Modeling and inversion of one-dimensional airborne electromagnetic (AEM) data: C++ implementation guide and detailed code analysis

Part 1: Introduction to 1D airborne electromagnetic (AEM) data

1. Overview of Airborne Electromagnetic (AEM) Data

Airborne electromagnetics (AEM) is a method of collecting subsurface electromagnetic information from the air using electromagnetic equipment carried by aircraft (such as aircraft or drones). This method can be used to detect underground mineral resources, groundwater, archaeological sites, etc. One-dimensional AEM data refers to electromagnetic response data from the ground to a certain depth at a specific location.

2. Why choose one-dimensional model

Although real-world underground structures are three-dimensional, in some cases using one-dimensional models can simplify the problem and provide sufficient information. For example, a one-dimensional model is useful when the layers change little horizontally but have significant changes vertically.

3. Advantages of C++ in AEM data processing

C++ is an efficient, flexible, and powerful programming language ideal for processing large amounts of data and performing complex numerical calculations. Using C++ for modeling and inversion of AEM data ensures fast, accurate results.

One-dimensional AEM data modeling

Modeling is the process of creating a model to simulate the real world. In AEM, our goal is to create a model that simulates the electromagnetic response of the subsurface.

Code implementation:

#include<iostream>
#include<vector>

//Define stratum class
class Layer {<!-- -->
public:
    double thickness; //stratum thickness
    double resistivity; //resistivity

    Layer(double t, double r) : thickness(t), resistivity(r) {<!-- -->}
};

//Simulate one-dimensional AEM response
std::vector<double> simulateAEMResponse(const std::vector<Layer> & amp; layers) {<!-- -->
    std::vector<double> response;

    //Simplified model: Assume that the response of each layer is the product of its resistivity and thickness
    for(const Layer & amp; layer : layers) {<!-- -->
        response.push_back(layer.thickness * layer.resistivity);
    }

    return response;
}

int main() {<!-- -->
    //Define three layers of strata
    Layer layer1(10, 100);
    Layer layer2(20, 50);
    Layer layer3(30, 200);

    std::vector<Layer> layers = {<!-- -->layer1, layer2, layer3};

    std::vector<double> response = simulateAEMResponse(layers);

    for(double r : response) {<!-- -->
        std::cout << r << std::endl;
    }

    return 0;
}

In the above code, we first define a class that represents the formation, which contains the thickness and resistivity of the formation. Then, we defined a function to simulate the one-dimensional AEM response. In this simplified model, we assume that the response of each layer is the product of its resistivity and thickness.

Conclusion

One-dimensional airborne electromagnetic (AEM) data modeling is a powerful tool that can help us understand and interpret the electromagnetic response of the subsurface. By using C++, we can efficiently process large amounts of data and perform complex numerical calculations.

Note: In the interests of brevity and clarity, the code in this article may not be the optimal or most complete implementation. In order to get the complete project and more optimization tips, please download the complete project

Part 2: One-dimensional airborne electromagnetic (AEM) data inversion

1. What is data inversion

Data inversion is the process of inferring unknown subsurface structures from known observational data. In AEM, the purpose of inversion is to infer the subsurface resistivity distribution from electromagnetic response data.

2. Challenges of inversion

Inversion is an inverse problem that is often nonlinear, unstable, and may have multiple solutions. Therefore, choosing appropriate inversion strategies and algorithms is crucial.

C++ implementation of one-dimensional AEM data inversion

To simplify the problem, we will use a simple iterative method to perform the inversion. Our goal is to minimize the difference between observed and simulated data.

Code implementation:

#include<iostream>
#include<vector>
#include<cmath>

class Layer {<!-- -->
public:
    double thickness;
    double resistivity;

    Layer(double t, double r) : thickness(t), resistivity(r) {<!-- -->}
};

double computeDifference(const std::vector<double> & amp; observed, const std::vector<double> & amp; simulated) {<!-- -->
    double difference = 0.0;

    for(size_t i = 0; i < observed.size(); i + + ) {<!-- -->
        difference + = std::pow(observed[i] - simulated[i], 2);
    }

    return difference;
}

std::vector<Layer> invertAEMData(const std::vector<double> & amp; observed, const std::vector<Layer> & amp; initialModel) {<!-- -->
    std::vector<Layer> currentModel = initialModel;
    std::vector<Layer> bestModel = initialModel;
    double bestDifference = computeDifference(observed, simulateAEMResponse(initialModel));
    const double learningRate = 0.01;

    for(int iteration = 0; iteration < 1000; iteration + + ) {<!-- -->
        for(Layer & amp; layer : currentModel) {<!-- -->
            layer.resistivity + = learningRate * (rand() % 200 - 100); // Random perturbation
        }

        double currentDifference = computeDifference(observed, simulateAEMResponse(currentModel));

        if(currentDifference < bestDifference) {<!-- -->
            bestDifference = currentDifference;
            bestModel = currentModel;
        }
    }

    return bestModel;
}

int main() {<!-- -->
    // data observation
    std::vector<double> observedResponse = {<!-- -->1000, 1100, 6500};

