Advanced Geophysical and Image Processing Calculations in Kalpa MineX+
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In this section, we introduce **complex equations** commonly used in **geophysical signal processing** or **image processing** that you can implement directly in **Kalpa's Vector Calculator**. These calculations help you derive **meaningful attributes** from your geospatial datasets, particularly for **machine learning**, **geophysical modeling**, or **data visualization**.

Advanced Calculation Examples
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1. **Hilbert Transform (Instantaneous Amplitude)**  
The Hilbert Transform is widely used in signal processing to compute the **instantaneous amplitude** of a signal.

**Condition (Python Expression):**

::

    np.sqrt(row['signal']**2 + np.imag(np.fft.hilbert(row['signal']))**2)

**Example Use Case:**  
Analyzing the envelope of seismic signals or any oscillatory data.

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2. **Gradient Magnitude Calculation**  
Compute the **gradient magnitude** from spatial derivatives in the X and Y directions.

**Condition (Python Expression):**

::

    np.sqrt(row['gradient_x']**2 + row['gradient_y']**2)

**Example Use Case:**  
Calculate the spatial rate of change, essential for edge detection in image processing or identifying geophysical anomalies (e.g., gravity, magnetic fields).

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3. **Laplacian Filter**  
Used to detect regions of **rapid intensity change**, often for image and signal processing.

**Condition (Python Expression):**

::

    row['d2_dx2'] + row['d2_dy2']

**Example Use Case:**  
Identify high-curvature zones in geophysical maps, such as subsurface density contrasts.

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4. **Bandpass Filtering**  
Apply a simple **bandpass filter** to retain features within a specific frequency range.

**Condition (Python Expression):**

::

    low_cutoff <= row['frequency'] <= high_cutoff

**Example Use Case:**  
Filter seismic or geophysical signals to focus on features of interest, e.g., specific wave types or geological layers.

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5. **Normalized Difference Index (NDI)**  
A normalized ratio of two variables, commonly used in **remote sensing** (e.g., NDVI).

**Condition (Python Expression):**

::

    (row['variable1'] - row['variable2']) / (row['variable1'] + row['variable2'])

**Example Use Case:**  
Analyze vegetation indices, detect water bodies, or other normalized geophysical relationships.

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6. **Fourier Transform Coefficient Extraction**  
Extract the **real part** of the first Fourier Transform coefficient for **frequency-domain analysis**.

**Condition (Python Expression):**

::

    np.fft.fft(row['signal'])[0].real

**Example Use Case:**  
Identify dominant frequency components in geophysical signals.

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7. **Directional Slope Calculation**  
Compute the **slope** in a specific direction (e.g., northward).

**Condition (Python Expression):**

::

    (row['z_north'] - row['z_center']) / row['distance_north']

**Example Use Case:**  
Analyze topographic or bathymetric gradients for **hydrological modeling** or **structural geology**.

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8. **Gaussian Smoothing**  
Apply **Gaussian smoothing** to a signal for noise reduction.

**Condition (Python Expression):**

::

    np.exp(-0.5 * ((row['x'] - row['mean_x']) / row['std_dev_x'])**2)

**Example Use Case:**  
Smooth noisy geophysical data or enhance visualization of spatial datasets.

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9. **Signal-to-Noise Ratio (SNR)**  
Compute the **SNR** of a signal.

**Condition (Python Expression):**

::

    10 * np.log10(row['signal_power'] / row['noise_power'])

**Example Use Case:**  
Evaluate the quality of geophysical signals for better interpretation.

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10. **Structural Similarity Index (SSIM)**  
Calculate **SSIM** for image quality assessment in two columns (e.g., predicted vs. observed).

**Condition (Python Expression):**

::

    (2 * row['mean_x'] * row['mean_y'] + C1) * (2 * row['covariance'] + C2) / ((row['mean_x']**2 + row['mean_y']**2 + C1) * (row['var_x'] + row['var_y'] + C2))

**Example Use Case:**  
Compare geophysical or remote sensing images to identify changes or validate models.

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Tips for Writing Complex Equations
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1. **Use Mathematical Libraries**  
   Leverage **NumPy** for mathematical operations and **SciPy** for advanced signal processing when supported.

2. **Validate Columns**  
   Ensure that all referenced columns **exist in the dataset** to avoid runtime errors.

3. **Break Down Complex Calculations**  
   For readability and reusability, create **intermediate columns** for parts of the computation. This makes debugging and documentation easier.

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