Climate Extremes Analysis - Run Theory

Overview

Run theory provides a framework for analyzing climate extreme events based on continuous periods beyond a threshold. This package implements comprehensive event analysis for both drought (dry) and wet (flood/excess) conditions with three operational modes.

Based on: Yevjevich (1967) run theory Applies to: Both dry events (drought) and wet events (flooding/excess precipitation) Provides: Event identification, time-series monitoring, period statistics

📘 Important Note on Terminology

All functions are neutral and work for both dry and wet extremes. This guide uses “drought” in some examples for historical consistency with run theory literature, but the same functions and concepts apply to wet events by simply using positive thresholds instead of negative ones.

Threshold Direction:

Negative threshold (e.g., -1.2): Identifies dry events (drought)
Positive threshold (e.g., +1.2): Identifies wet events (flooding/excess)

The mathematics and event characteristics (duration, magnitude, intensity, peak) work identically for both extremes.

Run Theory Concept

Figure 1: Run Theory Applied to Climate Extremes

This illustration demonstrates run theory using dry events (drought) as an example, where events occur when the index falls below the threshold (x₀).

⚠️ Important: The same framework applies identically to wet events (flooding/excess) by using a positive threshold—wet events occur when the index rises above the threshold. The mathematics, algorithms, and characteristic calculations remain unchanged.

Key characteristics:

Duration (D): Length of continuous period beyond threshold (D₁, D₂, D₃)
Magnitude (M): Total accumulated deviation from threshold (M₁, M₂, M₃)
Intensity (I): Average severity per time unit = M/D
Inter-arrival Time (T): Time between consecutive events (T₁, T₂)

The “Wet” and “Dry” regions are labeled to show that both extremes can be analyzed using the same methodology.

Three Modes of Analysis

Mode 1: Event-Based Analysis

Purpose: Identify and analyze complete climate extreme events

from runtheory import identify_events

events = identify_events(spi_ts, threshold=-1.2, min_duration=3)
# Returns: DataFrame with one row per event

Use for: Historical analysis, event comparison, detailed investigation

Mode 2: Time-Series Monitoring

Purpose: Month-by-month tracking with varying characteristics

from runtheory import calculate_timeseries

ts = calculate_timeseries(spi_ts, threshold=-1.2)
# Returns: DataFrame with one row per time step

Use for: Real-time monitoring, event evolution tracking, operational systems

Mode 3: Period Statistics (Gridded)

Purpose: Answer “What happened in 2023?” or “2021-2025?”

from runtheory import calculate_period_statistics

stats = calculate_period_statistics(spi, threshold=-1.2,
                                    start_year=2020, end_year=2024)
# Returns: xarray Dataset with (lat, lon) dimensions

Use for: Decision-making, reporting, regional analysis, mapping

Quick Start

Drought (Dry) Events

import xarray as xr
from runtheory import identify_events, calculate_period_statistics
from visualization import plot_events

# Load SPI
spi = xr.open_dataset('output/netcdf/spi_12.nc')['spi_gamma_12_month']

# Drought events (negative threshold)
spi_location = spi.isel(lat=50, lon=100)
dry_events = identify_events(spi_location, threshold=-1.2)
print(f"Found {len(dry_events)} dry events")

# Gridded dry event statistics
dry_stats = calculate_period_statistics(spi, threshold=-1.2,
                                        start_year=2023, end_year=2023)
dry_stats.num_events.plot(title='Dry Events 2023')

Wet (Flood/Excess) Events

# Wet events (positive threshold) - SAME FUNCTIONS!
wet_events = identify_events(spi_location, threshold=+1.2)
print(f"Found {len(wet_events)} wet events")

# Gridded wet event statistics
wet_stats = calculate_period_statistics(spi, threshold=+1.2,
                                        start_year=2023, end_year=2023)
wet_stats.num_events.plot(title='Wet Events 2023')

💡 Key Point: Same functions, different thresholds:

Dry (drought): threshold < 0 (e.g., -1.2, -1.5, -2.0)
Wet (flood/excess): threshold > 0 (e.g., +1.2, +1.5, +2.0)

Event Characteristics

Characteristic	Formula	Meaning
Duration (D)	end - start + 1	Months in event
Magnitude (Cumulative)	Σ(threshold - SPI)	Total water deficit
Magnitude (Instantaneous)	threshold - SPI[t]	Current severity
Intensity (I)	Magnitude / Duration	Average severity
Peak (P)	min(SPI)	Worst moment
Inter-arrival (T)	start[n+1] - start[n]	Time between events

Magnitude: Two Types

Cumulative (blue in plots):

Total accumulated deficit
Always increases during event
Like debt accumulation
Use for: Total impact, event comparison

Instantaneous (red in plots):

Current month’s severity
Varies with SPI (rises-peaks-falls)
Like NDVI crop phenology
Use for: Monitoring evolution, peak detection

See Magnitude Explained for detailed comparison.