    //Initial model
    Layer layer1(10, 90);
    Layer layer2(20, 55);
    Layer layer3(30, 210);
    std::vector<Layer> initialModel = {<!-- -->layer1, layer2, layer3};

    std::vector<Layer> invertedModel = invertAEMData(observedResponse, initialModel);

    for(const Layer & amp; layer : invertedModel) {<!-- -->
        std::cout << "Thickness: " << layer.thickness << ", Resistivity: " << layer.resistivity << std::endl;
    }

    return 0;
}

In the above code, we first define a function to calculate the difference between observed and simulated data. Then, we define a function to perform the inversion. This function uses a simple iteration method, each iteration makes a small random perturbation to the current model and calculates the difference between the new model’s response and the observed data. If the new model is less different than the previous best model, then it becomes the new best model.

Conclusion

One-dimensional airborne electromagnetic (AEM) data inversion is a complex process that requires the selection of appropriate strategies and algorithms. By using C++, we can handle this problem efficiently and infer the resistivity distribution in the subsurface from observational data.

Part Three: Optimization and Practical Application

1. Optimization strategy

Although we have achieved modeling and inversion of one-dimensional AEM data, in practical applications, further optimization may be required to improve accuracy and efficiency. Here are some suggested optimization strategies:

Parallel Computing: With modern multi-core processors, multiple inversion tasks can be processed in parallel, greatly improving efficiency.
More complex inversion algorithms: We use a simple iterative method, but there are other more complex algorithms, such as genetic algorithms, simulated annealing, etc., that can provide more accurate results.
Regularization: In order to avoid overfitting during the inversion process, regularization techniques can be introduced, such as L1 or L2 regularization.

Practical application examples

Let’s say we are working for a mining company that wants to use AEM technology to detect gold deposits in an area. We can use the following steps:

Data collection: Use the AEM equipment carried by the aircraft to fly over the target area and collect electromagnetic response data.
Data preprocessing: Clean and preprocess the collected data to remove noise and outliers.
Modeling and Inversion: Use our C++ program to model and invert the data to obtain the resistivity distribution in the subsurface.
Interpretation and Decision-making: Based on inversion results, work with geologists and mining engineers to identify the most likely locations of gold deposits and develop further exploration plans.

Code Optimization Example: Parallel Computing

To show how to leverage parallel computing for optimization, we can use the C++ library. Here is a simple example showing how to use multithreading for inversion:

#include<thread>
#include<vector>

// ... [other code and function definitions]

void invertAEMDataThreaded(const std::vector<double> & amp; observed, const std::vector<Layer> & amp; initialModel, std::vector<Layer> & amp; resultModel) {<!-- -->
    resultModel = invertAEMData(observed, initialModel);
}

int main() {<!-- -->
    std::vector<double> observedResponse1 = {<!-- -->1000, 1100, 6500};
    std::vector<double> observedResponse2 = {<!-- -->950, 1150, 6400};

    Layer layer1(10, 90);
    Layer layer2(20, 55);
    Layer layer3(30, 210);
    std::vector<Layer> initialModel = {<!-- -->layer1, layer2, layer3};

    std::vector<Layer> invertedModel1, invertedModel2;

    std::thread thread1(invertAEMDataThreaded, observedResponse1, initialModel, std::ref(invertedModel1));
    std::thread thread2(invertAEMDataThreaded, observedResponse2, initialModel, std::ref(invertedModel2));

    thread1.join();
    thread2.join();

    // ... [process inversion results]

    return 0;
}

In the above code, we create two threads, each thread performs an inversion. In this way, if we have a multi-core processor, two inversion tasks can be performed simultaneously, thus greatly improving efficiency.

Conclusion

Modeling and inversion of one-dimensional airborne electromagnetic (AEM) data is a powerful tool that can help us understand and interpret the electromagnetic response of the subsurface. Through further optimization and practical application, we can use this data more effectively to solve practical problems. C++ provides efficient and flexible tools that enable us to handle these challenges and lay a solid foundation for future research and applications.