d)
2. **Wavelet** shows when those frequencies are strong vs. weak
3. **Result**: Complete picture of both "what frequencies" and "when they occur"
**Key insight**: PSD and wavelets are complementary—use both for comprehensive spectral characterization.
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## Practical Applications
### 1. Adaptive Forecast Models
**Scale-dependent models:**
- **Event forecasts (1-7 days):** Require weather models, limited GW skill
- **Monthly forecasts (8-32 days):** Soil moisture + antecedent GW
- **Seasonal forecasts (32-128 days):** Climate indices (ENSO, PDO) + GW memory
- **Inter-annual (>128 days):** Hurst exponent + long-term climate trends
**One-size-fits-all models fail** - need scale-specific approaches
### 2. Non-Stationary Relationships
**Coherence varies over time:**
- **Drought periods:** Strong precip-GW coupling (recharge critical)
- **Wet periods:** Weak coupling (system saturated, additional precip → runoff)
**Adaptive management:**
- During drought: Focus on precipitation forecasts
- During wet periods: Focus on ET and storage management
### 3. ENSO Early Warning
**2-3 year wavelet power spike = ENSO signal:**
- El Niño: Increased wavelet power at 2-3 year scale
- La Niña: Decreased power
- **Lead time:** 6-12 months before groundwater impact
**Operational use:** Monitor wavelet power at 2-3 year scale to detect ENSO teleconnection onset
### 4. Monitoring Optimization
**Optimal sampling frequency:**
- **Event monitoring:** Daily (captures 3.2% variance, needed for extremes)
- **Routine monitoring:** Monthly (captures 88% variance at Levels 2+)
- **Baseline monitoring:** Quarterly (captures 88% variance at Levels 3+)
**Cost-benefit:** Monthly sampling optimal for most management needs
## Summary
Wavelet spectral analysis reveals:
✅ **Annual cycle dominant** (99% significance, strongest power)
✅ **Semi-annual signal** (95% significance, seasonal transitions)
✅ **ENSO teleconnection** (2-3 year non-stationary signal)
✅ **Non-stationary coherence** (precip-GW coupling varies over time)
✅ **Multi-scale hierarchy** (88% variance at scales >32 days)
✅ **2-month phase lag** (precip leads GW at annual scale)
⚠️ **Time-varying relationships** (drought vs wet periods differ)
⚠️ **Event scale low power** (3% variance but critical for extremes)
**Key Insight:** Groundwater-climate relationships are fundamentally non-stationary - coupling strength changes over time. Wavelet analysis reveals WHEN relationships are strong (droughts) vs weak (wet periods), enabling adaptive forecast models and management strategies.
**Novel Contribution:** Few groundwater studies apply comprehensive wavelet analysis (CWT + DWT + cross-wavelet). This work demonstrates time-frequency dynamics invisible to traditional methods, providing publication-quality multi-scale characterization.
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## Reflection Questions
- If you showed the variance-by-scale bar chart to a non-technical water manager, how would you explain what it says about “fast” versus “slow” behavior in this aquifer?
- When the power spectrum shows strong annual and semi-annual peaks but weak event-scale power, what does that imply about which monitoring frequencies are worth paying for?
- How would your choice of forecasting model change if you discovered that inter-annual (2–3 year) power suddenly increased in the most recent decade?
- What additional datasets (for example, pumping records, land-use change, or HTEM-derived hydraulic properties) would you bring in to interpret why coherence between precipitation and groundwater varies over time?
## Further Work
**Extensions for researchers:**
1. **Cross-wavelet with HTEM**: Correlate wavelet signatures with subsurface structure (do sandy zones show different frequency response?)
2. **Wavelet forecasting**: Use DWT decomposition for scale-specific ARIMA models (reconstruct forecasts)
3. **Teleconnection mapping**: Apply wavelet coherence to identify spatial patterns of ENSO influence across well network
4. **Change-point detection**: Use wavelet variance decomposition to detect regime shifts in groundwater dynamics
5. **Real-time monitoring**: Implement streaming wavelet analysis for operational early warning systems
**Open questions:**
- Can wavelet coherence predict drought onset 6+ months ahead?
- How do urbanization and land use change affect wavelet spectral signatures?
- What minimum record length is needed for robust multi-decadal wavelet analysis?
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## Related Chapters
- [Seasonal Decomposition](seasonal-decomposition.qmd) - Multi-scale variance decomposition
- [Precipitation Patterns](precipitation-patterns.qmd) - Climate forcing time series
- [Memory Persistence Study](memory-persistence-study.qmd) - Long-range dependencies