Mode 1: Event-Based Analysis

Basic Usage

from runtheory import identify_events

# Single location
spi_ts = spi.isel(lat=50, lon=100)
events = identify_events(spi_ts, threshold=-1.2, min_duration=3)

print(events)
# Columns: event_id, start_date, end_date, duration, magnitude,
#          intensity, peak, peak_date, interarrival

Event Summary Statistics

from runtheory import summarize_events

summary = summarize_events(events)
print(f"Total events: {summary['num_events']}")
print(f"Mean duration: {summary['mean_duration']:.1f} months")
print(f"Max magnitude: {summary['max_magnitude']:.2f}")
print(f"Most severe peak: {summary['most_severe_peak']:.2f}")

Visualization

from visualization import plot_events, plot_event_characteristics

# Timeline with events highlighted
plot_events(spi_ts, events, threshold=-1.2)

# Event characteristics analysis
plot_event_characteristics(events, characteristic='magnitude')

Mode 2: Time-Series Monitoring

Real-Time Tracking

from runtheory import calculate_timeseries

ts = calculate_timeseries(spi_ts, threshold=-1.2)

# Check current status
current = ts.tail(1)
if current['is_event'].values[0]:
    print(f"IN DRY EVENT")
    print(f"Duration: {current['duration'].values[0]} months")
    print(f"Cumulative magnitude: {current['magnitude_cumulative'].values[0]:.2f}")
    print(f"Current severity: {current['magnitude_instantaneous'].values[0]:.2f}")
else:
    print("No event detected")

Monitoring Evolution

# Is event worsening or easing?
recent = ts['magnitude_instantaneous'].tail(3)
if recent.is_monotonic_decreasing:
    print("EASING - severity decreasing")
elif recent.is_monotonic_increasing:
    print("WORSENING - severity increasing")

Visualization

from visualization import plot_event_timeline

# Shows both cumulative and instantaneous magnitude
plot_event_timeline(ts)

Mode 3: Period Statistics

Specific Time Period

from runtheory import calculate_period_statistics

# What happened in 2023?
stats_2023 = calculate_period_statistics(spi, threshold=-1.2,
                                         start_year=2023, end_year=2023)

# Map number of events
stats_2023.num_events.plot(cmap='YlOrRd')

# Map worst severity
stats_2023.worst_peak.plot(cmap='YlOrRd_r')

Year-by-Year Analysis

from runtheory import calculate_annual_statistics

# Calculate for each year
annual = calculate_annual_statistics(spi, threshold=-1.2)

# Access specific year
stats_2015 = annual.sel(year=2015)

# Trend over time
annual.num_events.mean(dim=['lat', 'lon']).plot()

Period Comparison

from runtheory import compare_periods

# Compare decades
comparison = compare_periods(
    spi,
    periods=[(1991, 2000), (2001, 2010), (2011, 2020), (2021, 2024)],
    period_names=['1990s', '2000s', '2010s', '2020s']
)

# Calculate change
diff = comparison.sel(period='2020s') - comparison.sel(period='1990s')
diff.num_events.plot(title='Change in Events: 2020s vs 1990s', cmap='RdBu_r')

Output Variables (Period Statistics)

All period statistics functions return xarray Dataset with:

Variable	Description	Units
`num_events`	Number of dry/wet events	count
`total_event_months`	Total months in events	months
`total_magnitude`	Sum of all event magnitudes	index units
`mean_magnitude`	Average magnitude per event	index units
`max_magnitude`	Largest single event	index units
`worst_peak`	Most severe SPI/SPEI value	index units
`mean_intensity`	Average intensity	index units/month
`max_intensity`	Maximum intensity	index units/month
`pct_time_in_event`	% of time in events	%

