I grabbed OpenSimplexNoise class and had to add my own functions to it so that I could use ‘octaves’ and ‘persistence’ values to add roughness to it. The extra functions which handle 2D and 3D noise are:

    public double OctavePerlin2(double x, double y, int octaves, double persistence) {
        double total = 0;
        double frequency = 1;
        double amplitude = 1;
        double maxValue = 0;  // Used for normalizing result to 0.0 - 1.0
        for(int i=0;i<octaves;i++) {
            total += eval(x * frequency, y * frequency) * amplitude;

            maxValue += amplitude;

            amplitude *= persistence;
            frequency *= 2;
        }

        return total/maxValue;
    }
    
    public double OctavePerlin3(double x, double y, double z, int octaves, double persistence) {
        double total = 0;
        double frequency = 1;
        double amplitude = 1;
        double maxValue = 0;  // Used for normalizing result to 0.0 - 1.0
        for(int i=0;i<octaves;i++) {
            total += eval(x * frequency, y * frequency, z * frequency) * amplitude;

            maxValue += amplitude;

            amplitude *= persistence;
            frequency *= 2;
        }

        return total/maxValue;
    }

Then I added a UI using Swing and got it all working including adding a water level.

[ADDED] And a better image:

All I need now is to break up the calculations into blocks and run each block on its own thread when I come to using this again.

The class which creates the bitmap:

public class BufferedImageNoise {
    
    int w, h;
    long seed;
    double scale;
    
    int off_x, off_y;
    
    int octaves;
    double persistence;
    
    double water;
    
    OpenSimplexNoise noise;
    
    public BufferedImageNoise(int width, int height, long seed, double scale, int octaves, double persistence, double water) {
        w = width; h = height;
        this.seed = seed;
        this.scale = scale;
        this.octaves = octaves;
        this.persistence = persistence;
        this.water = water;
        
        off_x = off_y = 0;
        
        noise = new OpenSimplexNoise(seed);
    }
    
    public BufferedImage getImage() {
        BufferedImage image = new BufferedImage(w, h, BufferedImage.TYPE_INT_RGB);
        for (int y = 0; y &lt h; y++)
        {
            for (int x = 0; x &lt w; x++)
            {
                double value = noise.OctavePerlin2(x / scale, y / scale, octaves, persistence) + 1.3;
                
                int rgb = value &gt water ? 0x010101 * (int)((value) * 127.5) : 0x000088;
                image.setRGB(x, y, rgb);
            }
        }
        return image;
    }
}