Decision-Maker Questions

“How many dry events in 2023?”

stats_2023 = calculate_period_statistics(spi, start_year=2023, end_year=2023)
stats_2023.num_events.plot(title='Dry Events in 2023')

“Where was the worst dry event in last 5 years?”

stats = calculate_period_statistics(spi, start_year=2020, end_year=2024)
stats.worst_peak.plot(title='Worst Severity 2020-2024', cmap='YlOrRd_r')

“How does recent compare to historical?”

comparison = compare_periods(spi,
    periods=[(1991, 2020), (2021, 2024)],
    period_names=['Historical', 'Recent'])

diff = comparison.sel(period='Recent') - comparison.sel(period='Historical')
diff.total_magnitude.plot(title='Change in Total Magnitude')

“Which year had the most events?”

annual = calculate_annual_statistics(spi)
worst_year = int(annual.year[annual.num_events.mean(dim=['lat','lon']).argmax()])
print(f"Worst year: {worst_year}")

Thresholds

Common dry event thresholds (use positive values for wet events):

Threshold	Category	Use
-0.5	Incipient dry	Early warning
-0.8	Mild dry	Agricultural concern
-1.0	Moderate dry	Standard threshold
-1.2	Moderate-severe	Conservative threshold
-1.5	Severe dry	Significant impacts
-2.0	Extreme dry	Crisis level

Recommendation: -1.0 or -1.2 for operational monitoring

Minimum Duration

Filter short-term fluctuations:

# Include all dry periods
events_all = identify_events(spi_ts, min_duration=1)

# Only sustained events
events_sustained = identify_events(spi_ts, min_duration=3)

# Long-term events
events_longterm = identify_events(spi_ts, min_duration=6)

Recommendation: min_duration=3 for SPI-12 (captures sustained events)

Performance

Function	Input Size	Time	Notes
`identify_events()`	Single location	<0.1 sec	Very fast
`calculate_timeseries()`	Single location	<0.1 sec	Very fast
`calculate_period_statistics()`	165×244 grid	~30 sec	Acceptable
`calculate_annual_statistics()`	165×244, 10 years	~5 min	Loops over years
`compare_periods()`	165×244, 2 periods	~1 min	Parallelizable

Optimization tips:

Use min_duration to filter events
Process regional subsets when possible
Save and reuse outputs

Complete Workflow Example

import xarray as xr
from runtheory import (identify_events,
                       calculate_period_statistics,
                       compare_periods)
from visualization import (plot_events,
                          plot_event_timeline)

# 1. Load SPI
spi = xr.open_dataset('output/netcdf/spi_12.nc')['spi_gamma_12_month']

# 2. Single location analysis
lat, lon = 50, 100
spi_loc = spi.isel(lat=lat, lon=lon)
events = identify_events(spi_loc, threshold=-1.2, min_duration=3)
ts = calculate_timeseries(spi_loc, threshold=-1.2)

# 3. Visualize
plot_events(spi_loc, events, threshold=-1.2)
plot_event_timeline(ts)

# 4. Regional statistics
stats_2020s = calculate_period_statistics(spi, threshold=-1.2,
                                          start_year=2020, end_year=2024)

# 5. Compare with historical
comparison = compare_periods(spi,
    periods=[(1991, 2020), (2021, 2024)],
    period_names=['Historical', 'Recent'])

# 6. Save outputs
events.to_csv(f'output/csv/dry_events_lat{lat}_lon{lon}.csv')
stats_2020s.to_netcdf('output/netcdf/dry_stats_2020-2024.nc')

# 7. Create maps
stats_2020s.num_events.plot()

References

Yevjevich, V. (1967). An objective approach to definitions and investigations of continental hydrologic droughts. Hydrology Papers 23, Colorado State University.
Sheffield, J., Wood, E.F., Roderick, M.L. (2012). Little change in global drought over the past 60 years. Nature, 491(7424), 435-438.

--- title: "Climate Extremes Analysis - Run Theory" --- ## Overview Run theory provides a framework for analyzing climate extreme events based on continuous periods beyond a threshold. This package implements comprehensive event analysis for **both drought (dry) and wet (flood/excess) conditions** with three operational modes. **Based on:** Yevjevich (1967) run theory **Applies to:** Both dry events (drought) and wet events (flooding/excess precipitation) **Provides:** Event identification, time-series monitoring, period statistics ## 📘 Important Note on Terminology > **All functions are neutral and work for both dry and wet extremes.** This guide uses "drought" in some examples for historical consistency with run theory literature, but the same functions and concepts apply to wet events by simply using positive thresholds instead of negative ones. **Threshold Direction:** - **Negative threshold** (e.g., -1.2): Identifies **dry events (drought)** - **Positive threshold** (e.g., +1.2): Identifies **wet events (flooding/excess)** The mathematics and event characteristics (duration, magnitude, intensity, peak) work identically for both extremes. ## Run Theory Concept ![Run Theory Illustration](../images/runtheory.svg) **Figure 1: Run Theory Applied to Climate Extremes** This illustration demonstrates run theory using **dry events (drought)** as an example, where events occur when the index falls **below** the threshold (x₀). **⚠️ Important:** The same framework applies identically to **wet events (flooding/excess)** by using a **positive** threshold—wet events occur when the index rises **above** the threshold. The mathematics, algorithms, and characteristic calculations remain unchanged. Key characteristics: - **Duration (D)**: Length of continuous period beyond threshold (D₁, D₂, D₃) - **Magnitude (M)**: Total accumulated deviation from threshold (M₁, M₂, M₃) - **Intensity (I)**: Average severity per time unit = M/D - **Inter-arrival Time (T)**: Time between consecutive events (T₁, T₂) The "Wet" and "Dry" regions are labeled to show that both extremes can be analyzed using the same methodology. --- ## Three Modes of Analysis ### Mode 1: Event-Based Analysis **Purpose:** Identify and analyze complete climate extreme events ```python from runtheory import identify_events events = identify_events(spi_ts, threshold=-1.2, min_duration=3) # Returns: DataFrame with one row per event ``` **Use for:** Historical analysis, event comparison, detailed investigation ### Mode 2: Time-Series Monitoring **Purpose:** Month-by-month tracking with varying characteristics ```python from runtheory import calculate_timeseries ts = calculate_timeseries(spi_ts, threshold=-1.2) # Returns: DataFrame with one row per time step ``` **Use for:** Real-time monitoring, event evolution tracking, operational systems ### Mode 3: Period Statistics (Gridded) **Purpose:** Answer "What happened in 2023?" or "2021-2025?" ```python from runtheory import calculate_period_statistics stats = calculate_period_statistics(spi, threshold=-1.2, start_year=2020, end_year=2024) # Returns: xarray Dataset with (lat, lon) dimensions ``` **Use for:** Decision-making, reporting, regional analysis, mapping ## Quick Start ### Drought (Dry) Events ```python import xarray as xr from runtheory import identify_events, calculate_period_statistics from visualization import plot_events # Load SPI spi = xr.open_dataset('output/netcdf/spi_12.nc')['spi_gamma_12_month'] # Drought events (negative threshold) spi_location = spi.isel(lat=50, lon=100) dry_events = identify_events(spi_location, threshold=-1.2) print(f"Found {len(dry_events)} dry events") # Gridded dry event statistics dry_stats = calculate_period_statistics(spi, threshold=-1.2, start_year=2023, end_year=2023) dry_stats.num_events.plot(title='Dry Events 2023') ``` ### Wet (Flood/Excess) Events ```python # Wet events (positive threshold) - SAME FUNCTIONS! wet_events = identify_events(spi_location, threshold=+1.2) print(f"Found {len(wet_events)} wet events") # Gridded wet event statistics wet_stats = calculate_period_statistics(spi, threshold=+1.2, start_year=2023, end_year=2023) wet_stats.num_events.plot(title='Wet Events 2023') ``` **💡 Key Point:** Same functions, different thresholds: - **Dry (drought):** `threshold < 0` (e.g., -1.2, -1.5, -2.0) - **Wet (flood/excess):** `threshold > 0` (e.g., +1.2, +1.5, +2.0) ## Event Characteristics | Characteristic | Formula | Meaning | |----------------|---------|---------| | **Duration (D)** | end - start + 1 | Months in event | | **Magnitude (Cumulative)** | Σ(threshold - SPI) | Total water deficit | | **Magnitude (Instantaneous)** | threshold - SPI[t] | Current severity | | **Intensity (I)** | Magnitude / Duration | Average severity | | **Peak (P)** | min(SPI) | Worst moment | | **Inter-arrival (T)** | start[n+1] - start[n] | Time between events | ### Magnitude: Two Types **Cumulative** (blue in plots): - Total accumulated deficit - Always increases during event - Like debt accumulation - Use for: Total impact, event comparison **Instantaneous** (red in plots): - Current month's severity - Varies with SPI (rises-peaks-falls) - Like NDVI crop phenology - Use for: Monitoring evolution, peak detection See [Magnitude Explained](magnitude.qmd) for detailed comparison. ## Mode 1: Event-Based Analysis ### Basic Usage ```python from runtheory import identify_events # Single location spi_ts = spi.isel(lat=50, lon=100) events = identify_events(spi_ts, threshold=-1.2, min_duration=3) print(events) # Columns: event_id, start_date, end_date, duration, magnitude, # intensity, peak, peak_date, interarrival ``` ### Event Summary Statistics ```python from runtheory import summarize_events summary = summarize_events(events) print(f"Total events: {summary['num_events']}") print(f"Mean duration: {summary['mean_duration']:.1f} months") print(f"Max magnitude: {summary['max_magnitude']:.2f}") print(f"Most severe peak: {summary['most_severe_peak']:.2f}") ``` ### Visualization ```python from visualization import plot_events, plot_event_characteristics # Timeline with events highlighted plot_events(spi_ts, events, threshold=-1.2) # Event characteristics analysis plot_event_characteristics(events, characteristic='magnitude') ``` ## Mode 2: Time-Series Monitoring ### Real-Time Tracking ```python from runtheory import calculate_timeseries ts = calculate_timeseries(spi_ts, threshold=-1.2) # Check current status current = ts.tail(1) if current['is_event'].values[0]: print(f"IN DRY EVENT") print(f"Duration: {current['duration'].values[0]} months") print(f"Cumulative magnitude: {current['magnitude_cumulative'].values[0]:.2f}") print(f"Current severity: {current['magnitude_instantaneous'].values[0]:.2f}") else: print("No event detected") ``` ### Monitoring Evolution ```python # Is event worsening or easing? recent = ts['magnitude_instantaneous'].tail(3) if recent.is_monotonic_decreasing: print("EASING - severity decreasing") elif recent.is_monotonic_increasing: print("WORSENING - severity increasing") ``` ### Visualization ```python from visualization import plot_event_timeline # Shows both cumulative and instantaneous magnitude plot_event_timeline(ts) ``` ## Mode 3: Period Statistics ### Specific Time Period ```python from runtheory import calculate_period_statistics # What happened in 2023? stats_2023 = calculate_period_statistics(spi, threshold=-1.2, start_year=2023, end_year=2023) # Map number of events stats_2023.num_events.plot(cmap='YlOrRd') # Map worst severity stats_2023.worst_peak.plot(cmap='YlOrRd_r') ``` ### Year-by-Year Analysis ```python from runtheory import calculate_annual_statistics # Calculate for each year annual = calculate_annual_statistics(spi, threshold=-1.2) # Access specific year stats_2015 = annual.sel(year=2015) # Trend over time annual.num_events.mean(dim=['lat', 'lon']).plot() ``` ### Period Comparison ```python from runtheory import compare_periods # Compare decades comparison = compare_periods( spi, periods=[(1991, 2000), (2001, 2010), (2011, 2020), (2021, 2024)], period_names=['1990s', '2000s', '2010s', '2020s'] ) # Calculate change diff = comparison.sel(period='2020s') - comparison.sel(period='1990s') diff.num_events.plot(title='Change in Events: 2020s vs 1990s', cmap='RdBu_r') ``` ## Output Variables (Period Statistics) All period statistics functions return xarray Dataset with: | Variable | Description | Units | |----------|-------------|-------| | `num_events` | Number of dry/wet events | count | | `total_event_months` | Total months in events | months | | `total_magnitude` | Sum of all event magnitudes | index units | | `mean_magnitude` | Average magnitude per event | index units | | `max_magnitude` | Largest single event | index units | | `worst_peak` | Most severe SPI/SPEI value | index units | | `mean_intensity` | Average intensity | index units/month | | `max_intensity` | Maximum intensity | index units/month | | `pct_time_in_event` | % of time in events | % | ## Decision-Maker Questions ### "How many dry events in 2023?" ```python stats_2023 = calculate_period_statistics(spi, start_year=2023, end_year=2023) stats_2023.num_events.plot(title='Dry Events in 2023') ``` ### "Where was the worst dry event in last 5 years?" ```python stats = calculate_period_statistics(spi, start_year=2020, end_year=2024) stats.worst_peak.plot(title='Worst Severity 2020-2024', cmap='YlOrRd_r') ``` ### "How does recent compare to historical?" ```python comparison = compare_periods(spi, periods=[(1991, 2020), (2021, 2024)], period_names=['Historical', 'Recent']) diff = comparison.sel(period='Recent') - comparison.sel(period='Historical') diff.total_magnitude.plot(title='Change in Total Magnitude') ``` ### "Which year had the most events?" ```python annual = calculate_annual_statistics(spi) worst_year = int(annual.year[annual.num_events.mean(dim=['lat','lon']).argmax()]) print(f"Worst year: {worst_year}") ``` ## Thresholds Common dry event thresholds (use positive values for wet events): | Threshold | Category | Use | |-----------|----------|-----| | -0.5 | Incipient dry | Early warning | | -0.8 | Mild dry | Agricultural concern | | -1.0 | Moderate dry | Standard threshold | | -1.2 | Moderate-severe | Conservative threshold | | -1.5 | Severe dry | Significant impacts | | -2.0 | Extreme dry | Crisis level | **Recommendation:** -1.0 or -1.2 for operational monitoring ## Minimum Duration Filter short-term fluctuations: ```python # Include all dry periods events_all = identify_events(spi_ts, min_duration=1) # Only sustained events events_sustained = identify_events(spi_ts, min_duration=3) # Long-term events events_longterm = identify_events(spi_ts, min_duration=6) ``` **Recommendation:** `min_duration=3` for SPI-12 (captures sustained events) ## Performance | Function | Input Size | Time | Notes | |----------|------------|------|-------| | `identify_events()` | Single location | <0.1 sec | Very fast | | `calculate_timeseries()` | Single location | <0.1 sec | Very fast | | `calculate_period_statistics()` | 165×244 grid | ~30 sec | Acceptable | | `calculate_annual_statistics()` | 165×244, 10 years | ~5 min | Loops over years | | `compare_periods()` | 165×244, 2 periods | ~1 min | Parallelizable | **Optimization tips:** - Use `min_duration` to filter events - Process regional subsets when possible - Save and reuse outputs ## Complete Workflow Example ```python import xarray as xr from runtheory import (identify_events, calculate_period_statistics, compare_periods) from visualization import (plot_events, plot_event_timeline) # 1. Load SPI spi = xr.open_dataset('output/netcdf/spi_12.nc')['spi_gamma_12_month'] # 2. Single location analysis lat, lon = 50, 100 spi_loc = spi.isel(lat=lat, lon=lon) events = identify_events(spi_loc, threshold=-1.2, min_duration=3) ts = calculate_timeseries(spi_loc, threshold=-1.2) # 3. Visualize plot_events(spi_loc, events, threshold=-1.2) plot_event_timeline(ts) # 4. Regional statistics stats_2020s = calculate_period_statistics(spi, threshold=-1.2, start_year=2020, end_year=2024) # 5. Compare with historical comparison = compare_periods(spi, periods=[(1991, 2020), (2021, 2024)], period_names=['Historical', 'Recent']) # 6. Save outputs events.to_csv(f'output/csv/dry_events_lat{lat}_lon{lon}.csv') stats_2020s.to_netcdf('output/netcdf/dry_stats_2020-2024.nc') # 7. Create maps stats_2020s.num_events.plot() ``` ## References - Yevjevich, V. (1967). An objective approach to definitions and investigations of continental hydrologic droughts. Hydrology Papers 23, Colorado State University. - Sheffield, J., Wood, E.F., Roderick, M.L. (2012). Little change in global drought over the past 60 years. Nature, 491(7424), 435-438. ## See Also - [Magnitude Explained](magnitude.qmd) - Cumulative vs Instantaneous - [Visualization Guide](visualization.qmd) - All plotting options - [SPI Guide](spi.qmd) - Index calculation - [SPEI Guide](spei.qmd) - Temperature-inclusive index